# Counting compatible indexing systems for $C_{p^{n}}$

Michael A. Hill  UCLA Department of Mathematics, Los Angeles, CA
<EMAIL_ADDRESS>,  Jiayun Meng  Department of Mathematics, University
of Texas, Austin, TX<EMAIL_ADDRESS>and  Nan Li  Department of
Mathematics, University of Minnesota, Minneapolis, MN<EMAIL_ADDRESS>
###### Abstract.

We count the number of compatible pairs of indexing systems for the cyclic
group $C_{p^{n}}$. Building on work of Balchin–Barnes–Roitzheim, we show that
this sequence of natural numbers is another family of Fuss–Catalan numbers. We
count this two different ways: showing how the conditions of compatibility
give natural recursive formulas for the number of admissible sets and using an
enumeration of ways to extend indexing systems by conceptually simpler pieces.

This material is based upon work supported by the National Science Foundation
under Grant No. 2105019

##  1\. Introduction

Recent work in equivariant algebra has studied the beautiful variety of
different multiplicative structures that can arise when we mix in the action
of a finite group. Blumberg–Hill showed that this is a fundamentally
combinatorial structure [2]. The various multiplicative norms or additive
transfers are entirely governed by certain particularly well-behaved
subcategories of finite $G$-sets.

This classification was reformulated in independent work of
Balchin–Barnes–Roitzheim and of Rubin, where they further underscored the
combinatorial structure by showing the norms and transfers are encoded in
“transfer systems”, certain refinements of the poset of subgroups of $G$ under
inclusion [1] [8].

######  Definition 1.1 ([8, Definition 3.4], [1, Definition 7]).

A transfer system for a finite group $G$ is a partial order $\to$ relation on
the set $\operatorname{Sub}(G)$ of subsets of $G$ such that

  1. (1)

if $K\to H$, then $K\leq H$ (so this is a “weak subposet”),

  2. (2)

if $K\to H$ and $g\in G$, then $gKg^{-1}\to gHg^{-1}$, and

  3. (3)

if $K\to H$ and $J\leq H$, then $K\cap J\to J$.

Let $\operatorname{Tran}(G)$ denote the set of transfer systems for $G$.

The set of all transfer systems for $G$ itself has a partial order: we say
$\mathcal{O}\leq\mathcal{O}^{\prime}$ if the identity map is order-preserving.

######  Example 1.2.

For $G=C_{p^{n}}$, the subgroup lattice is order isomorphic to the linear
order

$\\{1\leq 2\dots\leq n+1\\}=[n+1].$

The conjugation condition is always satisfied, and the restriction condition
here can be rephrased as saying that if $C_{p^{i}}\to C_{p^{j}}$ and $i\leq
k\leq j$, then $C_{p^{i}}\to C_{p^{k}}$.

Balchin–Barnes–Roitzheim showed that the poset of transfer systems for
$C_{p^{n}}$ is order isomorphic to the Tamari lattice [1, Theorem 25].

In this paper, we will focus on the groups $C_{p^{n}}$. Example 1.2 stresses
that we could equivalently look at “transfer systems for the poset $[n+1]$” in
the sense of Franchere–Ormsby–Osorno–Qin–Waugh.

######  Definition 1.3 ([6, Definition 4.1]).

A transfer system on

$[n]=\\{1\leq\dots\leq n\\}$

is a weak sub-poset of $[n]$ with partial order $\to$ that contains all the
elements and which satsifies the “restriction condition”: if $i\to j$ and
$i\leq k\leq j$, then $i\to k$.

It is helpful to view these as a graded set in which we allow $n$ to vary, as
this will help encode certain natural operations.

######  Definition 1.4.

For each $n\in\mathbb{N}$, let

$T_{n}=\begin{cases}\operatorname{Tran}([n])&n\geq 1\\\
\\{\emptyset\\}&n=0.\end{cases}$

Balchin–Barnes–Roizheim counted the number of transfer systems for
$C_{p^{n}}$:

######  Theorem 1.5 ([1, Theorem 20]).

For $n$, we have

$|T_{n}|=\frac{1}{2n+1}\binom{2n+1}{n}=\operatorname{Cat}(n).$

The numbers “$\operatorname{Cat}(n)$” are the Catalan numbers. These are
ubiquitous in combinatorics, parameterizing structructures from binary rooted
trees to Dyck paths. This sequence fits into a bivariant family of sequences.

######  Definition 1.6.

The _Fuss–Catalan numbers_ are defined by

$A_{n}(p,r)=\frac{r}{np+r}\binom{np+r}{n}$

for non-negative $n$ and positive $p$ and $r$.

The Catalan numbers arise here:

$\operatorname{Cat}(n)=A_{n}(2,1).$

In this paper, we show the next term of the sequence is also related to
equivariant algebra.

Blumberg–Hill studied what kind of compatibility conditions arise if we allow
both the additive transfers and multiplicative norms to each be structured by
various transfer systems [3]. As a slogan, “the presence of certain
multiplicative norms forces some additive transfers”. These conditions were
simplified by Chan who gave a definition internal to transfer systems [4].
When the conditions are satisfied, we say that
$(\mathcal{O}_{a},\mathcal{O}_{m})$ is compatible. Counting these for
$C_{p^{n}}$ is the main result of this paper.

######  Theorem 1.7.

For $C_{p^{n-1}}$, there are

$A_{n}(3,1)$

compatible pairs of transfer systems.

We present two proofs of this theorem in Sections 4 and 5. Each of which
underscores a different combinatorial feature of the number of compatible
pairs. The mathematics in this paper arose from an REU project in Summer 2021.
The two junior coauthors each came up with a distinct solution to this
counting problem, so we include here both of their solutions. A third solution
by Henry Ma will appear separately.

### Acknowledgements

The authors thank Bridget Tenner, Kyle Ormsby, Andrew Blumberg, and Angelica
Osorno for helpful conversations during this.

The authors would like to thank the Hausdorff Research Institute for
Mathematics for the hospitality in the context of the Trimester program
Spectral Methods in Algebra, Geometry, and Topology, funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-2047/1 – 390685813.

##  2\. Operations on Transfer Systems

###  2.1. Concatination

A key piece of structure on transfer systems for $[n]$ is the ability to
“concatinate” a transfer system for $[k]$ and one for $[n-k]$ to build one for
$[n]$. This makes computations with the entire graded set easier to
understand.

######  Definition 2.1.

Let $\mathcal{O}_{L}$ be a transfer system for $[k]$ and $\mathcal{O}_{R}$ be
a transfer system for $[n-k]$. Then we define a relation
$\mathcal{O}=\mathcal{O}_{L}\oplus\mathcal{O}_{R}$ on $[n]$ by saying
$i\xrightarrow{\mathcal{O}}j$ if and only if

  1. (1)

$j\leq k$ and $i\xrightarrow{\mathcal{O}_{L}}j$ or

  2. (2)

$i\geq(k+1)$ and $(i-k)\xrightarrow{\mathcal{O}_{R}}(j-k)$.

We call $\mathcal{O}$ the _direct sum_ or _concatination_ of $\mathcal{O}_{L}$
and $\mathcal{O}_{R}$.

######  Remark 2.2.

As a poset, this is just the disjoint union of the two posets
$\mathcal{O}_{L}$ and $\mathcal{O}_{R}$. The additional data is the map to
$[n]$ in this case.

######  Proposition 2.3.

The concatination of transfer systems is a transfer system.

###### Proof.

The definition of $\mathcal{O}$ is that of the disjoint union of posets, so
$\mathcal{O}$ is a poset, and it visibly maps to the usual inclusions.

We need only check the restriction condition, and here, it suffices to check
the restriction of $i+k\to j+k$ along $m\leq j+k$ for some $m\leq k$. In this
case, both the source and target are $m$. ∎

The direct sum is a graded operation here:

$T_{n}\times T_{m}\xrightarrow{\oplus}T_{n+m}.$

We extend this to $T_{0}$ by declaring

$\emptyset\oplus\mathcal{O}=\mathcal{O}\oplus\emptyset=\mathcal{O}$

for any transfer system $\mathcal{O}$ or the empty set.

$1$$2$$3$$4$$5$$6$$7$$\mathcal{O}_{L}$$\mathcal{O}_{R}$$\mathcal{O}_{L}\oplus\mathcal{O}_{R}$
Figure 1. Example of concatination of $\mathcal{O}_{L}$ and $\mathcal{O}_{R}$

###  2.2. Restriction

Given a transfer system for $[n]$, we have two natural ways to build transfer
systems for smaller natural numbers. These both arise from the inclusion of
subposets.

######  Definition 2.4.

If $k\leq n$, then let

$\iota\colon[k]\to[n]$

be the map sending $i\to i$, and let

$\phi\colon[n-k]\to[n]$

be the map sending $i$ to $i+k$.

We can restrict a transfer system on $[n]$ along either of these inclusions.

######  Definition 2.5.

Let $\mathcal{O}$ be a transfer system for $[n]$, and let $k\leq n$.

Let $i_{{k}}^{\ast}\mathcal{O}$ be defined by saying for all $i\leq j\leq k$,

$i\xrightarrow{i_{{k}}^{\ast}\mathcal{O}}j\text{ if and only if
}i\xrightarrow{\mathcal{O}}j.$

Let $\Phi^{k}\mathcal{O}$ be defined by saying for all $i\leq j\leq(n-k)$,

$i\xrightarrow{\Phi^{k}\mathcal{O}}j\text{ if and only if
}{(i+k)}\xrightarrow{\mathcal{O}}{(j+k)}.$

######  Proposition 2.6.

For a transfer system on $\mathcal{O}$, $i_{k}^{\ast}\mathcal{O}$ is a
transfer system on $[k]$ and $\Phi^{k}\mathcal{O}$ is a transfer system on
$[n-k]$.

###### Proof.

The construction gives wide, subposets of $[k]$ and $[n-k]$ respectively, by
observation. Since the two inclusions $\iota$ and $\phi$ are interval
inclusions, the restriction condition is easily checked. ∎

The two restrictions can be visualized as simply throwing away any transfers
that start or end outside of the given range. An example picture of this is
shown in Figure 2.

######  Remark 2.7.

The map $i_{k}^{\ast}$ is induced by the inclusion of the subgroup lattice
corresponding to the inclusion of $C_{p^{k}}$ into $C_{p^{n}}$. Topologically,
this models the restriction to $C_{p^{k}}$. The map $\Phi^{k}$ is induced by
the inclusion of subgroup lattices corresponding to the quotient map from
$C_{p^{n}}$ to $C_{p^{n-k}}$. Topologically, this models the fixed points.

$1$$2$$3$$4$$5$$6$$7$$i_{4}^{\ast}\mathcal{O}$$\Phi^{4}\mathcal{O}$ Figure 2.
Example of $i_{k}^{\ast}\mathcal{O}$ and $\Phi^{k}\mathcal{O}$. The dashed
transfers are forgotten to form $i_{4}^{\ast}\mathcal{O}$ and
$\Phi^{4}\mathcal{O}$.

###  2.3. Wrapped and saturated systems

We single out two families of transfer systems that are useful for our counts.

######  Definition 2.8.

A transfer system $\mathcal{O}$ for $[n]$ is _wrapped_ if
$1\xrightarrow{\mathcal{O}}n$.

######  Example 2.9.

The transfer systems $\mathcal{O}_{L}$ and $\mathcal{O}_{R}$ from Figure 1 are
wrapped; the transfer system $\mathcal{O}_{L}\oplus\mathcal{O}_{R}$ is not.

######  Example 2.10.

The transfer system $\mathcal{O}$ in Figure 2 is wrapped, as is
$i_{4}^{\ast}\mathcal{O}$. The transfer system $\Phi^{4}\mathcal{O}$ is not.

By the restriction condition, in a wrapped transfer system, we have $1\to j$
for all $j$, and we think of the largest transfer as “wrapping” the rest of
the transfer system. Additionally, we can always “wrap” any given transfer
system, using the circle-dot product of [1]. We use a single case of their
construction here.

######  Definition 2.11.

If $\mathcal{O}\in T_{n}$, let $w(\mathcal{O})$ be the transfer system for
$[n+1]$ with

  1. (1)

for all $j$, $1\to j$ in $w(\mathcal{O})$, and

  2. (2)

$\Phi^{1}\big{(}w(\mathcal{O})\big{)}=\mathcal{O}$.

This is implicit in Balchin–Barnes–Roitzheim’s “circle-dot product” and their
decomposition theorem.

######  Theorem 2.12 ([1, Corollary 21]).

Any transfer system for $[n]$ with $n\geq 1$ decomposes uniquely as

$\mathcal{O}\odot\mathcal{O}^{\prime}=\mathcal{O}\oplus
w(\mathcal{O}^{\prime}),$

for some transfer systems $\mathcal{O}\in T_{m}$ and $\mathcal{O}^{\prime}\in
T_{n-m-1}$.

######  Corollary 2.13.

Any transfer system $\mathcal{O}$ for $[n]$ can be written uniquely as

$\mathcal{O}=w(\mathcal{O}_{1})\oplus\dots\oplus w(\mathcal{O}_{k})$

for some transfer systems $\mathcal{O}_{1},\dots,\mathcal{O}_{k}$.

A special case of wrapped transfer systems is given by complete ones.

######  Definition 2.14.

The _complete_ transfer system for $[n]$ is the one for which $i\to j$ for all
$i\leq j$. Denote it $\mathcal{O}^{cpt}_{n}$.

######  Remark 2.15.

The complete transfer system for $[n]$ is one for which the partial order is
just the inclusion. This means that complete transfer systems are maximal
elements in the poset of transfer systems.

These complete transfer systems will be especially useful building blocks for
us.

######  Definition 2.16.

A transfer system $\mathcal{O}$ is _saturated_ if it can be written as a
direct sum of complete transfer systems.

Given any transfer system on $[n]$, there is a minimal saturated transfer
system that contains it. This is immediate from the observation that the
intersection of two saturated transfer systems is again a saturated transfer
system. For concreteness, we spell this out directly here.

######  Proposition 2.17.

If $\mathcal{O}$ is any transfer system, then there is a minimal saturated
transfer system that contains $\mathcal{O}$.

###### Proof.

Write $\mathcal{O}$ as a direct sum of wrapped transfer systems
$\mathcal{O}=\mathcal{O}_{n_{1}}\oplus\dots\mathcal{O}_{n_{r}}$, with
$\mathcal{O}_{n_{i}}\in T_{n_{i}}$. Then minimal saturated transfer system
containing it is simply

$\mathcal{O}_{n_{1}}^{cpt}\oplus\dots\mathcal{O}^{cpt}_{n_{r}},$

where we replace each summand with the complete one of that size. ∎

######  Definition 2.18.

If $\mathcal{O}$ is a transfer system for $[n]$, let
$\operatorname{hull}(\mathcal{O})$ denote the minimal saturated transfer
system that contains it. This is the _saturated hull_.

For transfer systems for $[n]$, there is a kind of dual notion of a maximal
saturated transfer systems inside any given transfer system.

Complete transfer systems have two useful properties:

  1. (1)

Whenever $i\to j$, we also have $k\to j$ for all $i\leq k\leq j$, and

  2. (2)

they are generated as a poset as the transitive closure of the relation
$i\to{i+1}$ for all $1\leq i\leq n-1$.

These two properties give us two different ways to repackage the condition of
being saturated.

######  Proposition 2.19.

A transfer system $\mathcal{O}$ for $n$ is saturated if and only if whenever
$i\to j$ with $i<j$, we have ${{(j-1)}}\to{j}$.

###### Proof.

If $\mathcal{O}$ is saturated, then by definition of the direct sum, if $i\to
j$, then $i$ and $j$ correspond to subgroups from the same direct summand. By
completeness, we therefore have all intermediate transfers.

Using the decomposition of a transfer system into wrapped ones (Corollary
2.13), we see that it suffices to show that if a wrapped transfer system that
has the property that $i\to j$ implies $(j-1)\to j$, then the transfer system
is complete. This follows from downward induction on $n$, using that $1\to n$
by the wrapped assumption, and hence $1\to j$ for all $j$ by restriction. ∎

######  Corollary 2.20.

A transfer system $\mathcal{O}$ is saturated if and only if it is generated as
a partial order by relations $i\to{{(i+1)}}$ for some collection of positive
$i$ at most $n-1$.

The possibly surprising part here is that we only need the partial order: the
other parts of being a transfer system come along for free in this case, since
we are generating by a covering condition. This gives us a second kind of
structural result.

######  Definition 2.21.

If $\mathcal{O}$ is a transfer system let the _core_ of $\mathcal{O}$, denoted
$\operatorname{core}{\mathcal{O}}$, be the partial order generated by
${i}\to{{(i+1)}}$, where $i$ ranges over the integers from $1$ to $n-1$ such
that $i\to(i+1)$.

######  Example 2.22.

In Figure 3, we see the core of the transfer system $\mathcal{O}$ from Figure
2.

$1$$2$$3$$4$$5$$6$$7$ Figure 3. Core of $\mathcal{O}$ from Figure 2

##  3\. Compatible pairs

Our main object of study is the notion of compatible pairs. These were defined
by Blumberg–Hill to describe “compatibility” between equivariant norms and
transfers in an abstract, categorical way [3]. Chan reformulated this in the
language of transfer system, giving a purely combinatorial formulation [4,
Theorem 4.10]. We use that here for the special case of $C_{p^{n}}$.

######  Definition 3.1 ([4, Definition 4.6]).

A pair of transfer system $(\mathcal{O}_{a},\mathcal{O}_{m})$ for $[n]$ are
compatible if for whenever $i\xrightarrow{m}j$, we have $k\xrightarrow{a}j$
for all $i\leq k\leq j$.

Note here that the conditions are asymmetrical: arrows in $\mathcal{O}_{m}$
force those in $\mathcal{O}_{a}$. Moreover, this is a kind of relative
saturation condition, with an arrow in $\mathcal{O}_{m}$ actually forcing
$\mathcal{O}_{a}$ to look saturated in a range. This gives us two equivalent
forms.

######  Proposition 3.2.

A pair $(\mathcal{O}_{a},\mathcal{O}_{m})$ is compatible if and only if the
following equivalent comparisons hold

  1. (1)

$\mathcal{O}_{m}\leq\operatorname{core}(\mathcal{O}_{a})$,

  2. (2)

$\operatorname{hull}(\mathcal{O}_{m})\leq\mathcal{O}_{a}$, and

  3. (3)

$\operatorname{hull}(\mathcal{O}_{m})\leq\operatorname{core}(\mathcal{O}_{a})$.

###### Proof.

The conditions of Definition 3.1 are a restatement of the condition that the
saturated hull of $\mathcal{O}_{m}$ is less than or equal to
$\mathcal{O}_{a}$. For the equivalence of the three conditions, we use that
the core of $\mathcal{O}$ is the largest saturated transfer system less than
or equal to $\mathcal{O}$ and the hull is the smallest saturated transfer
system greater than or equal to $\mathcal{O}$. ∎

######  Corollary 3.3.

Let $(\mathcal{O}_{a},\mathcal{O}_{m})$ be a compatible pair of transfer
systems for $[n]$. If for some $1\leq k\leq n-1$, we have
$k\not\xrightarrow{a}(k+1)$, then for all $j\leq k$ and $\ell\geq(k+1)$, we
must have $j\not\xrightarrow{m}\ell$.

Put another way, we see that $\mathcal{O}_{m}$ must break apart at $k$, and
this must be compatible with $\mathcal{O}_{a}$.

######  Corollary 3.4.

Let $(\mathcal{O}_{a},\mathcal{O}_{m})$ be a compatible pair of transfer
systems for $[n]$. If for some $1\leq k\leq(n-1)$, we have
$k\not\xrightarrow{a}(k+1)$, then

  1. (1)

$\mathcal{O}_{m}=i_{k}^{\ast}\mathcal{O}_{m}\oplus\Phi^{k}\mathcal{O}_{m}$,
and

  2. (2)

the pairs

$(i_{k}^{\ast}\mathcal{O}_{a},i_{k}^{\ast}\mathcal{O}_{m})\text{ and
}(\Phi^{k}\mathcal{O}_{a},\Phi^{k}\mathcal{O}_{m})$

are compatible.

######  Definition 3.5.

Let

$\mathcal{D}^{(0)}_{n}=\big{\\{}(\mathcal{O}_{a},\mathcal{O}_{m})\mid\text{compatible}\big{\\}}\subset
T_{n}\times T_{n}.$

We have projection maps

$p_{a},p_{m}\colon\mathcal{D}^{(0)}_{n}\to T_{n}$

which take a pair $(\mathcal{O}_{a},\mathcal{O}_{m})$ to $\mathcal{O}_{a}$ or
$\mathcal{O}_{m}$ respectively. Our main goal is to find the cardinality of
$\mathcal{D}^{(0)}_{n}$ for all $n$. We solve this in several different ways
using different aspects of Proposition 3.2.

##  4\. Solving the recurrence relations

###  4.1. Decomposition and Recurrence Relation

The wrapping map $w$ defines a natural filtration on the collection of
transfer systems. We can use this to build a recursive relation describing
compatible pairs.

######  Definition 4.1.

For each $i\geq 0$, let

$(\mathcal{F}^{i}T)_{n}=Im(w^{\circ i})\subset T_{n},$

viewed as a graded subset.

######  Definition 4.2.

Let

$\mathcal{D}^{(i)}_{n}=p_{a}^{-1}\big{(}(\mathcal{F}^{i}T)_{n}\big{)}$

be the set of composable pairs with $\mathcal{O}_{a}\in\mathcal{F}^{i}T$.

Let $d(n,i)=|\mathcal{D}^{(i)}_{n}|$ be the corresponding cardinality.

We deduce our recursive formulae from the Balchin–Barnes–Roitzheim
decomposition theorem (Theorem 2.12). We restate the result here to set up our
decomposition.

######  Proposition 4.3.

If $\mathcal{O}$ is a transfer system in $(\mathcal{F}^{i}T)_{n}$, then there
is

  1. (1)

a unique natural number $1\leq j\leq n-i$,

  2. (2)

a unique wrapped transfer system $w\mathcal{O}_{R}$ in $T_{j}$,

  3. (3)

and a unique transfer system $\mathcal{O}_{L}$ in $T_{n-i-j}$

such that

$\mathcal{O}=w^{i}\big{(}\mathcal{O}_{L}\oplus w\mathcal{O}_{R}\big{)}.$

######  Notation 4.4.

Let $(\mathcal{F}^{i}T)_{n,j}$ be the set of transfer systems in
$(\mathcal{F}^{i}T)_{n}$ which decompose as

$\mathcal{O}=w^{i}\big{(}\mathcal{O}_{L}\oplus w\mathcal{O}_{R}\big{)}$

with $w\mathcal{O}_{R}\in T_{j}$ a wrapped transfer system.

######  Proposition 4.5.

The map

$\left(\coprod_{j=1}^{n-i-1}\mathcal{D}^{(i)}_{n-j}\times\mathcal{D}^{(1)}_{j}\right)\amalg\mathcal{D}^{(i+1)}_{n}\to\mathcal{D}^{(i)}_{n}$

given on $\mathcal{D}^{(i)}_{n-j}\times\mathcal{D}^{(1)}_{j}$ by

$\big{(}(w^{i}\mathcal{O}_{a}^{\prime},\mathcal{O}_{m}^{\prime}),(w\mathcal{O}_{a}^{\prime\prime},\mathcal{O}_{m}^{\prime\prime})\big{)}\mapsto\big{(}w^{i}(\mathcal{O}_{a}^{\prime}\oplus
w\mathcal{O}_{a}^{\prime\prime}),\mathcal{O}_{m}^{\prime}\oplus\mathcal{O}_{m}^{\prime\prime}\big{)}$

and on the last summand by the natural inclusion, is a bijection.

###### Proof.

We use Proposition 4.3 to further break up $\mathcal{D}^{(i)}_{n}$, since the
decomposition here gives a disjoint union decomposition

$(\mathcal{F}^{i}T)_{n}=\coprod_{j=1}^{n-i}(\mathcal{F}^{i}T)_{n,j}.$

This decomposition induces a decomposition of $\mathcal{D}^{(i)}_{n}$:

$\mathcal{D}^{(i)}_{n,j}=p_{1}^{-1}(\mathcal{F}^{i}T)_{n,j}.$

Since $T_{0}=\\{\emptyset\\}$, the unit for $\oplus$, we have

$\mathcal{D}^{(i)}_{n,n-i}=\mathcal{D}^{(i+1)}_{n},$

given by the usual inclusion. Now let $1\leq j<n-i$, and consider an element
$(\mathcal{O}_{a},\mathcal{O}_{m})$ in $\mathcal{D}^{(i)}_{n,j}$. By
definition, we have

$\mathcal{O}_{a}=w^{i}\big{(}\mathcal{O}_{a,L}\oplus
w\mathcal{O}_{a,R}\big{)},$

with $\mathcal{O}_{a,L}\neq\emptyset$ and $w\mathcal{O}_{a,R}$ wrapped, and
hence we are missing the transfer

$(i+n-j)\to(i+n-j+1)$

in $\mathcal{O}_{a}$. This means that $\mathcal{O}_{m}$ breaks up into a
direct sum

$\mathcal{O}_{m}^{\prime}\oplus\mathcal{O}_{m}^{\prime\prime},$

where $\mathcal{O}_{m}^{\prime}\in T_{i+n-j}$ and
$\mathcal{O}_{m}^{\prime\prime}\in T_{j}$, by Corollary 3.4. Moreover, we know
that the pairs

$\big{(}i_{i+n-j}^{\ast}\mathcal{O}_{a},\mathcal{O}_{m}^{\prime}\big{)}\text{
and }\big{(}\Phi^{i+n-j}\mathcal{O}_{a},\mathcal{O}_{m}^{\prime\prime}\big{)}$

are compatible. The result follows, since

$i_{i+n-j}^{\ast}\mathcal{O}_{a}=w^{i}\mathcal{O}_{a,L}\text{ and
}\Phi^{i+n-j}\mathcal{O}_{a}=w\mathcal{O}_{a,R}.$

∎

######  Corollary 4.6.

We have a recursive formula

$d(n,i)=d(n,i+1)+\sum_{j=1}^{n-i-1}d(n-j,i)d(j,1).$

The base case here is actually $\mathcal{D}^{(n)}_{n}$, which is
$p_{a}^{-1}(\mathcal{O}_{n}^{cpt})$. Every transfer system is compatible with
the additive one. Note also that we have an important edge case:

$(\mathcal{F}^{n-1}T)_{n}=(\mathcal{F}^{n}T)_{n},$

since both correspond to the unique complete transfer system on $[n]$.

######  Proposition 4.7 ([1, Theorem 20]).

For each $n$, we have

$d(n,n)=d(n,n-1)=\operatorname{Cat}(n).$

###  4.2. Rewriting the Recurrence Relation

We now can solve the recurrence relation, giving our first proof of the main
theorem. Recall the definition of the Fuss–Catalan numbers:

$A_{n}(p,r)=\frac{r}{np+r}\binom{np+r}{n}.$

We begin with some helpful properties of the Fuss-Catalan number.

######  Proposition 4.8.

For any positive integer $p$ and positive $n,r$, the following properties
hold.

  1. (1)

$A_{n}(3,r)=\sum_{j=0}^{n}A_{j}(2,1)A_{n-j}(3,j+r-1)$.

  2. (2)

$A_{n}(p,r)=A_{n}(p,r-1)+A_{n-1}(p,p+r-1)$.

  3. (3)

$A_{n+1}(p,1)=A_{n}(p,p)$.

  4. (4)

$A_{n}(p,s+r)=\sum_{j=0}^{n}A_{j}(p,r)A_{n-j}(p,s)$.

  5. (5)

$A_{n}(3,2)=\sum_{j=1}^{n}A_{j+1}(2,1)A_{n-j}(3,j)$.

###### Proof.

Four of these formulae are from work of Młotkowski. Formula 1 is a special
case of [7, Proposition 2.1]. The proof 2 is [7, Equation 2.2], 3 is [7,
Equation 2.3], and 4 is [7, Equation 2.4].

We will only prove 5. It can be done by induction on $n$. The base case $n=1$
is a straightforward check. Take $n\geq 2$.

Denote

$s=\sum_{j=1}^{n}A_{j+1}(2,1)A_{n-j}(3,j).$

Replacing $j$ with $j+1$, we can rewrite this as

$s=\sum_{j=2}^{n+1}A_{j}(2,1)A_{n-j+1}(3,j-1).$

Applying 2 twice, we have

$\displaystyle A_{n-j+1}(3,j+1)$ $\displaystyle=A_{n-j+1}(3,j)+A_{n-j}(3,j+3)$
$\displaystyle=\big{(}A_{n-j+1}(3,j-1)+A_{n-j}(3,j+2)\big{)}+A_{n-j}(3,j+3).$

With the help of 1, we can expand $A_{n+1}(3,2)$ to

$A_{n+1}(3,2)=A_{n+1}(3,1)+A_{n}(3,2)+\sum_{j=2}^{n+1}A_{n-j+1}(3,j+1)A_{j}(2,1),$

and substituting in for $A_{n-j+1}(3,j+1)$, we can rewrite this as

$A_{n+1}(3,2)=A_{n+1}(3,1)+A_{n}(3,2)+s+\\\
\sum_{j=2}^{n}A_{n-j}(3,j+2)A_{j}(2,1)+\sum_{j=2}^{n}A_{n-j}(3,j+3)A_{j}(2,1).$

The last two sums also show up in the expansions of $A_{n}(3,3)$ and
$A_{n}(3,4)$, respectively, by 1:

$\displaystyle\sum_{j=2}^{n}A_{n-j}(3,j+2)A_{j}(2,1)=A_{n}(3,3)-A_{n}(3,2)-A_{n-1}(3,3),$
$\displaystyle\sum_{j=2}^{n}A_{n-j}(3,j+3)A_{j}(2,1)=A_{n}(3,4)-A_{n}(3,3)-A_{n-1}(3,4).$

This gives an equality

(4.9) $A_{n+1}(3,2)=A_{n+1}(3,1)+A_{n}(3,2)+s\\\
+\big{(}A_{n}(3,3)-A_{n}(3,2)-A_{n-1}(3,3)\big{)}+\big{(}A_{n}(3,4)-A_{n}(3,3)-A_{n-1}(3,4)\big{)}.$

We can use 3 to rewrite Equation 4.9 as

$A_{n+1}(3,2)=A_{n+1}(3,1)+A_{n}(3,4)+\big{(}s-A_{n}(3,1)-A_{n-1}(3,4)\big{)}.$

However, applying 2 to $A_{n+1}(3,2)$, we get exactly

$A_{n+1}(3,2)=A_{n+1}(3,1)+A_{n}(3,4),$

so we deduce

$s=A_{n}(3,1)+A_{n-1}(3,4).$

By 2 again, we get $s=A_{n}(3,2)$. ∎

######  Theorem 4.10.

For any $n\in\mathbb{N}$, $A_{n-1}(3,2)$ is $d(n,1)$.

Before proving the theorem, we first discuss the strategy of the proof.
Instead of focusing on the recurrence steps for a specific $n$, we put all
$d(k,j)$ together which forms a triangle as follows.

$d(2,1)$$d(3,1)$$d(4,1)$$d(5,1)$…$d(3,2)$$d(4,2)$$d(5,2)$$d(4,3)$$d(5,3)$$d(5,4)$

Recall Corollary 4.6. For $i\in\mathbb{N}_{+}$, the elements in the set
$\\{d(j+i,j):j\geq 1\\}$ share the same recursive formula. In fact, we can
generate the above triangle hypotenuse by hypotenuse from outside to inside.
To begin with, we generate the second outermost hypotenuse $d(3,1)$, $d(4,2)$,
$d(5,3)$… by the outermost hypotenuse $d(2,1)$, $d(3,2)$, $d(4,3)$… according
to the recursive formula for $d(n,n-2)$. In other word, $d(n,n-2)$ is a linear
combination of $d(j,j-1)$ with coefficients 1 or $d(1,1)$. Similarly, we can
generate the third outermost hypotenuse by the outermost and the second
outermost hypotenuse according to the recursive formula for $d(n,n-3)$. Now
the coefficients are from the space spanned by $d(2,1)$, $d(1,1)$ and 1.

As the whole triangle can be generated hypotenuse by hypotenuse, we can
conclude that any $d(k,l)$ in the triangle can be written as a linear
combination of $d(j,j-1)$ where $l+1\leq j\leq k$ with coefficients from the
space spanned by $1$ and $d(i,1)$ where $1\leq i\leq k-l-1$. Graphically,
$d(k,l)$ is a linear combination $d(j,j-1)$ in the blue area.

$d(k,l)$$d(j,j-1)$ Figure 4. Showing what contributes to each term: $d(k,l)$
is a linear combination of $d(j,j-1)$ from the blue shaded region. And a given
$d(j,j-1)$ contributes only in the orange region.

###### Proof of Theorem 4.10.

It is easy to check for $d(2,1)$. Then by induction, it suffices to show the
theorem is true for $d(n,1)$ if the theorem holds for $d(i,1)$ for any $2\leq
i\leq n-1$.

Fix $j$ such that $2\leq j\leq n$. For our convenience, we denote the
coefficient of $d(j,j-1)$ for $d(k,l)$ as $c(k,l)$. Our goal is to determine
$c(n,1)$. We will achieve this by induction.

First, we figure out the coefficients of $d(j,j-1)$ for the vertical line
$d(j+l,j-1)$ where $0\leq l\leq n-j$. We know $c(j,j-1)=1=A_{0}(3,1)$ and for
$0\leq l\leq n-j$,

$c(j+l,j-1)=\sum_{i=1}^{l}d(i,1)\,c(j+l-i,j-1).$

As $d(i,1)$ is $A_{i-1}(3,2)$ for $2\leq i\leq n-1$, we get
$c(j+l,j-1)=A_{l}(3,1)$ by 4 of Proposition 4.8 and the equality
$A_{n}(3,3)=A_{n+1}(3,1)$ for $0\leq l\leq n-j$ inductively.

In fact, we claim that for $1\leq i\leq j-1$, $c(j+l,j-i)=A_{l}(3,i)$ for
$0\leq l\leq n-j$. We have shown the case for $i=1$. Now it is sufficient to
show the case $k$ provided the claim holds for cases $1\leq i\leq k-1$.
Similarly, we know $c(j,j-k)=1=A_{0}(3,k)$ and for $0\leq l\leq n-j$,

$c(j+l,j-k)=c(j+l,j-(k-1))+\sum_{i=1}^{l}d(i,1)\,c(j+l-i,j-k).$

As $d(i,1)$ is $A_{i-1}(3,2)$ for $2\leq i\leq n-1$, we get
$c(j+l,j-k)=A_{l}(3,k)$ by 2 and 4 of Proposition 4.8 for $0\leq l\leq n-j$
inductively. Thus, we get $c(n,1)=A_{n-j}(3,j-1)$. As $d(n,1)$ depends only on
$d(j,j-1)$ for $2\leq j\leq n$,

$d(n,1)=\sum_{j=2}^{n}A_{n-j}(3,j-1)d(j,j-1)=\sum_{j=2}^{n}A_{n-j}(3,j-1)A_{j}(2,1),$

where the last equality is Proposition 4.7. By 5 of Proposition 4.8, we
conclude $d(n,1)=A_{n-1}(3,2)$. ∎

######  Theorem 4.11.

There are $A_{n}(3,1)$ compatible transfer systems for $[n]$.

###### Proof.

We will prove this by induction. It is easy to check the case for $[1]$.
Assume the theorem is true for $[i]$ where $1\leq i<n$. We want to show it is
true for $[n]$.

Corollary 4.6 shows that the number of compatible systems for $[n]$ is

$d(n,0)=d(n,1)+\sum_{j=1}^{n-1}d(j,0)d(n-j,1).$

Theorem 4.10 and the inductive hypothesis let us rewrite this as

$d(n,0)=A_{n-1}(3,2)+\sum_{j=1}^{n-1}A_{j}(3,1)A_{n-j-1}(3,2)=\sum_{j=0}^{n-1}A_{j}(3,1)A_{n-j-1}(3,2).$

By 4 of Proposition 4.8, we deduce

$d(n,0)=A_{n-1}(3,3),$

which by 3 of Proposition 4.8, equals $A_{n}(3,1)$. ∎

##  5\. Extensions of saturated systems

###  5.1. Counting using additive cores

Instead of using the filtration by powers of $w$ on the additive indexing
system, we can use the first part of Proposition 3.2. This says that
compatibility of $(\mathcal{O}_{a},\mathcal{O}_{m})$ is the same question as
compatibility of $(\operatorname{core}\mathcal{O}_{a},\mathcal{O}_{m})$, since
both reduce to the comparison

$\mathcal{O}_{m}\leq\operatorname{core}\mathcal{O}_{a}.$

Since the core breaks up as a direct some of complete transfer systems, this
last condition is really the same as asking that if

$\operatorname{core}\mathcal{O}_{a}=\mathcal{O}_{n_{1}}^{cpt}\oplus\dots\oplus\mathcal{O}_{n_{k}}^{cpt},$

then we have a direct sum decomposition

$\mathcal{O}_{m}=\mathcal{O}_{n_{1}}^{\prime}\oplus\dots\oplus\mathcal{O}_{n_{k}}^{\prime},$

where $\mathcal{O}_{n_{i}}^{\prime}\in T_{n_{i}}$ for all $i$. This is the
only condition here, so we deduce the following proposition.

######  Proposition 5.1.

If $\mathcal{O}_{a}$ is a transfer system with

$\operatorname{core}\mathcal{O}_{a}=\mathcal{O}_{n_{1}}^{cpt}\oplus\dots\oplus\mathcal{O}_{n_{k}}^{cpt},$

then there are

$\prod_{i=1}^{k}\operatorname{Cat}(n_{i})$

transfer systems $\mathcal{O}_{m}$ such that the pair
$(\mathcal{O}_{a},\mathcal{O}_{m})$ is compatible.

This reduces our question of the number of compatible pairs to two parts:

  1. (1)

Enumerate all of the transfer systems with a fixed core, then

  2. (2)

evaluate the corresponding sum.

######  Notation 5.2.

Let $\vec{k}=(k_{1},\dots,k_{n})$ be a sequence of positive integers. For each
$1\leq s\leq n$, let

$K_{s}=k_{1}+\dots+k_{s}.$

Let

$\mathcal{O}_{\vec{k}}^{sat}=\mathcal{O}_{k_{1}}^{cpt}\oplus\dots\oplus\mathcal{O}_{k_{n}}^{cpt}.$

######  Definition 5.3.

For a sequence $\vec{k}=(k_{1},\dots,k_{n})$ with $k=k_{1}+\dots+k_{n}$, let

$\mathcal{E}_{\vec{k}}=\big{\\{}\mathcal{O}\in
T_{k}\mid\operatorname{core}{\mathcal{O}}=\mathcal{O}_{\vec{k}}^{sat}\big{\\}},$

and let

$e_{\vec{k}}=|\mathcal{E}_{\vec{k}}|.$

###  5.2. Catalan tuples

The enumeration and exact sum were analyzed by de Jong, Hock, and Wulkenhaar
[5] in a slightly different guise. They consider certain sequences which they
call Catalan tuples.

######  Definition 5.4 ([5, Definition 3.1]).

For each positive integer $n$, a _Catalan tuple of length $n$_ is a a sequence
of non-negative integers

$\vec{s}=(s_{0},\dots,s_{r})$

with three properties:

  1. (1)

for all $j$, we have

$\sum_{i=0}^{j}s_{i}>j,$

  2. (2)

at the end,

$\sum_{i=0}^{r}s_{i}=n,$

  3. (3)

and if $n>0$, then $s_{r}>0$.

Let $\mathcal{S}_{n}$ be the set of all Catalan tuple of length $n$.

######  Remark 5.5.

We have slightly modified the definition here to ignore trailing zeros. This
removes our ability to predict the length of the string, but it will better
connect with the extensions.

We can restate the conditions in Definition 5.4 slightly to start connecting
with extensions.

######  Definition 5.6.

The _excess_ of a Catalan tuple $\vec{s}=(s_{1},\dots,s_{r})$ of length $n$ is

$e(\vec{s}):=n-r-1.$

######  Remark 5.7.

Note that the edge condition in Definition 5.4 of $j=r$ implies the inequality
$n>r$. This means the excess is always non-negative.

######  Proposition 5.8.

Let $\vec{s}$ be a Catalan tuple of length $n$ and excess $e=n-r-1$. Then for
any $k>0$, the sequence

$\vec{s}_{k}(\ell)=(s_{0},\dots,s_{r},\underbrace{0,\dots,0}_{\ell},k)$

is a Catalan tuple if and only if

$0\leq\ell\leq e.$

###### Proof.

For $0\leq j\leq r$, the Catalan tuple condition holds since it does for
$\vec{s}$. If $\ell=0$, then we have

$\sum_{i=0}^{r}s_{i}+k=n+k>r+1,$

since $n>r$ and $k\geq 1$. Assume now that $\ell>0$, and consider a $1\leq
j\leq\ell$. We have

$\sum_{i=0}^{r+j}s_{i}=n=r+e+1.$

On the other hand, this is greater than $r+j$ if and only if $j<e+1$. This
gives the bounds on $\ell$. Finally, note that the analysis for the case
$\ell=0$ now also implies the case $j=\ell+1$. ∎

######  Proposition 5.9.

Let $\vec{s}$ be a Catalan tuple of length $n$ and excess $e=n-r-1$. Then for
any $k>0$ and $0\leq\ell\leq e$, the excess of the Catalan tuple

$\vec{s}_{k}(\ell)=(s_{0},\dots,s_{r},\underbrace{0,\dots,0}_{\ell},k)$

is

$e\big{(}\vec{s}_{k}(\ell)\big{)}=k-1+e-\ell=n+k-(r+\ell+2).$

###### Proof.

The Catalan tuple given has length $n+k$. The sequence $\vec{s}$ has length
$(r+1)$, and we added $\ell+1$ new terms to form $\vec{s}_{k}(\ell)$. ∎

This lets us rewrite Catalan tuples using only the non-zero entries.

######  Definition 5.10.

If $\vec{s}$ is a Catalan tuple, then let $\operatorname{core}(\vec{s})$ be
the subsequence of non-zero entries of $\vec{s}$. Given a partition $\vec{k}$
of $k$, let $\mathcal{S}_{\vec{k}}$ be the subset of $\mathcal{S}_{k}$ of
Catalan tuples with core $\vec{k}$:

$\mathcal{S}_{\vec{k}}:=\\{\vec{s}\in\mathcal{S}_{k}\mid\operatorname{core}{\vec{s}}=\vec{k}\\}.$

######  Corollary 5.11.

Catalan tuples with core $\vec{k}$ are those sequences of the form

$k_{1},\underbrace{0,\dots,0}_{\ell_{1}},k_{2},\underbrace{0,\dots,0}_{\ell_{2}},k_{3},\dots,k_{n-1},\underbrace{0,\dots,0}_{\ell_{n-1}}k_{n}$

such that for all $1\leq j\leq n-1$, we have

$\sum_{i=1}^{j}(k_{i}-\ell_{i})\geq j.$

The excess of such a sequence is

$(k_{n}-1)+\sum_{i=1}^{n-1}(k_{i}-1-\ell_{i}).$

In their work, de Jong, Hock, and Wulkenhaar consider certain collections of
Catalan tuples.

######  Definition 5.12 ([5, Definition 4.1]).

A _nested Catalan tuple of length $n$_ is a sequence of Catalan tuples
$(\vec{s}_{i_{1}},\dots,\vec{s}_{i_{r}})$ such that
$\vec{s}_{i_{j}}\in\mathcal{S}_{i_{j}}$ and the sequence

$(i_{1}+1,i_{2},\dots,i_{r})$

is a Catalan tuple of length $n$.

A key result in [5] is the cardinality of the number of nested Catalan tuple
that begin with $(0)$. For this, we need a straightforward lemma.

######  Lemma 5.13.

The map $\Sigma\colon\mathcal{S}_{n}\to\mathcal{S}_{n+1}$ given by

$(s_{0},\dots,s_{r})\mapsto(1,s_{0},\dots,s_{r})$

is an injection with image those sequences which begin with $1$.

######  Proposition 5.14 ([5, Corollary 4.6]).

The number of nested Catalan tuples of length $(n+1)$ with first term $(0)$ is

$\sum_{\vec{s}\in\mathcal{S}_{n}}\prod_{i}\operatorname{Cat}(s_{i})=\frac{1}{2n+1}\binom{3n}{n}=\frac{1}{3n+1}\binom{3n+1}{n}=A_{n}(3,1).$

We will produce an explicit bijection

$\sigma\colon T_{n}\to\mathcal{S}_{n}$

by building bijections between $\mathcal{E}_{\vec{k}}$ and
$\mathcal{S}_{\vec{k}}$.

###  5.3. Enumerating Extensions by a complete transfer system

It is helpful to think of elements of $\mathcal{E}_{\vec{k}}$ also as various
“extensions” of the complete transfer systems
$\mathcal{O}_{k_{1}}^{cpt}$,…$\mathcal{O}_{k_{n}}^{cpt}$. For our count, it is
easier to instead consider a more general class.

######  Definition 5.15.

An _extension of $\mathcal{O}^{\prime}\in T_{m}$ by
$\mathcal{O}^{\prime\prime}\in T_{k}$_ is a transfer system $\mathcal{O}\in
T_{m+k}$ such that

  1. (1)

$i_{m}^{\ast}\mathcal{O}=\mathcal{O}^{\prime}$,

  2. (2)

$\Phi^{m}\mathcal{O}=\mathcal{O}^{\prime\prime}$, and

an extension $\mathcal{O}$ of $\mathcal{O}^{\prime}$ by
$\mathcal{O}^{\prime\prime}$ is _core-preserving_ if moreover:

$\operatorname{core}{\mathcal{O}}=\operatorname{core}{\mathcal{O}^{\prime}}\oplus\operatorname{core}{\mathcal{O}^{\prime\prime}}.$

Note that since by assumption we have specified $i_{m}^{\ast}$ and $\Phi^{m}$
in an extension, we need only determine the transfers with source $i\leq m$
and target $j>m$.

######  Definition 5.16.

Any transfer $i\to j$ with $i\leq m$ and $j>m$ in an extension of
$\mathcal{O}^{\prime}\in T_{m}$ by $\mathcal{O}^{\prime\prime}$ is a
_crossing-transfer_.

######  Proposition 5.17.

Let $\mathcal{O}$ be an extension of $\mathcal{O}^{\prime}\in T_{m}$ by
$\mathcal{O}^{\prime\prime}\in T_{k}$. Then following are equivalent:

  1. (1)

The extension is core-preserving.

  2. (2)

If there is a transfer $m\to j$, then $j=m$.

###### Proof.

Note that the existence of a nontrivial transfer $m\to j$ is equivalent to the
existence of a transfer $m\to(m+1)$, by the restriction axiom. If we have

$\operatorname{core}{\mathcal{O}^{\prime}}=\mathcal{O}_{n_{1}}^{cpt}\oplus\dots\oplus\mathcal{O}_{n_{j}}^{cpt}\text{
and
}\operatorname{core}{\mathcal{O}^{\prime\prime}}=\mathcal{O}_{m_{1}}^{cpt}\oplus\dots\mathcal{O}_{m_{i}}^{cpt},$

then by construction of the core, the existence of the transfer $m\to(m+1)$ is
equivalent to the core of $\mathcal{O}$ be

$\operatorname{core}{\mathcal{O}}=\mathcal{O}_{n_{1}}^{cpt}\oplus\dots\oplus\mathcal{O}_{n_{j-1}}^{cpt}\oplus\mathcal{O}_{n_{j}+m_{1}}^{cpt}\oplus\mathcal{O}_{m_{2}}\oplus\dots\oplus\mathcal{O}_{m_{i}}.$

The result follows. ∎

Since we want to enumerate transfer systems with a fixed core, we now
restriction attention to core-preserving extensions of $\mathcal{O}$ by a
complete transfer system $\mathcal{O}^{\prime}$. This significantly simplifies
our combinatorics.

######  Lemma 5.18.

Let $\mathcal{O}$ be a core-preserving extension of $\mathcal{O}^{\prime}\in
T_{m}$ by $\mathcal{O}_{k}^{cpt}$. Then for each $i\leq m$, the following are
equivalent

  1. (1)

We have a crossing transfer $i\to m+j$ for some $j>0$.

  2. (2)

We have crossing transfers $i\to m+j$ for all $0\leq j\leq k$.

###### Proof.

One direction is immediate. For the other, if we have a transfer $i\to m+j$,
then by the restriction axiom, we have transfers $i\to m$ and $i\to m+1$.
Since $\mathcal{O}_{k}^{cpt}$ is complete, in our extension, we have transfers
$m+1\to m+j$ for any $1\leq j\leq k$, which gives the second result. ∎

######  Remark 5.19.

We singled out the transfer $i\to m$ since this constrains the number of
possible sources for a transfer from $[m]$ up to $[k]$. Any crossing transfer
has source an element of $[m]$ that transfers up to $m$ in
$\mathcal{O}^{\prime}$.

Moreover, however, we have a kind of “non-decreasing” property.

######  Lemma 5.20.

Let $\mathcal{O}$ be a core-preserving extension of $\mathcal{O}^{\prime}\in
T_{m}$ by $\mathcal{O}_{k}^{cpt}$, and let

$\\{d_{1}<d_{2}<\dots<d_{r}<d_{r+1}=m\mid d_{i}\to m\\}$

be the set of elements of $[m]$ which transfer to $m$ in
$\mathcal{O}^{\prime}$. In $\mathcal{O}$, if we have a transfer

$d_{j}\to m+k,$

then for all $1\leq i\leq j$, we have transfers

$d_{i}\to m+k.$

###### Proof.

By the restriction axiom, whenever $i\leq j$, we have a transfer $d_{i}\to
d_{j}$. The result follows from transitivity. ∎

######  Definition 5.21.

Let $\mathcal{O}^{\prime}\in T_{m}$, and let

$\\{d_{1}<d_{2}<\dots<d_{r}<d_{r+1}=m\mid d_{i}\to m\\}$

be the set of elements of $[m]$ which transfer to $m$ in
$\mathcal{O}^{\prime}$. For each $k>0$ and for each $0\leq\ell\leq r$, define
a relation $\to_{\ell}$ on $[m+k]$ that refines the partial order $\leq$ by
saying

  1. (1)

if $i\leq j\leq m$, then $i\to_{\ell}j$ if and only if $i\to j$ in
$\mathcal{O}^{\prime}$,

  2. (2)

if $m<i\leq j$, then $i\to_{\ell}j$ if and only if $i\leq j$, and

  3. (3)

if $i\leq m<j$, then $i\to_{\ell}j$ if and only if $i=d_{s}$ for some $1\leq
s\leq r-\ell$.

######  Proposition 5.22.

The relation $\to_{\ell}$ is a transfer system on $[n+k]$ that is a core-
preserving extension of $\mathcal{O}^{\prime}$ by $\mathcal{O}_{k}^{cpt}$.

######  Definition 5.23.

Let $\mathcal{O}_{\mathcal{O}^{\prime},k}(\ell)$ denote the transfer system
$\to_{\ell}$ on $[n+k]$.

######  Remark 5.24.

There is a special case of the extensions: $\ell=r$. In this case, we have the
direct sum $\mathcal{O}^{\prime}\oplus\mathcal{O}_{k}^{cpt}$.

There is a crucial observation about the number of transfers to $[m+k]$ here.

######  Proposition 5.25.

Let $\mathcal{O}^{\prime}$ be a transfer system on $[m]$ in which $r+1$
elements transfer up to $m$, and let $0\leq\ell\leq r$. Then in
$\mathcal{O}_{\mathcal{O}^{\prime},k}(\ell)$, we have $k+r-\ell$ elements
which transfer up to $m+k$.

###### Proof.

All of the $k$ elements of $[k]$ transfer up, and by construction, the
$r-\ell$ elements $d_{1},\dots,d_{r-\ell}$ are the only elements from $[m]$
which also transfer up to $m+k$. ∎

Putting these together gives a complete classification of the core-preserving
extensions.

######  Theorem 5.26.

Let $\mathcal{O}\in T_{m}$ be a transfer system, and let

$\\{d_{1}<d_{2}<\dots<d_{r}<d_{r+1}=m\mid d_{i}\to m\\}$

be the set of elements of $[m]$ which transfer to $m$ in
$\mathcal{O}^{\prime}$. Then there are $(r+1)$ core-preserving extensions of
$\mathcal{O}$ by $\mathcal{O}_{k}^{cpt}$ given by
$\mathcal{O}_{\mathcal{O}^{\prime},k}(\ell)$ for $0\leq\ell\leq r$.

###### Proof.

Lemma 5.18 and Lemma 5.20 show that any core-preserving extension has this
form. The converse is the content of Proposition 5.22. ∎

###  5.4. Enumerating transfer systems with a fixed core

Now let $\mathcal{O}$ be a transfer system with

$\operatorname{core}{\mathcal{O}}=\mathcal{O}_{\vec{k}}^{sat}.$

Write $\vec{k}=(k_{1},\dots,k_{n})$. We can immediately identify $\mathcal{O}$
inductively as a type considered in the previous section.

######  Proposition 5.27.

The transfer system $\mathcal{O}$ is a core-preserving extension of
$i_{K_{n-1}}^{\ast}\mathcal{O}$ by $\mathcal{O}_{k_{n}}^{cpt}$.

This turns our problem into an inductive one, working down on the number of
summands in the partition of $k$. We can now build our bijection.

######  Definition 5.28.

Let

$\sigma\colon\mathcal{E}_{\vec{k}}\to\mathcal{S}_{\vec{k}}$

be defined inductively by the following procedure. If $\vec{k}=(k)$, then
$\mathcal{O}=\mathcal{O}_{k}^{cpt}$, and we define

$\sigma(\mathcal{O}_{k}^{cpt})=k.$

For a general $\vec{k}=(k_{1},\dots,k_{n})$ with $n>1$ and
$\mathcal{O}\in\mathcal{E}_{\vec{k}}$, let $\ell_{n-1}$ be the unique number
$0\leq\ell_{n-1}$ such that

$\mathcal{O}=\mathcal{O}_{\mathcal{O}^{\prime},k_{n}}(\ell_{n-1}),$

where $\mathcal{O}^{\prime}=i_{K}^{\ast}\mathcal{O}$, and define

$\sigma(\mathcal{O})=\sigma(\mathcal{O}^{\prime}),\underbrace{0,\dots,0}_{\ell_{n-1}},k_{n}.$

######  Example 5.29.

For $\mathcal{O}$ the transfer system in Figure 2, we have

$\sigma\mathcal{O}=(4,0,1,2).$

We need to verify that $\sigma$ actually lands in the set
$\mathcal{S}_{\vec{k}}$.

######  Proposition 5.30.

For any $\mathcal{O}$ with $\operatorname{core}{\mathcal{O}}=\vec{k}$, we have

  1. (1)

$\sigma(\mathcal{O})\in\mathcal{S}_{\vec{k}}$, and

  2. (2)

$e\big{(}\sigma(\mathcal{O})\big{)}$ is the number of elements $j$ between $1$
and $K_{n}-1$ that transfer up to $K_{n}$ in $\mathcal{O}$.

###### Proof.

We show this by induction on $n$. The base case of $n=1$ is immediate by the
definitions of $\sigma$ and the excess, so assume this is true for partitions
with fewer than $n$ terms.

Let $\mathcal{O}^{\prime}=i_{K_{n-1}}^{\ast}\mathcal{O}$, and let $r$ be the
number of $j<K_{n-1}$ which transfer up to $K_{n-1}$ in
$\mathcal{O}^{\prime}$. By the inductive hypothesis,
$\sigma\mathcal{O}^{\prime}$ is a Catalan tuple with core
$(k_{1},\dots,k_{n-1})$ and we also have

$e\big{(}\sigma\mathcal{O}^{\prime}\big{)}=r.$

Now if $0\leq\ell_{n}\leq r$ is such that
$\mathcal{O}=\mathcal{O}_{\mathcal{O}^{\prime},k_{n}}(\ell_{n-1})$, then since
$\ell_{n-1}\leq e\big{(}\sigma\mathcal{O}^{\prime}\big{)}$, the first claim is
Proposition 5.8. For the second part, Proposition 5.9 shows that the excess of
$\sigma\mathcal{O}$ is

$e(\sigma\mathcal{O})=e-\ell_{n-1}+(k_{n}-1).$

Proposition 5.25 shows this quantity is exactly the number of elements smaller
than $K_{n}$ which transfer up to $K_{n}$. ∎

This gives us the final piece for our argument.

######  Corollary 5.31.

The map $\sigma$ is a bijection
$\mathcal{E}_{\vec{k}}\to\mathcal{S}_{\vec{k}}$.

###### Proof.

By induction on $n$, we see that there are exactly as many extensions of
$i_{K_{n-1}}^{\ast}\mathcal{O}$ by $\mathcal{O}_{k_{n}}$ as there are
extensions of the Catalan tuple $\sigma(i_{K_{n-1}}^{\ast}\mathcal{O})$ to a
Catalan tuple ending with $k_{n}$, and the map $\sigma$ gives a bijection
between these. ∎

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1

# A Highly Variable Magnetized Environment in a Pulsar Binary resembling Fast
Radio Bursts

Dongzi Li${}^{\ref{aff:caltech}}$<EMAIL_ADDRESS>orcid.org/0000-0001-7931-0607  Anna Bilous${}^{\ref{aff:astron}}$ Scott
Ransom${}^{\ref{aff:nrao}}$ Robert Main${}^{\ref{aff:mpifr}}$ Yuan-Pei
Yang${}^{\ref{aff:swifa}}$

###### Abstract

Fast radio bursts (FRBs) are short, intense extragalactic radio bursts of
unknown origin [1]. Recent polarimetric studies have shown that a noticeable
fraction of the repeating FRBs display irregular, short-time variations of the
Faraday rotation measure (RM) [2, 3, 4]. Moreover, evidence for rare
propagation effects such as Faraday conversion and polarized attenuation is
seen in at least one FRB repeater [2]. Together, they suggest a highly
variable magneto-active circum-burst environment. In this paper, we report
similar behavior in a globular cluster pulsar binary system PSR B1744$-$24A
[5]. We observe irregular fast changes of RM with both signs at random orbital
phases as well as profile changes of the circular polarization when the pulsar
emission passes close to the companion. The latter provides strong evidence
for Faraday conversion and circularly polarized attenuation. These
similarities between PSR B1744$-$24A and some FRB repeaters, as well as the
possible binary-produced long-term periodicity of two active repeaters [6, 7],
and the discovery of a nearby FRB in a globular cluster[8, 9], where pulsar
binaries are common, all suggest that some fraction of FRBs have binary
companions.

Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, 1216 E California Boulevard, Pasadena, CA 91125, USA

ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA
Dwingeloo, The Netherlands

National Radio Astronomy Observatory, Charlottesville, VA, USA

Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn,
Germany

South-Western Institute For Astronomy Research, Yunnan University, Yunnan
650504, P. R. China

Located in the globular cluster Terzan 5, PSR B1744$-$24A (Ter5A) is a
11.56-ms “redback” pulsar in a 1.82-hr tight orbit with a
$\sim$0.09$M_{\odot}$ companion [5]. For more than 15 years, it has been
routinely observed with the 100-m Green Bank Telescope (GBT) with a cadence of
a few months. The emission from Ter5A is usually eclipsed around pulsar
superior conjunction, i.e. orbital phase $\Phi=0.25$, when the pulsar is
behind the companion. Small irregular eclipses at other orbital phases and
long eclipses spanning the full orbit are also common [5]. These features can
usually last several orbits. While the detailed analysis of Ter5A’s radio
properties from the whole observational sample will be presented elsewhere,
here we report on a couple of observations which show two interesting kinds of
polarized propagation phenomena - rapid RM variation at random orbital phases
and a strong change in the circular polarization near superior conjunction.

In Fig. 1 left panel, the observation at 1.5 GHz has shown a sudden 50%
decrease of RM at orbital phases of 0.65 and 1.8, which is $140\degree$ and
$160\degree$ from superior conjunction, when the companion is almost behind
the pulsar. At orbital phase 1.9, $\sim 120\degree$ away from superior
conjunction, the RM has a sudden 50% increase. Similar levels of RM variation
are seen at 2 GHz, although with larger error bars (Fig 1 right panel). No
significant dispersion measure (DM) variation up to $0.01$ pc cm-3 is detected
during the RM jumps, implying a lower limit of the average parallel magnetic
field in the material of the companion wind $\langle
B\rangle=12~{}\mathrm{mG}\,(\Delta\mathrm{RM}/100~{}\mathrm{rad}~{}\mathrm{cm}^{-2})/(\Delta\mathrm{DM}/0.01~{}\mathrm{pc}\,\mathrm{cm}^{-3})>12~{}\mathrm{mG}$.
Large RM variation of $>3000$ rad m-2 is expected at $\Phi=0.9$ from this
lower limit of magnetic field, because significant $\Delta$
DM$=0.3\,\mathrm{pc}\,\mathrm{cm}^{-3}$ is observed. However, the linear
polarization (L) is depolarized here. While our frequency resolution should
enable us to measure RM up to $10^{4}~{}$rad m-2, fast RM variation of $\sim
100$ rad m-2 is most commonly seen, despite a much larger value expected from
the DM variation. Linear depolarization often appears at a significant
fraction of the orbit, especially during the DM variation. This is best
explained by the depolarization because pulses with large varying RMs are
summed together (see Methods) The commonly observed random RM variation and
linear depolarization show a highly variable magnetized environment.

During another observation, significant variation of circularly polarized
emission $V$ was detected near superior conjunction in three consecutive
orbits recorded at 2 GHz (Figure 2). Near superior conjunction, $V$ has an
opposite sign compared to the normal $V$ profile, slowly turning to zero and
then flipping back to the normal profile as the pulsar moves away from behind
the companion. The synthetic sign reversal of $V$ across the pulse profile is
a definite evidence of Faraday conversion when the radio wave adiabatically
tracks a reversal in the parallel magnetic field $B_{z}$, an effect also known
as mode coupling/tracking [10, 11, 12].

Faraday conversion is considered an important mechanism to explain the
recently observed circular polarization behavior of FRB 20201124A [2, 13].
Ter5A, with which the Faraday conversion is first seen on a pulsar, has the
unique advantage that the propagation effects can be isolated from intrinsic
polarization behavior, as the latter is known owing to the observation away
from superior conjunction. Propagation will affect all parts of the spin phase
equally, and the complicated $V$ profile give multiple independent
measurements of the parameter of the radio transfer function when compared to
the unaffected $V$ profile. We show that the $I$, $V$ profile across frequency
and spin phase can be well reproduced with the normal profile going through a
reversal of circular polarization and circularly polarized absorption (Figure
2 panel d).

The Faraday conversion is best explained with the radio wave passing through
the poloidal field of the companion magnetosphere with
$B>10\,G\,(\Delta\mathrm{DM}/0.1\,\mathrm{pc\,cm}^{-3})^{-1/3}(f/2\,\mathrm{GHz})^{-4/3}$
(see Methods). A reversal of parallel magnetic field $B_{z}$ will be stably
experienced due to the geometry of the large scale poloidal field (Figure 3).
A plausible scenario for the circularly polarized absorption is the
synchrotron-cyclotron absorption, which suggests $B\sim 100$ G given the
observed increase of $V$ fraction, consistent with the lower limit from the
Faraday conversion (see Methods). With the model, we give a qualitative
prediction of $V$ against orbital phase for higher frequency (Methods,
Extended data Figure 1). Birefringence of lensed single pulses[14, 15] near
superior conjunction will also be expected (see Methods). Both the Faraday
conversion and the circularly polarized absorption indicates a highly
magnetized companion. The change of the polarization profile against orbital
phase could provide a measure of the orbital inclination angle, as the orbital
geometry strongly affects the observed values of B throughout the orbit ([16]
and see Methods).

Magnetized companions have been observed to introduce RM variations or
depolarization of linear fraction near periastron or superior conjunction [17,
18, 19]. Our observations of Ter5A further show that in some systems, the
magnetized windy companion can introduce order-one RM variation even when the
companion is almost behind the pulsar. When the binary system is observed
edge-on, RM/DM variations close to phase 0.25 will be seen; while if this
binary system is observed near face-on, irregular RM variation of both signs
would be seen depending on the intra-binary weather, which contains little
information about the orbital phases.

The irregular large RM changes are similar to what has been observed in
repeating fast radio bursts (FRBs). Four out of six FRB repeaters with more
than two published RMs have shown RM variations [20, 2, 3, 4, 21]. Most
prominently, FRB 20190520B and FRB 20201124A have shown prominent fast
irregular RM changes, which can be modelled with the existence of a companion
[22]. Moreover, FRB 20201124A has shown changes of circular polarization on a
small fraction of the bursts, which is attributed to the polarized absorption
and Faraday conversion[2, 13]. We show that these effects can also appear near
a companion. Moreover, the absorption mechanism responsible for the increase
of $V$ fraction at Ter5A may also be responsible for the rare large $V$
observed in some FRBs [23, 2, 24]. Near superior conjunction, the estimated
$\gtrsim 10$ G field from Faraday conversion and $\Delta\mathrm{DM}\approx
0.2$ pc cm-3yields an RM of $\gtrsim 10^{6}$ m-2. This is comparable to the RM
of FRB 20121102A [20], which has one of the largest RMs of any astronomical
source. The limited detection of large RM variation in pulsar binaries is
likely due to depolarization arising from averaging pulses with different RMs
and potential scattering (see Methods). We will be able to observe much larger
RM and RM variation if single pulses, similar to FRBs, can be detected in the
binary system.

Apart from the similarity in the observed polarimetry, there is other
independent evidence suggesting the possibility that some FRBs reside in a
binary system. FRB 20180916A is observed to have a 16-day period [6]; while
FRB 20121102A potentially has a periodicity of 160 days[7, 25]. Binary orbit
has been proposed as an origin for the long-term periodicity[26, 27, 28, 29,
30]. Additionally, the closest extra-galactic FRB 20200120E is localized to a
globular cluster[8, 9] only 3.6 kpc away. Globular clusters host old stellar
populations where traditional supernovae-formed magnetars will be too old to
stay active. Even considering the dynamic formation channels of the magnetars,
the merging rate require magnetic activity lifetimes longer than empirically-
constrained lifetimes of Galactic magnetars [31, 32]. On the other hand,
binary pulsars are overabundant in globular clusters. Together, they call for
consideration of the existence of a companion for a fraction of FRBs.

Figure 1: Highly variable Ter5A polarization versus orbital phase. (a) Pulse
intensity ($I$) against orbital and spin phase at 1.5 GHz (top panel) and 2
GHz (bottom panel). (b) Linear polarization ($L$) with Faraday rotation
corrected using local RM value. (c) Circular polarization ($V$), with red/blue
corresponding to positive/negative value. Panel a,b,c are both dedispersed
with constant $\mathrm{DM_{0}}=242.36~{}\mathrm{pc}\,\mathrm{cm}^{-3}$, folded
with 512 phase bins and averaged over 20-s. (d)
$\delta\mathrm{DM}=\mathrm{DM}-\mathrm{DM}_{0}$ in 1-min integrations (black
dots) and $\mathrm{RM}$ in 20-s integrations (red dots). The superior
conjunction of the pulsar is indicated with dashed lines. Irregular, fast
variations of RM of both signs, as well as depolarization are seen at random
orbital phases due to the magnetized plasma from the companion.

Figure 2: Ter5A Faraday conversion near superior conjunction. (a,b): Zoom-ins
of the $I$, $V$ profiles in the bottom panel of Fig 1. Emission observed in
the selected orbital phase range is averaged to produce the pulse profiles in
panel c. The $V$ (blue curve) first disappears and then changes sign when
approaching superior conjunction. (d): The unaffected (i.e. normal) $I$
(black), $V$ (blue) and polarization fraction $P$ (orange) against spin phase
and frequency. (e): Observations (points) and model (lines) of the $I$ and $V$
profiles near superior conjunction averaged over three orbits. The observed
profiles are reproduced by the normal profile in panel d going through Faraday
conversion and circularly polarized absorption (see Methods).

Figure 3: The model of Faraday conversion in the companion poloidal magnetic
field. (a): The geometry of the system. The pulsar is located at (0,0). (b):
The magnetic strength $|B|$ and parallel magnetic field $B_{z}$ in the
companion magnetosphere along the line of sight (LOS). $B_{z}$ has changed its
sign in the shaded region. (c) The change of Stokes Q, U, V along the line of
sight (LOS) for the shaded region in (b). The $V$ changes its sign after
passing the region, because the radio wave adiabatically traces the field
reversal. For this illustrative example, we assume the companion has a dipole
field with a dipole moment of $2\times 10^{32}$ $\mathrm{G}\,\mathrm{cm}^{3}$
(300 G surface field at the equator), orbital inclination angle
$i=75^{\circ}$, and an electron density
$n_{e}\approx\Delta\mathrm{DM}/a\approx 5\times 10^{6}\mathrm{cm}^{-3}$, where
$a$ is the orbital separation. (b,c) are shown for orbital phase 0.3 at 2 GHz.
A reversal of $B_{z}$ will be experienced in the poloidal field as long as the
LOS is not parallel to the magnetic axis. The predicted behavior of $V$
against frequency and orbital phase is shown in Extended Data Fig 1.

##  0.1 Data recording and reduction.

The observations of Ter5A were recorded with the GUPPI backend on Oct 14th
2014 and the VEGAS backend on Jun 21st 2020 in a 800-MHz band centered at 1500
MHz (L-band, session 141014) or 2000 MHz (S-band, session 200621) in coherent
dedispersion mode with time resolution of 10.24 $\mu$s. The signal was
dedispersed in each of 512 1.56-MHz channels with the average DM of the
cluster, $\mathrm{DM}=238.0$ pc cm-3. The raw data were folded modulo
topocentric pulse spin period and integrated within every 20 s with dspsr
software. The folded archives contained four Stokes parameters, 512 frequency
channels, and 512 spin phase bins (corresponding to $t_{\mathrm{res}}=22.6$
$\mu$s).

The initial ephemerides for folding were obtained using observations spanning
2010-2014 [14]. For the 2020 session, ephemerides were iteratively refined
using pulse times of arrivals (TOAs) away from eclipses for seven sessions in
2019-2020. TOAs were determined using single-frequency profile templates
constructed from the average uneclipsed profile in the sessions with brightest
S/N. Pulsar spin period, period derivative and DM were updates using all seven
observations, fitting for arbitrary phase offset between L- and S- TOAs.
Subsequently, the epoch of the ascending node, DM and projected semimajor axis
were updated on per-session basis using simple BT binary orbit.

We performed full-Stokes calibration of the folded archives using standard
techniques. Prior to each observation, a pulsed calibration signal was
recorded which was used together with quasar B1442+101 as unpolarized flux
calibrator to correct for the instrumental response of the receiver system.
Polarization calibration for L-band was conducted using predetermined Mueller
matrix solutions, which described the cross-coupling between orthogonal
polarizations in the receivers (van Straten 2004). The Mueller matrix was
determined using the PSRCHIVE task pcm based on observations of PSR B0450+55.
For polarization calibration in S-band we assumed that the feed is ideal and
consists of two orthogonally polarized receptors. Such technique was
previously used for S-band observations of another MSP [33] and it proved to
be adequate when comparing to more rigorous polarization calibration with
predetermined Mueller matrix solutions.

DM was measured with tempo2 using single-frequency template on a variety of
timescales ranging from 10 s to 4 min if a subintergation had sufficient S/N
of pulsed emission in each of 4 subbands. RM was measured with rmfit on 20-s
subintegrations with 256 subbands.

##  0.2 RM variation and linear depolarization

In the quiescent phase, Ter5A’s radio emission has a considerable level of
fractional linear polarization (see Figure 2 “normal” pulse profile), which
allows measuring RM via Faraday rotation. In an observation carried on 2014
with 800 MHz bandwidth centered at 1.5 GHz, irregular fast RM changes are
observed (Figure 1). Various single-integration (20 s) RM outliers are seen in
different orbital phases. And at orbital phase 1.9, i.e. $\sim 120^{\circ}$
away from the inferior conjunction, the RM suddenly increases by $\sim 100$
rad m-2 in less than 20 s and lasts for $\sim 6$% of the orbit. No significant
dispersion measure (DM) variation $>0.01$ pc cm-3 is detected during the RM
variation, giving a lower limit of magnetic field of $\langle
B\rangle=12~{}\mathrm{mG}\,(\Delta\mathrm{RM}/100~{}\mathrm{rad}\,\mathrm{m}^{-2})/(\Delta\mathrm{DM}/0.01~{}\mathrm{pc}\,\mathrm{cm}^{-3})>12~{}\mathrm{mG}$
at orbital phase 1.9 where the pulsar is not behind the companion. The large
change in RM happening in less than $\Delta t\sim 20$ s suggests either
turbulent activities at extremely short timescale, or, more likely, a small-
scale structure in electron column density or magnetic field, with spatial
scale $\Delta L\lesssim v\Delta t=0.01R_{\odot}$, where $v\approx 800$ km s-1
is the orbital velocity assuming the neutron star mass of $1.4M_{\odot}$.

Multiple regions of excess DM are observed at various orbital phases, around
which the linear polarization disappears. This is expected from the underlying
fast RM variation. To obtain enough S/N, each data point is the sum of 20 s of
pulses. If the pulses have passed through an RM gradient of
$\Delta\mathrm{RM}$ in the 20s, the linear polarization will drop to
$L_{\mathrm{obs}}/L_{\mathrm{emi}}=\sin(\Delta RM\lambda^{2})/(\Delta
RM\lambda^{2})$ of the original level. At 1.5 GHz, an RM gradient of $\sim
100$ rad m-2 in the integrated pulses will reduce the linear polarization to
20% of the original level. Therefore, RM changes greater than 100 rad m-2 in
one sub-integration time will leads to depolarization of the observed linear
polarization. This has nicely explained why RM variation of $\sim 100-200$ rad
m-2 is most commonly seen in this system, while larger value is expected from
the DM variation and the frequency resolution should enable RM measurement all
the way to $10^{4}$ rad m-2. The linear polarization will drop faster if there
is random RM variation in the integrated pulses. With a standard deviation of
$\sigma_{\mathrm{RM}}$, the linear fraction will drop as
$L_{\mathrm{obs}}/L_{\mathrm{emi}}=\exp{(-2\sigma_{\mathrm{RM}}^{2}\lambda^{4})}$[34].
Therefore, the linear polarization will disappear if
$\sigma_{\mathrm{RM}}\gtrsim 25~{}$rad m-2. Near orbital phase 0.9, the
$\Delta$ DM rapidly increases to $\sim 0.3\,\mathrm{pc}\,\mathrm{cm}^{-3}$,
suggesting a quick change of $\Delta\mathrm{RM}>3000\,\mathrm{rad/cm}^{2}$
given the lower limit of $\langle B\rangle$ at phase 1.9, which will certainly
lead to the observed depolarization of linear polarization. In this case, if
single FRB-like pulses can be detected from this system, orders larger RM
variations, at least 3000 rad m-2, will likely be seen.

Apart from summation of pulses of different RMs, summation of scattered light
passing through paths of different RMs will also lead to depolarization. This
has been used to explain the decrease of polarization fraction in lower
frequency for several FRBs[35, 36]. Estimated from the scintillation time of
Ter5A[14], the separation of scattered light at the distance of the companion
due to the interstellar medium scattering should be of the order of 10 m,
which will unlikely cause depolarization. However, strong plasma lensed pulses
have been observed in this system[14], suggesting small scale DM variation in
the companion wind, which can also cause scattering. The separation of the
scattered light $\Delta x_{\mathrm{sct}}$ can be estimated from the scattering
time $\tau$ with $\Delta
x_{\mathrm{sct}}=\sqrt{2ac\tau}=0.01R_{\odot}\sqrt{\tau/0.1~{}\mathrm{ms}}$.
Therefore, if some pulses have scattering time longer than 0.1 ms, the
separation of the scattered light will be similar to the distance the pulsar
have moved in the 20 s integration time, and hence some single pulses may also
experience depolarization, similar to FRBs.

##  0.3 Faraday conversion in cold plasma

In the S-band observation, when the pulsar is behind the companion, the DM
increases by 0.2 pc cm-3and the circular polarization appears to have a sign
reversal at 2 GHz. The changes in the circular polarization indicates that the
radio flux has gone through a Faraday conversion caused by the plasma and
magnetic field of the companion.

In the cold plasma, assume wave propagates along $z$ direction, the
polarization vector $\vec{\mathrm{P}}=(Q,U,V)$ change follows:

$\frac{d\vec{\mathrm{P}}}{dz}=\vec{\Omega}\times\vec{\mathrm{P}}$ (1)

where $\vec{\Omega}=(\rho_{Q},\rho_{U},\rho_{V})$. $\rho_{V}$ is the Faraday
rotation rate:

$\rho_{V}=-\frac{2\pi}{c}\frac{f_{p}^{2}f_{B}}{f^{2}}\hat{B}_{z}$ (2)

and $\rho_{Q},\rho_{U}$ is the Faraday conversion rate:

$\displaystyle\rho_{L}=\rho_{Q}+i\rho_{U}=-\frac{\pi}{c}\frac{f_{p}^{2}f_{B}^{2}}{f^{3}}(\hat{B}_{x}+i\hat{B}_{y})^{2}$
(3)

where $\hat{B}\equiv\vec{B}/B$; $f_{p}=\sqrt{n_{e}e^{2}/\pi
m_{e}}=9.0\,\mathrm{kHz}(n_{e}/\mathrm{cm}^{-3})^{1/2}$ is the plasma
frequency, $f_{B}$ is the cyclotron frequency $f_{B}=eB/2\pi
m_{e}c=2.8\,\mathrm{MHz}(B/\mathrm{G})$. This procedure can be visualized as a
rotation of polarization vector around the natural axis in the direction of
$\vec{\Omega}$ in the Poincare sphere.

Define

$\tilde{z}\equiv 2\frac{f}{f_{B}}\hat{B}_{z}=2\frac{f}{f_{B}}\sin\alpha$ (4)

where $\alpha$ is the pitch angle between the wave vector and the magnetic
field.

When $\tilde{z}\gg 1$, i.e. $\rho_{V}\gg\rho_{L}$, the natural axis
$\vec{n}=\vec{\Omega}/|\vec{\Omega}|$ is pointing towards the $V$ direction,
resulting in only $Q$, $U$ rotation – an effect known as Faraday rotation.
Faraday conversion happens when the conversion rate is similar or higher than
Faraday rotation rate, i.e. $\tilde{z}\lesssim 1$. In this case, the natural
axis $\Omega$ is pointing close to the linear axis, and the wave modes become
quasi-linear, resulting in circular/linear conversion.

As can be seen from Equation 2,3, the Faraday rotation rate is proportional to
$f^{-2}$, while the Faraday conversion rate is proportional to $f^{-3}$ for
cold plasma.

In order for the Faraday conversion to happen, i.e. $\tilde{z}\lesssim 1$,
$f_{B}\gtrsim 2f\sin\alpha$ (Equation 4),

$B\gtrsim 1400G\left(\frac{f}{2\mathrm{GHz}}\right)\cos\alpha$ (5)

For a random pitch angle of $\alpha$, an extremely large magnetic field is
required for the Faraday conversion to happen. The phase difference between
two linear eigenmodes $\theta_{f}$ (Faraday conversion angle) is:

$\displaystyle\theta_{f}$
$\displaystyle=\int\rho_{L}dz\approx\frac{\pi}{c}\langle\frac{f_{p}^{2}f_{B}^{2}}{2f^{3}}\rangle
L$ (6) $\displaystyle\sim
10^{6}\langle\left(\frac{\Delta\mathrm{DM}}{0.1\mathrm{pc}\,\mathrm{cm}^{-3}}\right)\left(\frac{B}{1000\,G}\right)\rangle\left(\frac{f}{2\,\mathrm{GHz}}\right)^{-3}$
(7)

Given the observed DM excess of $\sim$ 0.1 pc cm-3, the conversion angle is
$\gg 1$ rad. Therefore, if $f_{B}\sim f$, the circular polarization will
oscillate fast against frequency, resulting in depolarized circular
polarization with our frequency resolution. Moreover, it will be fine-tuning
to require it to persist stable $180\degree$ conversion angle across
$40\degree$ of the orbital phase. Therefore, this scenario is inconsistent
with the observation (Figure 2).

To avoid the fast change of conversion angle, we require $f\gg f_{B}$. In this
case, the Faraday conversion will happen when the magnetic field is almost
perpendicular to the LOS, i.e. $\hat{B}_{z}=\cos\alpha\ll 1$ in Equation 5.
This would happen when there is magnetic reversal.

Consider a field reversal happening at $z=0$, we can Taylor expand
$\rho_{V}=\rho_{V}^{\prime}z$. Choose the $x$-axis to be the direction of
perpendicular magnetic field at the reversal, the angular frequency of the
rotation can be simplified as $\Omega=(\rho_{L},0,\rho_{V}^{\prime}z)$. Define

$\displaystyle\xi$
$\displaystyle\equiv\frac{\rho_{L}^{2}}{\rho_{V}^{\prime}}\approx\frac{\pi
L}{2c}\frac{f_{p}^{2}f_{B}^{3}}{f^{4}}$ (8)

where $L$ is the spatial scale where the magnetic field changes direction
(i.e. $|\Delta B_{z}|\sim|B|$). $\hat{B}_{x}$ is omitted in the expression
because $\hat{B}_{x}\sim 1$ near field reversal.

Near field reversal $\rho_{L}^{\prime}/\rho_{L}\ll\rho_{V}^{\prime}/\rho_{V}$,
so Equation 1 can be simplified as

$\frac{d\vec{\mathrm{P}}}{\xi
d\tilde{z}}=(1,0,\tilde{z})\times\vec{\mathrm{P}}$ (9)

Assume $\rho_{Q}$, $\rho_{V}^{\prime}$ can be considered as constants near the
field reversal for $-\tilde{z}_{0}<\tilde{z}<\tilde{z}_{0}$, where
$\tilde{z}_{0}\gg 1$. Then the conversion angle $\theta_{f}(\xi)$ depends only
on $\xi$ [12].

$\theta_{f}(\xi)=\arccos(2\exp(-\pi\xi/2)-1)$ (10)

With $\xi\ll 1$, $\theta_{f}\sim\sqrt{2\pi\xi}$, while when $\xi\gg 1$, the
circular polarization would flip sign.

As shown in Figure 2, the sign of circular polarization of the pulsar flux has
reversed near the orbital phase 0.25, suggesting $\xi\gg 1$ near the superior
conjunction where pulsar is behind the companion. The sign reversal can
persist for more than 10% of the orbital phase, corresponding to a spatial
scale of $L\sim 0.6R_{\odot}$ at the distance of the companion. This spatial
scale is of the same order as the size of the companion, therefore, we
attribute the sign reversal to the large scale structure of the companion. For
a companion with a poloidal field, e.g. a dipole field, any LOS passing
through the magnetosphere will experience a field reversal (Figure 3), usually
near the magnetic pole. We can estimate the required field strength from
Equation 8. The dispersion measure (DM) changes by $\sim
0.1\mathrm{pc}\,\mathrm{cm}^{-3}$ during the conversion, giving a mean density
of $n_{e}\approx N_{e}/a=5\times 10^{6}$ cm-3; where $a=0.85R_{\odot}$ is the
orbital separation. It corresponds to plasma frequency $f_{p}=0.02$ GHz. We
use the spatial scale $L$, where the conversion persists, as an approximate
for the scale where magnetic field changes direction. Therefore we require the
average $B\gtrsim 10$ G in the companion’s magnetosphere.

The nature of the 0.089 $M_{\odot}$ companion is unknown, however, whether it
is an M dwarf or white dwarf, it is common to have a large surface magnetic
field[37]. Moreover, a magnetic field of the order of 10 G at the contact
discontinuity is expected from the pressure equilibrium with the pulsar wind.
The companion has been seen to eclipse the pulsar for a whole orbit, requiring
a large pressure in the companion wind to withhold the pulsar wind. The
pressure in the pulsar wind at the distance of the companion can be estimated
from the spin down energy $P_{\mathrm{pw}}=\pi I\dot{P}/P^{3}a^{2}c$, where
I$\sim 10^{45}$ g cm2 is the pulsar’s moment of inertia, $\dot{P}$ is the
period derivative, and $c$ is the speed of light. The observed period
derivative of Ter5A is $\dot{P}=-1.5\times 10^{-20}$ s/s which is dominated by
the acceleration in the globular cluster. Given the location of the pulsar in
the Terzan 5 cluster, the maximum contribution of the $\dot{P}$ from the
cluster dynamic is $\approx 2\times 10^{-19}$ s s-1, and both the mean and
median value estimated from all the reasonable cluster models are greater than
half of this value [38]. Taking the cluster contribution to be $\dot{P}\sim
10^{-19}$ s s-1, the pressure of the pulsar wind at the distance companion is
$P_{\mathrm{pw}}\sim 2$ erg cm-3. To balance the pulsar wind with magnetic
pressure $P_{\mathrm{pw}}=B_{\mathrm{cp}}^{2}/8\pi$, a field strength of
$B_{\mathrm{cp}}\sim 7$ G in the contact discontinuity would be expected.
Therefore, it is reasonable to expect the magnetic field in companion
magnetosphere to be greater than this value, and hence satisfy the condition
for the Faraday conversion scenario detailed above.

##  0.4 Modeling the IV spectrum

As shown in Figure 2, the change of circular polarization at the top of the
band ($\sim 2.3$ GHz) roughly resembles the normal $V$ profile with an
opposite sign. However, in the lower half of the band ($\lesssim 2$ GHz), in
additional to the flip of the sign, the V profile is shifted systematically
towards the positive value. Selecting orbital phases far from the superior
conjunction, where stable RM has been measured, we can measure the normal
linear polarization $L$ of the pulsar, and hence can calculate the total
polarization $P=\sqrt{L^{2}+V^{2}}$ (shown in orange line in left panel d at
Figure 2). Around the spin phase 0.35, the circular polarization fraction near
superior conjunction exceeds the total polarization fraction in the normal
profile, and this trend becomes more obvious as the frequency decreases. At
1.7 GHz, the total polarization fraction $P/I$ at spin phase 0.35 is only 15%
in the normal profile; however, near the superior conjunction $V/I$ is around
50% at the same spin phase, suggesting $P\geq 50$%, i.e. the total
polarization fraction has changed. As shown in Equation 1, the total
polarization fraction $P$ is a conserved quantity in both Faraday rotation and
Faraday conversion. The change of total polarization fraction against
frequency, accompanied by the decrease of flux, is best explained by
circularly polarized attenuation. Therefore, the radiative transfer equation
(Equation 1) becomes

$\displaystyle\frac{d\vec{\mathrm{S}}}{ds}=-R\vec{\mathrm{S}}$ (11)
$\displaystyle R=\begin{pmatrix}\eta&0&0&\eta_{V}\\\ 0&\eta&\rho_{V}&0\\\
0&-\rho_{V}&\eta&\rho_{Q}\\\ \eta_{V}&0&-\rho_{Q}&\eta\end{pmatrix}$

where $\vec{\mathrm{S}}=(I,Q,U,V)$; $\eta$, $\eta_{V}$ are the isotropic, and
circular absorption. The circular absorption happening during the adiabatic
field reversal are largely cancelled. For the circular absorption happening
before/after the conversion, the integrated form will be:

$\displaystyle\begin{pmatrix}I^{\prime}\\\
V^{\prime}\end{pmatrix}=e^{-\tau}\begin{pmatrix}\cosh{\tau_{v}}&-\sinh{\tau_{v}}\\\
-\sinh{\tau_{v}}&\cosh{\tau_{v}}\end{pmatrix}\begin{pmatrix}I\\\
c\,V\end{pmatrix}$ (12)

where $\tau=\eta L$, $\tau_{v}=\eta_{v}L$ are the optical depth,
$c=\cos\theta_{f}$ is the conversion angle.

The optical depth should be frequency dependent, there we parameterize them as
$\tau=A(f/f_{0})^{-\alpha}$, $\tau_{v}=A_{v}(f/f_{0})^{-\alpha_{v}}$, where
$f_{0}=2$ GHz. Near the superior structure, the mode tracking is visually
complete at the top of the band, therefore we fix $c=-1$ for all of the
frequencies. Using the $I$, $V$ at the normal phase as template $I$, $V$, we
fit for the $I^{\prime},V^{\prime}$ near the superior structure by varying
$A,\alpha,A_{v},\alpha_{v}$. The best fit parameters are $A=1.08\pm
0.01,\alpha=-3.5\pm 0.1,A_{v}=-0.21\pm 0.01,\alpha_{v}=-3.7\pm 0.4$, where the
errorbars are 1 $\sigma$ error. The fitted curves are shown in Figure 2 panel
d right column as solid lines.

##  0.5 Synchrotron-cyclotron absorption

Given the large magnetic field required for the Faraday conversion, the
synchrotron-cyclotron absorption is able to account for the circularly
polarized absorption and varying absorption index at different orbital phase.

In the classic synchrotron limit, assume the relativistic electron following
an isotropic powerlaw distribution $n_{r}(\gamma)\propto\gamma^{-p}$ for
$\gamma_{\mathrm{min}}<\frac{3}{2}f_{B}\sin\alpha<\gamma_{\mathrm{max}}$, the
isotropic and circular absorption index $\eta,\eta_{v}$ can be estimated with
[39]:

$\displaystyle\eta=$
$\displaystyle\frac{0.08}{R_{\odot}^{-1}}\frac{n_{r}}{[\mathrm{cm}^{-3}]}\frac{(p-1)}{\gamma_{\mathrm{min}}^{1-p}}$
$\displaystyle\Gamma(\frac{3p+2}{12})\Gamma(\frac{3p+22}{12})\Big{(}\frac{3f_{B}\sin\alpha}{f}\Big{)}^{p/2+1}\frac{[\mathrm{GHz}]}{f}$
$\displaystyle\eta_{V}=$
$\displaystyle\frac{0.03}{R_{\odot}^{-1}}\frac{n_{r}}{[\mathrm{cm}^{-3}]}\frac{(p-1)}{\gamma_{\mathrm{min}}^{1-p}}$
(13)
$\displaystyle\frac{p+3}{p+1}\Gamma(\frac{3p+7}{12})\Gamma(\frac{3p+11}{12})\Big{(}\frac{3f_{B}\sin\alpha}{f}\Big{)}^{p/2+3/2}\frac{[\mathrm{GHz}]}{f}$

Therefore $\tau_{v}=\eta_{V}L\propto f^{-p/2-5/2}$ has steeper spectrum
compared to the isotropic optical depth $\tau=\eta L\propto f^{-p/2-2}$. The
ratio $\eta_{v}/\eta\approx 0.6\sqrt{f_{B}\sin\alpha/f}$.

In the methods, we have fitted $\tau=A(f/f_{0})^{-\alpha}$,
$\tau_{v}=A_{v}(f/f_{0})^{-\alpha_{v}}$ for the pulsar flux. Near the superior
conjunction, the fitted $\alpha,\alpha_{v}$ are consistent with $p=3\pm 0.2$.
The ratio $\tau_{v}/\tau=A_{v}/A\approx 0.2$ at 2 GHz requires
$B\sin\alpha\sim 100$ G. Assume $\gamma_{\mathrm{min}}=2$ and
$L=0.6R_{\odot}$, the required relativistic electron density for the eclipse
is $n_{r}=20$ cm-3 near superior conjunction. The required magnetic field
$B\sim 100$ G is consistent with the required $10\mathrm{G}\lesssim B<700$ G
from the observed Faraday conversion.

The relativistic electrons also inevitably introduce linear absorption
$\eta_{L}$ and Faraday conversion $\rho_{L}^{r}$, however, given $n_{r}\ll
n_{e}$ and $f_{B}<f$, these two terms will be much smaller than the Faraday
rotation and conversion introduced by cold plasma (Equation 2,3) and hence
their effect will not be visible.

##  0.6 Constraining the orbital inclination angle

The stable conversion indicates that the LOS is passing through the
magnetosphere of the companion, where a dipole field is expected. A wide-band
observation would provide constraints on the excess DM at different orbital
phase. In this way we could model the magnetic strength variation against
orbital phase, and by comparing with the dipole magnetic field, we would be
able to constrain the orbital inclination angle.

For a dipole field,

$|B|=|M|\frac{(1+3\cos^{2}\theta)^{1/2}}{r^{3}}$ (14)

Here $|M|=B_{0}r_{0}^{3}$ is the dipole moment, and $B_{0}$ is the surface
field at the equator, $r_{0}$ is the radius of the companion. Despite the weak
dependence (less than a factor of two) on polar angle $\theta$, the change of
$|B|$ mainly depends on $r^{-3}$.

For a binary orbit at an inclination angle $i$ (where $i=90\degree$ is edge
on), and an orbital phase $\Psi=2\pi(\Phi-0.25)$ (when $\Psi=0$ the pulsar is
behind the companion), the LOS has the shortest radial distance to companion
$r_{\mathrm{min}}=d\sqrt{1-\sin^{2}i\cos^{2}\Phi}$ (happening at a distance
$R$ from the pulsar $R=d\sin i\cos\Phi$), where d is orbital separation. When
the orbit is close to edge on $i\sim 90\degree$, $r_{\mathrm{min}}\sim
d\sin\phi$, the field strength changes fast against orbital phase $B\propto
r^{-3}\propto\sin^{-3}\Phi$, while for a face-on orbit $i\sim 0\degree$, the
field strength is independent of the orbital phase. Therefore, the relative
field strength against orbital phase mainly depends on the inclination angle
with mild dependence on the polar angle. Extended Data Figure 2 shows the
relative field strength against orbital phase at different inclination angle
when the magnetic pole aligned with the orbital axis. Comparing $i=85\degree$
to $i=75\degree$, the relative strength of $B_{0.35}/B_{0.25}$ reduced by an
order. Therefore, measuring the field strength against orbital phase will
provide a great constraint on the inclination angle. We defer the detailed
modeling to a future paper.

##  0.7 Observation prediction

In the scenario of Faraday conversion in the slow field reversal, for the
majority of the paths, the wave is going through Faraday rotation. We can
estimate the RM given the estimated magnetic field:

$\frac{<\Delta\mathrm{RM}>}{[\mathrm{rad}~{}m^{-2}]}\sim\frac{<\Delta\mathrm{DM}>}{[\mathrm{pc/cm}^{-3}]}\frac{B}{[\mu
G]}$ (15)

Given the $\Delta\mathrm{DM}$ of around 0.2 pc/cm3, the $\Delta\mathrm{RM}$
would be $10^{6}$ rad m-2 for 10-G magnetic field. The large RM would lead to
a delay $\tau_{F}$ between the two circular polarization, which may show up in
the stokes V profile:

$<\tau_{F}>=\frac{4<f_{B_{\parallel}}>}{f}<\tau_{p}>$ (16)

where $\tau_{p}$ is the dispersive delay, which is $\sim 0.2$ ms. The
$\tau_{F}$ would be $\sim 10\,\mu$s for 10-G field. For B strength of $10-100$
G, this may lead to a suppression of stokes in the phase profile where V
switches sign and have order 1 modulation in the timescale of $\sim
100\,\mu$s.

Observations of single pulses near superior conjunction will be the smoking
gun to verify the magnetic strength. Plasma lensing events have been seen in
Ter5A [14], where single pulses can occasionally be magnified significantly.
We will be able to measure the RM value of the single pulses given enough
frequency resolution. Moreover, the presence of a large magnetic field will
result in the left and right circular polarization of the pulse being
magnified at different frequency[15]. The lensing profile will be offset by a
cyclotron frequency $f_{c}=2.8\,\mathrm{MHz}(B/\mathrm{G})$, which is $\sim
30$ MHz for $B=10$ G and $\sim 0.3$ GHz for $B=100$ G, and hence should be
measurable.

Last but not least, the observation suggests that the parameter quantifying
Faraday conversion $\xi\gg 1$ at 2 GHz (Equation 8) and hence $|B|\gtrsim 10$
G near the superior conjunction. It also suggests $f_{B}\ll f$ for most of the
path, and hence $|B|\ll 700$ G. Since $\xi\propto f^{-4}$ has strong
dependence on frequency, the transition from full conversion to negligible
conversion will happen quickly. Given $B\ll 700$ G, and the conversion would
stops happening with $\xi\ll 1$, the transition from full conversion to
negligible conversion must happen between 2 GHz to 20 GHz (referred to as
transition frequency $f_{T}$). $\xi\propto f_{B}^{3}\propto B^{3}$ is very
sensitive to the magnetic field strength. At different orbital phase, the LOS
is passing through different part of the magnetosphere, therefore, we expect
the transition frequency $f_{T}$ to decrease as the pulsar moves away from the
superior conjunction. We have observed $f_{T}=2$ GHz at around orbital phase
0.35, where $V$ starts to change sign. At higher frequency we will be able to
observe $f_{T}$ closer to $\Phi=0.25$. Therefore, at higher frequency, there
will be a narrower window in the orbital phase where $V$ changes sign.

To demonstrate the model and the prediction, we configured a toy model with a
magnetized companion with a dipole moment of $2\times 10^{32}$
$\mathrm{G}\,\mathrm{cm}^{3}$ (300 G surface field at the equator, 0.12
$R_{\odot}$ companion radius). With orbital inclination angle $i=75^{\circ}$,
and an electron density estimated with $n_{e}\approx\Delta\mathrm{DM}/a\approx
5\times 10^{6}\mathrm{cm}^{-3}$, where $a$ is the orbital separation, we can
reproduce the Faraday conversion behavior of the pulsar against orbital phase
for 2.3 GHz and show the behavior of $V$ at higher frequency (Extended Data
Fig 1).

### Data availability

Data are available at request.

### Code availability

DSPSR (http://dspsr.sourceforge.net)

PSRCHIVE (http://psrchive.sourceforge.net)

AVB is supported by the European Research Council under the European Union’s
Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 617199
(‘ALERT’) and by Vici research programme ‘ARGO’ with project number
639.043.815, financed by the Dutch Research Council (NWO). YPY is supported by
NSFC grant No. 12003028 and the China Manned Spaced Project (CMS-
CSST-2021-B11). SMR is a CIFAR Fellow and is supported by the NSF Physics
Frontiers Center awards 1430284 and 2020265. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. The Green Bank
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc.

DZL modelled and interpreted the data, and prepared the majority of the
manuscript. AVB performed data pre-processing, calibration and intermediate
measurements. SMR led the data acquisition and discovered the variability of
circularly polarized emission. RM and YPY helped to revise the draft and
provided valuable comments.

The authors declare no competing financial interests.

Requests for materials should be addressed to D. Z. Li
<EMAIL_ADDRESS>

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Figure 1: Predicted $V$ change against orbital phase at higher frequency with
the model in Fig 3. Here we demonstrate the $V$ behavior for a single spin
phase, with red and blue representing different signs. Around the superior
conjunction, $\phi=0.25$, $V$ will experience Faraday conversion resulting in
the change of the sign. The window for Faraday conversion will become narrower
at higher frequency. The modeling of circularly polarized absorption has more
uncertainty due to the unknown distribution of mildly relativistic electron
against orbital phase. However, this absorption is observed to decrease fast
against frequency with $\tau_{v}\propto f^{-4}$, and become negligible at
$f=2.3$ GHz. Therefore it is safe to ignore it at higher frequency. Figure 2:
The peak magnetic field encountered for LOS at different orbital phases
compared to the one at orbital phase 0.25. The larger the inclination angle
$i$ is, the faster the field strength changes against orbital phases. Here we
assume the LOS has passed through a dipole companion magnetosphere with
magnetic axis aligned with the orbital axis. The result will only have minor
changes with other magnetic axis alignment.