# Bijections, generalizations, and other properties of sequentially congruent
partitions

Ezekiel Cochran  LeTourneau University
Longview, TX, 75602, U.S.A<EMAIL_ADDRESS>,  Madeline Locus Dawsey
Department of Mathematics
University of Texas at Tyler
Tyler, TX 75799, U.S.A<EMAIL_ADDRESS>,  Emma Harrell  Mount Holyoke
College
South Hadley, MA, 01075, U.S.A<EMAIL_ADDRESS>and  Samuel Saunders
University of Texas at Tyler
Tyler, TX 75799, U.S.A<EMAIL_ADDRESS>
###### Abstract.

Recently, Schneider and Schneider defined a new class of partitions called
sequentially congruent partitions, in which each part is congruent to the next
part modulo its index, and they proved two partition bijections involving
these partitions. We introduce a new partition notation specific to
sequentially congruent partitions which allows us to more easily study these
bijections and their compositions, and we reinterpret them in terms of Young
diagram transformations. We also define a generalization of sequentially
congruent partitions, and we provide several new partition bijections for
these generalized sequentially congruent partitions. Finally, we investigate a
question of Schneider–Schneider regarding how sequentially congruent
partitions fit into Andrews’ theory of partition ideals. We prove that the
maximal partition ideal of sequentially congruent partitions has infinite
order and is therefore not linked, and we identify its order 1 subideals.

###### Key words and phrases:

Partition, Young diagram, Partition bijection, Partition ideal

###### 2010 Mathematics Subject Classification:

05A17, 11P84

##  1\. Introduction

A partition $\lambda$ of a nonnegative integer $n$ is a sequence
$(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ of positive integers, called the
parts of $\lambda$, such that $\sum_{i=1}^{r}\lambda_{i}=n$ and
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{r}$. Sometimes it is useful
to refer to a partition $\lambda$ by its frequency notation $\lambda=\langle
1^{f_{1}},2^{f_{2}},3^{f_{3}},\dots\rangle$, where the frequency $f_{i}$ (also
called the multiplicity) is the number of times the part $i$ appears in
$\lambda$. Throughout this paper, frequency notation does not include parts
with frequency zero, unless we explicitly say otherwise. The size of a
partition $\lambda$ of $n$ is $|\lambda|=n$, and the length of $\lambda$,
denoted $\ell(\lambda)$, is the number of parts that comprise $\lambda$.

The Young diagram of a partition $\lambda$ is an array of $|\lambda|$ nodes
(or boxes) with $\ell(\lambda)$ rows, where the $i$th row has $\lambda_{i}$
nodes. The conjugate of a partition is obtained by reflecting the Young
diagram over its main (upper left to lower right) diagonal. For example, the
partition $(7,5,5,4,1)$ has the Young diagram below, and its conjugate is
shown to the right.

mathmode, boxsize=1em, centertableaux &

(a)

mathmode, boxsize=1em, centertableaux &

(b)

Figure 1. Young diagrams of $\lambda=(7,5,5,4,1)$ and its conjugate

Counting the number of partitions of a certain size and type is a question
that has intrigued mathematicians for centuries. A partition bijection is a
bijection between two sets of partitions, which implies that the two sets
contain the same number of partitions. The first famous partition bijection
was proved by Euler in the 1700s: the number of partitions of a natural number
$n$ into odd parts is equal to the number of partitions of $n$ into distinct
parts. In the late 19th and early 20th century, Rogers and Ramanujan
discovered the following two q-series identities, now known as the
Rogers–Ramanujan identities, which lead to two more sophisticated partition
bijections:

$\displaystyle\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})},$
$\displaystyle\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-2})(1-q^{5n-3})}.$

The first identity implies that the number of partitions of a natural number
$n$ in which adjacent parts differ by at least 2 equals the number of
partitions of $n$ into parts congruent to 1 or 4 modulo 5. The second identity
implies that the number of partitions of $n$ in which adjacent parts differ by
at least 2 and whose smallest part is at least 2 is the same as the number of
partitions of $n$ into parts congruent to 2 or 3 modulo 5. Partitions in which
adjacent parts differ by at least 2, as mentioned in the bijection implied by
the first identity, are sometimes called Rogers–Ramanujan partitions. By
rearranging the Young diagrams, it is straightforward to see that
Rogers–Ramanujan partitions are also in bijection with partitions with no
parts below the Durfee square, which is the largest square that fits in the
top left corner of a Young diagram. For example, the partition
$\lambda=(7,6,3,3,1)$ shown below has a $3\times 3$ Durfee square.

mathmode, boxsize=1em, centertableaux *(lightgray) & *(lightgray) *(lightgray)
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Figure 2. Durfee square of $\lambda=(7,6,3,3,1)$

A partition with no parts below its Durfee square can also be thought of as a
partition whose smallest part is greater than or equal to the length of the
partition.

Recently, Schneider and Schneider [12] defined a new type of partition.

######  Definition 1.1.

A _sequentially congruent partition_ $\lambda=(\lambda_{1},\dots,\lambda_{r})$
is a partition such that

  1. (1)

$\lambda_{i}\equiv\lambda_{i+1}\pmod{i}$ for all $1\leq i<r$, and

  2. (2)

$\lambda_{r}\equiv 0\pmod{r}$.

We denote the set of sequentially congruent partitions by $\mathcal{S}$. An
example of a sequentially congruent partition is $(21,16,14,8)$, since
$1|(21-16)$, $2|(16-14)$, $3|(14-8)$, and $4|8$. Note that the conjugate of a
sequentially congruent partition is of the form
$\left<1^{f_{1}},2^{f_{2}},3^{f_{3}},\dots\right>$, where $f_{i}\equiv
0\pmod{i}$. That is to say, the part $i$ occurs in the partition a multiple of
$i$ times. Schneider and Schneider refer to the conjugates of sequentially
congruent partitions as _frequency congruent partitions_ [12].

In the two papers so far on sequentially congruent partitions [12, 13],
several partition bijections between the set of sequentially congruent
partitions and other sets of partitions have been found, many of which use the
set of frequency congruent partitions as an intermediary. Some open questions
arise in these papers, such as a description of the composition of these
bijections, how to define analogous bijections with generalized sequentially
congruent partitions, and the connection, if it exists, between sequentially
congruent partitions and partition ideals as defined by Andrews [1] (for a
more concise treatment of partition ideals, see [2, Chapter 8]). We answer
these questions in this work.

First, we introduce some notation that will simplify our study of sequentially
congruent partitions. For a partition $\pi$, we consider $\pi_{i}=0$ for all
$i>\ell(\pi)$. We define an operation $\star$ such that for any two partitions
$\lambda$ and $\gamma$, we have that $\lambda\star\gamma:=\mu$, where
$\mu=(\mu_{1},\mu_{2},\dots)$ is the partition such that
$\mu_{i}=\lambda_{i}+\gamma_{i}$ for all $i\in\mathbb{N}$. In other words,
this operation “adds” two partitions, componentwise from the left. For
example, $(5,3,2,2)\star(3,2,1)=(8,5,3,2)$. Note that
$|\lambda\star\gamma|=|\lambda|+|\gamma|$ and
$\ell(\lambda\star\gamma)=\max\\{\ell(\lambda),\ell(\gamma)\\}$. We also
define $c(\lambda_{1},\dots,\lambda_{r})=(c\lambda_{1},\dots,c\lambda_{r})$,
so that
$c\lambda=\underbrace{\lambda\star\lambda\star\dots\star\lambda}_{\text{$c$
times}}$. Using $\star$, we observe a new, simple way to write a sequentially
congruent partition.

######  Theorem 1.2.

A partition $\lambda$ is sequentially congruent if and only if it can be
written uniquely in the form

$\displaystyle\lambda$ $\displaystyle=c_{1}(1)\star c_{2}(2,2)\star
c_{3}(3,3,3)\star\cdots$

for some nonnegative integers $c_{1},c_{2},c_{3},\dots$, where finitely many
$c_{i}$ are nonzero.

######  Remark 1.3.

A sequentially congruent partition $\lambda=c_{1}(1)\star c_{2}(2,2)\star
c_{3}(3,3,3)\star\cdots$ can be written in standard notation as
$\lambda=(c_{1}+2c_{2}+3c_{3}+\dots+rc_{r},2c_{2}+3c_{3}+\dots+rc_{r},\dots,rc_{r})$,
where $r$ is the largest integer so that $c_{r}$ is nonzero.

Theorem 1.2 leads to a new partition notation that is surprisingly natural for
sequentially congruent partitions and simplifies the results and proofs in
this paper. This notation, which we call _$c$ -notation_, will be discussed in
more detail in Section 2.

Schneider, Sellers, and Wagner [13] prove a bijection between sequentially
congruent partitions and partitions whose parts are squares. This bijection
also follows easily from Theorem 1.2.

######  Corollary 1.4.

The number of sequentially congruent partitions of size $n$ equals the number
of partitions of $n$ whose parts are perfect squares.

For a set $Q$ of partitions, let $p(Q,n)$ be the number of partitions in $Q$
of size $n$. It is known that partitions into squares, and thus sequentially
congruent partitions, have the generating function

(1)
$\sum_{n=0}^{\infty}p(\mathcal{S},n)q^{n}=\prod_{n=1}^{\infty}\frac{1}{1-q^{n^{2}}}.$

Sums of squares have been studied for centuries, and although (1) does not
have nice modular transformation properties like many other generating
functions related to partitions, asymptotics are known. For example, see [3,
7, 8, 9, 10, 11] for background and more details on partitions into squares.

In [12], Schneider and Schneider define the bijections $\pi$ and $\sigma$
between $\mathcal{P}$, the set of all partitions, and $\mathcal{S}$. The
bijection $\pi:\mathcal{P}\to\mathcal{S}$ maps partitions of size $n$ to
sequentially congruent partitions with largest part $n$. For a partition
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})\in\mathcal{P}$ of size
$n$, they define $\pi(\lambda)=\lambda^{\prime}$, where
$\lambda^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots,\lambda_{r}^{\prime})$
is the partition with

(2) $\lambda_{i}^{\prime}=i\lambda_{i}+\sum_{j=i+1}^{r}\lambda_{j}$

for all $1\leq i\leq r$. The bijection $\sigma:\mathcal{S}\to\mathcal{P}$ maps
sequentially congruent partitions with largest part $n$ to partitions of size
$n$. For a sequentially congruent partition
$\phi=(\phi_{1},\phi_{2},\dots,\phi_{r})$, they define

(3)
$\sigma(\phi)=\big{<}1^{\phi_{1}-\phi_{2}},2^{(\phi_{2}-\phi_{3})/2},3^{(\phi_{3}-\phi_{4})/3},\dots,r^{\phi_{r}/r}\big{>}.$

We now give examples of the maps $\pi$ and $\sigma$. For the partition
$(12,8,4,3,3)\in\mathcal{P}$ of size $30$, we have

$\displaystyle\pi((12,8,4,3,3))=(30,26,18,15,15)\in\mathcal{S}$ and
$\displaystyle\sigma((30,26,18,15,15))=\langle
1^{4},2^{4},3^{1},5^{3}\rangle\in\mathcal{P}.$

Schneider and Schneider find that the composition
$\sigma\circ\pi:\mathcal{P}\rightarrow\mathcal{P}$ is equivalent to
conjugation [12]. Based on their result, the following proposition becomes
apparent. While conjugation is traditionally defined in terms of Young
diagrams reflected about the main diagonal, the composition $\sigma\circ\pi$
gives an alternate definition of conjugation in terms of the parts of
partitions.

######  Proposition 1.5.

The conjugate of any partition
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ is the partition

$\big{\langle}1^{\lambda_{1}-\lambda_{2}},2^{\lambda_{2}-\lambda_{3}},\dots,(r-1)^{\lambda_{r-1}-\lambda_{r}},r^{\lambda_{r}}\big{\rangle}.$

######  Corollary 1.6.

A partition $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ is self-
conjugate if and only if

$\lambda_{i}=\begin{cases}r&\;\;\text{ when }\;i\leq\lambda_{r},\\\
r-1&\;\;\text{ when }\;\lambda_{r}<i\leq\lambda_{r-1},\\\ \vdots&\\\
2&\;\;\text{ when }\;\lambda_{3}<i\leq\lambda_{2},\\\ 1&\;\;\text{ when
}\;\lambda_{2}<i\leq\lambda_{1}.\end{cases}$

The proofs of Proposition 1.5 and Corollary 1.6 are immediate from the
definitions of $\pi$ and $\sigma$.

Schneider and Schneider also briefly investigate the composition
$\pi\circ\sigma$ [12] and observe that $\pi\circ\sigma$ is not equivalent to
conjugation. Instead, they suggest that this composition is some analogue of
conjugation specific to sequentially congruent partitions. We identify this
analogue in the following theorem.

######  Theorem 1.7.

Let $\phi=(\phi_{1},\phi_{2},\dots,\phi_{r})$ be a sequentially congruent
partition. Then we have that

$(\pi\circ\sigma)(\phi)=\left\langle\left(\sum_{j=1}^{1}\sum_{i=j}^{r}\frac{\phi_{i}-\phi_{i+1}}{i}\right)^{\phi_{1}-\phi_{2}},\left(\sum_{j=1}^{2}\sum_{i=j}^{r}\frac{\phi_{i}-\phi_{i+1}}{i}\right)^{\frac{\phi_{2}-\phi_{3}}{2}},\dots,\left(\sum_{j=1}^{r}\sum_{i=j}^{r}\frac{\phi_{i}-\phi_{i+1}}{i}\right)^{\frac{\phi_{r}}{r}}\right\rangle.$

Through the lens of Theorem 1.2 and $c$-notation, we will more clearly
describe $\pi\circ\sigma$ in Section 3.

In Section 2, we prove Theorem 1.2 and Corollary 1.4. In Section 3, we discuss
bijections between sequentially congruent partitions and other classes of
partitions, and we prove Theorem 1.7. In Section 4, we introduce
generalizations of sequentially congruent partitions and of the bijections
discussed in Section 3. In Section 5, we explore the question of
Schneider–Schneider regarding how sequentially congruent partitions fit into
Andrews’ theory of partition ideals.

##  2\. $c$-Notation

In this section, we prove Theorem 1.2 and then introduce notation that is
useful for working with sequentially congruent partitions. This new notation
will be used frequently throughout the paper.

###### Proof of Theorem 1.2.

Let $\lambda$ be a partition of the form

$\displaystyle\lambda$ $\displaystyle=c_{1}(1)\star c_{2}(2,2)\star
c_{3}(3,3,3)\star\dots\star c_{r}(\underbrace{r,\dots,r}_{r\text{ times}})$
$\displaystyle=(c_{1}+2c_{2}+\dots+rc_{r},2c_{2}+\dots+rc_{r},\dots,rc_{r}).$

It is clear that $\ell(\lambda)=r$ and that $rc_{r}\equiv 0\pmod{r}$. For any
$\lambda_{k}$ with $1\leq k<r$, we have that
$\lambda_{k}=kc_{k}+(k+1)c_{k+1}+\dots+rc_{r}$ and
$\lambda_{k+1}=(k+1)c_{k+1}+\dots+rc_{r}$, and hence
$\lambda_{k}-\lambda_{k+1}=kc_{k}$. Then
$\lambda_{k}\equiv\lambda_{k+1}\pmod{k}$. Therefore, $\lambda$ is a
sequentially congruent partition.

Now let $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ be a
sequentially congruent partition. Then $\lambda_{r}\equiv 0\pmod{r}$ and
$\lambda_{k}\equiv\lambda_{k+1}\pmod{k}$ for all $1\leq k<r$. Thus
$\lambda_{r}=rc_{r}$, $\lambda_{r-1}-\lambda_{r}=(r-1)c_{r-1}$, …,
$\lambda_{2}-\lambda_{3}=2c_{2}$, and $\lambda_{1}-\lambda_{2}=c_{1}$ for some
nonnegative integers $c_{1},\dots,c_{r}$. Then note that we can rewrite the
parts as follows:

$\displaystyle\lambda_{r-1}$ $\displaystyle=(r-1)c_{r-1}+\lambda_{r}$
$\displaystyle=(r-1)c_{r-1}+rc_{r}$ $\displaystyle\lambda_{r-2}$
$\displaystyle=(r-2)c_{r-2}+\lambda_{r-1}$
$\displaystyle=(r-2)c_{r-2}+(r-1)c_{r-1}+rc_{r}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\vdots$
$\displaystyle\lambda_{2}$
$\displaystyle=2c_{2}+3c_{3}+\dots+(r-1)c_{r-1}+rc_{r}$
$\displaystyle\lambda_{1}$
$\displaystyle=c_{1}+2c_{2}+3c_{3}+\dots+(r-1)c_{r-1}+rc_{r}.$

Therefore

$\displaystyle\lambda$
$\displaystyle=(c_{1}+2c_{2}+\dots+rc_{r},2c_{2}+\dots+rc_{r},\dots,rc_{r})$
$\displaystyle=c_{1}(1)\star c_{2}(2,2)\star c_{3}(3,3,3)\star\dots\star
c_{r}(\underbrace{r,\dots,r}_{r\text{ times}}).$

Thus any sequentially congruent partition can be written in the desired form.

Finally, suppose

$\displaystyle\lambda$ $\displaystyle=c_{1}(1)\star c_{2}(2,2)\star
c_{3}(3,3,3)\star\dots\star c_{r}(\underbrace{r,\dots,r}_{r\text{ times}})$
$\displaystyle=c_{1}^{\prime}(1)\star c_{2}^{\prime}(2,2)\star
c_{3}^{\prime}(3,3,3)\star\dots\star
c_{r}^{\prime}(\underbrace{r,\dots,r}_{r\text{ times}})$

are two representations of the same partition $\lambda$ in $c$-notation.
Observe that $\lambda_{r}=rc_{r}$ and $\lambda_{r}=rc_{r}^{\prime}$. Hence
$c_{r}=c_{r}^{\prime}$. Now, let $n\geq 1$ be an integer, and assume that
$c_{r-k}=c^{\prime}_{r-k}$ for all $0\leq k<n$. Then

$\displaystyle\lambda_{r-n}$
$\displaystyle=(r-n)c_{r-n}+(r-n+1)c_{r-n+1}+\dots+rc_{r}$
$\displaystyle=(r-n)c_{r-n}+(r-n+1)c_{r-n+1}^{\prime}+\dots+rc_{r}^{\prime}$
by the induction hypothesis, and $\displaystyle\lambda_{r-n}$
$\displaystyle=(r-n)c_{r-n}^{\prime}+(r-n+1)c_{r-n+1}^{\prime}+\dots+rc_{r}^{\prime}$

by assumption, so $c_{r-n}=c_{r-n}^{\prime}$. Thus, by induction, we have that
$c_{r-n}=c_{r-n}^{\prime}$ for all $0\leq n<r$, and the representation is
unique. ∎

######  Definition 2.1.

Since the representation in Theorem 1.2 is unique, we can denote any
sequentially congruent partition $\lambda=c_{1}(1)\star c_{2}(2,2)\star
c_{3}(3,3,3)\star\dots\star c_{r}(\underbrace{r,\dots,r}_{r\text{ times}})$
simply by $\lambda=[c_{1},c_{2},\dots,c_{r}]$. We refer to this notation as
$c$-notation, and we distinguish it from standard notation (which uses
parentheses) and frequency notation (which uses angle brackets) by using
square brackets.

For example, we write the partition $(8,6,4,4)$ in $c$-notation as

$(8,6,4,4)=2(1)\star 1(2,2)\star 0(3,3,3)\star 1(4,4,4,4)=[2,1,0,1].$

Note that a partition $\lambda=[c_{1},\dots,c_{r}]$ has length $r$ and size
$c_{1}+4c_{2}+\cdots+r^{2}c_{r}=\sum_{i=1}^{r}i^{2}c_{i}$. Furthermore,
$\lambda_{i}=\sum_{j=i}^{r}jc_{j}$ for each $1\leq i\leq r$. Then observe that
$\lambda_{i}-\lambda_{i+1}=ic_{i}$, and so
$c_{i}=(\lambda_{i}-\lambda_{i+1})/i$ for each $i$. Since we consider
$\lambda_{i}=0$ for all $i>\ell(\lambda)$, we have that $c_{r}=\lambda_{r}/r$
and $c_{i}=0$ for all $i>r$.

The Young diagram of a sequentially congruent partition is composed of squares
of varying sizes, as observed by Schneider–Sellers–Wagner [13]. This can be
seen in the example below, which is the Young diagram for the sequentially
congruent partition $(16,15,11,5,5)$.

mathmode, boxsize=1em, centertableaux

& *(lightgray) *(lightgray) *(lightgray) *(lightgray) *(lightgray)
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Figure 3. Young diagram of $(16,15,11,5,5)\in\mathcal{S}$

In fact, the number of $i\times i$ squares corresponds to the value of $c_{i}$
in the $c$-notation representation of the partition. For the above example,
note that $(16,15,11,5,5)=[1,2,2,0,1]$, and for each $1\leq i\leq 5$, the
$i\times i$ square occurs $c_{i}$ times. In general, the Young diagram of a
sequentially congruent partition looks like this:

$\cdots$$\cdots$$c_{5}$ times$\cdots$$c_{4}$ times$\cdots$$c_{3}$
times$\cdots$$c_{2}$ times$\cdots$$c_{1}$ times Figure 4. Young diagram of a
partition $[c_{1},c_{2},\dots]\in\mathcal{S}$

We now prove the bijection given in [13] between sequentially congruent
partitions and partitions whose parts are squares using $c$-notation.

###### Proof of Corollary 1.4.

Let $\mathcal{S}(n)$ denote the set of sequentially congruent partitions of
size $n$, and let $\mathcal{P}_{\mathbb{N}^{2}}(n)$ denote the set of
partitions of $n$ whose parts are perfect squares. We have from Theorem 1.2
that any partition $\phi\in\mathcal{S}$ can be uniquely written in
$c$-notation as $\phi=[c_{1},\dots,c_{r}]$. Additionally, any partition
$\lambda\in\mathcal{P}_{\mathbb{N}^{2}}$ can be uniquely written as
$\lambda=\big{\langle}{(1^{2})}^{d_{1}},{(2^{2})}^{d_{2}},\dots,{(r^{2})}^{d_{r}}\big{\rangle}$,
for some nonnegative integers $d_{1},\dots,d_{r}$. Because both of these
descriptions are unique, and each $c_{i}$ can take precisely the same values
as $d_{i}$, we have a bijection
$\psi:\mathcal{S}\to\mathcal{P}_{\mathbb{N}^{2}}$ defined by

$\psi\left([c_{1},c_{2},\dots,c_{r}]\right):=\left\langle{(1^{2})}^{c_{1}},{(2^{2})}^{c_{2}},\dots,{(r^{2})}^{c_{r}}\right\rangle.$

Recall that the size of the partition $\phi=[c_{1},\dots,c_{r}]\in\mathcal{S}$
is $\sum_{i=1}^{r}i^{2}c_{i}$. Furthermore, the size of the output of $\psi$
is $(1^{2})c_{1}+(2^{2})c_{2}+\dots+(r^{2})c_{r}=\sum_{i=1}^{r}i^{2}c_{i}$.
Therefore, $\psi$ is a size-preserving bijection. ∎

This bijection is easily understood through Young diagrams. We have shown that
sequentially congruent partitions have Young diagrams composed of squares. To
map a sequentially congruent partition to a partition whose parts are squares,
we simply transform each of the $i\times i$ squares into a row with $i^{2}$
boxes, as shown below.

mathmode, boxsize=.99em, centertableaux & *(lightgray) *(lightgray)
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*(lightgray) *(lightgray) *(lightgray)
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(a)

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*(lightgray) *(lightgray) *(lightgray) *(lightgray) *(lightgray) *(lightgray) *(lightgray)*(lightgray) *(lightgray)

*(lightgray)

*(lightgray)

(b)  Figure 5. The bijection $\psi:\mathcal{S}\to\mathcal{P}_{\mathbb{N}^{2}}$

$\hskip 7.11317pt\xmapsto{\;\;\psi\;\;}$

##  3\. Sequentially Congruent Partition Bijections

Let $\mathcal{P}(n)$ denote the set of partitions of $n$, and let
$\mathcal{S}_{\operatorname{lg}=n}$ denote the set of sequentially congruent
partitions with largest part $n$. Recall that Schneider and Schneider [12]
defined the bijections
$\pi:\mathcal{P}(n)\to\mathcal{S}_{\operatorname{lg}=n}$ and
$\sigma:\mathcal{S}_{\operatorname{lg}=n}\to\mathcal{P}(n)$ as in (2) and (3).
Here we rewrite $\pi$ and $\sigma$ using $c$-notation and describe how they
transform the Young diagrams of partitions.

######  Proposition 3.1.

Let $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ be any partition. We
have that $\pi(\lambda)=[c_{1},\dots,c_{r}]$, where
$c_{i}=\lambda_{i}-\lambda_{i+1}$ for all $1\leq i<r$ and $c_{r}=\lambda_{r}$.

###### Proof.

Let $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ be a partition. Then
we have that $\pi(\lambda)=\lambda^{\prime}$, where

$\displaystyle\lambda^{\prime}_{i}$
$\displaystyle=i\lambda_{i}+\sum_{j=i+1}^{r}\lambda_{j}$
$\displaystyle=i\lambda_{i}+\lambda_{i+1}+\dots+\lambda_{r-1}+\lambda_{r}.$

We rewrite each coefficient of 1 and simplify as follows.

$\displaystyle\lambda^{\prime}_{i}$
$\displaystyle=i\lambda_{i}+(-i+(i+1))\lambda_{i+1}+\dots+(-(r-2)+(r-1))\lambda_{r-1}+(-(r-1)+r)\lambda_{r}$
$\displaystyle=i\lambda_{i}-i\lambda_{i+1}+(i+1)\lambda_{i+1}-(i+1)\lambda_{i+2}+\dots+(r-1)\lambda_{r-1}-(r-1)\lambda_{r}+r\lambda_{r}$
$\displaystyle=i(\lambda_{i}-\lambda_{i+1})+(i+1)(\lambda_{i+1}-\lambda_{i+2})+\dots+(r-1)(\lambda_{r-1}-\lambda_{r})+r\lambda_{r}$
$\displaystyle=\sum_{j=i}^{r-1}j(\lambda_{j}-\lambda_{j+1})+r\lambda_{r}.$

Then we define $c_{i}=\lambda_{i}-\lambda_{i+1}$ for all $1\leq i<r$ and
$c_{r}=\lambda_{r}$, so that

$\displaystyle\lambda^{\prime}_{i}$
$\displaystyle=\sum_{j=i}^{r-1}jc_{j}+rc_{r}=\sum_{j=i}^{r}jc_{j}$

for all $1\leq i<r$ and $\lambda_{r}^{\prime}=rc_{r}$. Therefore, by Theorem
1.2, we have that $\lambda^{\prime}=[c_{1},\dots,c_{r}]$. ∎

To see why $\pi:\mathcal{P}(n)\to\mathcal{S}_{\operatorname{lg}=n}$ maps size
to largest part, we can think of its action on the Young diagram as
transforming each column into the top row of a square in the resulting
partition.

*(lightgray) & *(gray) *(lightgray) *(gray) *(lightgray) *(gray)
*(lightgray) *(gray) *(lightgray) *(gray)
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(a)

*(lightgray)&*(lightgray) *(lightgray) *(lightgray) *(gray)*(gray)*(gray)*(gray)*(lightgray) *(lightgray) *(lightgray) *(gray) *(gray)*(lightgray) *(gray)

(b)

$\hskip 7.11317pt\xmapsto{\;\;\pi\;\;}$

Alternatively, we can describe this action by “stretch”: the map $\pi$
transforms each $i\times 1$ column in the Young diagram into an $i\times i$
square.

& *(lightgray) *(lightgray) *(lightgray)
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(c)

& *(lightgray)*(lightgray)*(lightgray)*(lightgray) *(lightgray) *(lightgray)
*(lightgray)
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(d)

$\hskip 7.11317pt\xmapsto{\;\;\pi\;\;}$

We now turn to the map
$\sigma:\mathcal{S}_{\operatorname{lg}=n}\to\mathcal{P}(n)$. Since any
partition $\phi=(\phi_{1},\phi_{2},\dots,\phi_{r})\in\mathcal{S}$ can be
written in $c$-notation as $\phi=[c_{1},c_{2},\dots,c_{r}]$ with
$c_{i}=(\phi_{i}-\phi_{i+1})/i$ for all $1\leq i<r$ and $c_{r}=\phi_{r}/r$, we
immediately obtain the following description of $\sigma$ using $c$-notation.

######  Proposition 3.2.

Let $\phi=[c_{1},c_{2},\dots,c_{r}]$ be any sequentially congruent partition.
We have that $\sigma(\phi)=\left<1^{c_{1}},2^{c_{2}},\dots,r^{c_{r}}\right>$.

In the Young diagram, we can think of $\sigma$ as treating the top row of each
square as a part in the resulting partition.

*(lightgray) & *(lightgray) *(lightgray) *(gray) *(gray) *(gray) *(lightgray) *(lightgray) *(gray) *(lightgray) *(gray)

(e)

*(lightgray) & *(lightgray) *(lightgray)
*(gray)*(gray)*(gray)
*(lightgray) *(lightgray)
*(gray)
*(lightgray)
*(gray)

(f)

$\hskip 7.11317pt\xmapsto{\;\;\sigma\;\;}$

Alternatively, we can describe this action by “squish-flip”: each $i\times i$
square is reduced to an $i\times 1$ column (“squish”), and then we take the
conjugate (“flip”).

& *(lightgray) *(lightgray) *(lightgray) *(lightgray) *(lightgray)
*(lightgray) *(lightgray) *(lightgray)
*(lightgray) *(lightgray) *(lightgray)

(g)

&*(lightgray) *(lightgray) *(lightgray)
*(lightgray)
*(lightgray)

(h)

&
*(lightgray)*(lightgray)*(lightgray)

*(lightgray)

*(lightgray)

(i)

$\rightarrow{}$ $\rightarrow{}$

$\sigma$

These visual interpretations also give insight into a key difference between
$\pi$ and $\sigma$. The map $\pi$ treats the size of its input as the sum of
the lengths of the columns, whereas $\sigma$ treats the size of its output as
the sum of the lengths of the rows. This difference will be highlighted again
in the generalized bijections defined in Section 4.

Using Propositions 3.1 and 3.2, we can more easily describe $\pi\circ\sigma$,
and we use this description to prove Theorem 1.7.

###### Proof of Theorem 1.7.

Let $\phi=(\phi_{1},\phi_{2},\dots,\phi_{r})=[c_{1},\dots,c_{r}]$ be a
sequentially congruent partition. Then we have that

$\displaystyle\pi(\sigma(\phi))$
$\displaystyle=\pi(\sigma([c_{1},c_{2},\dots,c_{r}]))$
$\displaystyle=\pi(\left<1^{c_{1}},2^{c_{2}},\dots,r^{c_{r}}\right>)$
$\displaystyle=\pi((\underbrace{r,\dots,r}_{c_{r}\text{
times}},\underbrace{r-1,\dots,r-1}_{c_{r-1}\text{
times}},\dots,\underbrace{1,\dots,1}_{c_{1}\text{ times}}))$
$\displaystyle=[\underbrace{0,\dots,0,1}_{c_{r}\text{
terms}},\underbrace{0,\dots,0,1}_{c_{r-1}\text{
terms}},\dots,\underbrace{0,\dots,0,1}_{c_{1}\text{ terms}}].$

We think of this as moving $c_{i}$ spaces and then adding a one to the
$c$-notation representation of the partition, so if any of the $c_{i}$ are
zero, we simply increment the number in the previous location of an added one.
Recall that $c_{r}$ is nonzero by definition. We now rewrite
$(\pi\circ\sigma)(\phi)$ in frequency notation to obtain

$\displaystyle\pi(\sigma(\phi))$
$\displaystyle=\left<\left(\sum_{i=1}^{r}c_{i}\right)^{c_{1}},\left(\sum_{i=1}^{r}c_{i}+\sum_{i=2}^{r}c_{i}\right)^{c_{2}},\dots,\left(\sum_{i=1}^{r}c_{i}+\sum_{i=2}^{r}c_{i}+\dots+\sum_{i=r}^{r}c_{i}\right)^{c_{r}}\right>$
$\displaystyle=\left<\left(\sum_{j=1}^{1}\sum_{i=j}^{r}c_{i}\right)^{c_{1}},\left(\sum_{j=1}^{2}\sum_{i=j}^{r}c_{i}\right)^{c_{2}},\dots,\left(\sum_{j=1}^{r}\sum_{i=j}^{r}c_{i}\right)^{c_{r}}\right>.$

Substituting $c_{i}=(\phi_{i}-\phi_{i+1})/i$ for $1\leq i<r$ and
$c_{r}=\phi_{r}/r$ yields the desired result. ∎

It will often be more convenient to think of $\pi\circ\sigma$ in terms of the
Young diagram, where we simply perform the “squish-flip” of $\sigma$ followed
by the “stretch” of $\pi$. All together, $\pi\circ\sigma$ transforms the Young
diagram of a sequentially congruent partition by “squish-flip-stretch”. These
graphical transformations are defined well by our $c$-notation definitions of
$\pi$ and $\sigma$ in Propositions 3.1 and 3.2, and in Section 4 we will apply
these same definitions to partitions that are sequentially congruent in a more
general sense, with $c$-notation appropriately redefined, to transform the
Young diagrams in a similar way.

##  4\. Generalizations

In this section, we define a more general notion of sequentially congruent
partitions, and we describe several generalizations of the bijections $\pi$
and $\sigma$. We also describe several generalizations of a bijection of
Schneider, Sellers, and Wagner [13] between sequentially congruent partitions
and partitions into $k$th powers.

###  4.1. Generalizations of $\mathcal{S}$, $\pi$, and $\sigma$

In [12], Schneider and Schneider propose a more general set of frequency
congruent partitions based on a set
$B=\\{b_{1},b_{2},b_{3},\dots\\}\subseteq\mathbb{N}$ and a sequence of
positive integers $A=(a_{1},a_{2},a_{3},\dots)$. They define the set of
partitions with parts from $B$ where the $b_{i}$th part occurs a multiple of
$a_{i}$ times, denoted $\mathcal{F}_{B}(A)$. Partitions in
$\mathcal{F}_{B}(A)$ are of the form $\langle
b_{1}^{n_{1}a_{1}},b_{2}^{n_{2}a_{2}},b_{3}^{n_{3}a_{3}},\dots\rangle$, where
all $n_{i}\geq 0$, and finitely many $n_{i}$ are nonzero. They then define
$\mathcal{S}_{B}(A)$ as the set of conjugates of partitions in
$\mathcal{F}_{B}(A)$ and state that $\mathcal{S}_{B}(A)$ is a generalization
of $\mathcal{S}$, but they do not further describe the partitions in
$\mathcal{S}_{B}(A)$. We provide a description here, and then we define
generalizations of $\pi$ and $\sigma$ for these generalized sequentially
congruent partitions.

The Young diagram of a partition in $\mathcal{S}_{B}(A)$ consists of
rectangles of width $a_{i}$ and height $b_{i}$, each occurring $n_{i}$ times.
These rectangles are analogous to the squares within the Young diagrams of
partitions in $\mathcal{S}$. More explicitly, the Young diagram of a partition
in $\mathcal{S}_{B}(A)$ is of the following form.

$b_{1}$$a_{1}$$\cdots$$n_{1}$ times$b_{2}$$a_{2}$$\cdots$$n_{2}$
times$\cdots$$b_{r}$$a_{r}$$\cdots$$n_{r}$ times Figure 6. Young diagram of a
partition $[n_{1},\dots,n_{r}]\in\mathcal{S}_{B}(A)$

Applying a similar argument to that in the proof of Theorem 1.2, we can show
that any $\lambda\in S_{B}(A)$ can be uniquely written as

(4) $\displaystyle\lambda$
$\displaystyle=n_{1}(\underbrace{a_{1},\dots,a_{1}}_{b_{1}\text{ times}})\star
n_{2}(\underbrace{a_{2},\dots,a_{2}}_{b_{2}\text{ times}})\star
n_{3}(\underbrace{a_{3},\dots,a_{3}}_{b_{3}\text{ times}})\star\cdots$

with finitely many nonzero $n_{i}$. From this, we can define an analogue of
$c$-notation, which we call $n$-notation to distinguish between partitions
involving squares and partitions involving rectangles.

######  Definition 4.1.

For any partition $\lambda\in\mathcal{S}_{B}(A)$, we have that $\lambda$ is of
the form (4), and we write $\lambda=[n_{1},n_{2},\dots,n_{r}]_{A,B}$. We refer
to this notation as _$n$ -notation_.

For reference, we can write a general partition $\lambda\in\mathcal{S}_{B}(A)$
in standard notation as follows:

$\lambda=[n_{1},n_{2},\dots,n_{r}]_{A,B}=\left(\underbrace{\sum_{i=1}^{r}a_{i}n_{i},\dots,\sum_{i=1}^{r}a_{i}n_{i}}_{b_{1}\text{
times}},\underbrace{\sum_{i=2}^{r}a_{i}n_{i},\dots,\sum_{i=2}^{r}a_{i}n_{i}}_{b_{2}-b_{1}\text{
times}},\dots,\underbrace{a_{r}n_{r},\dots,a_{r}n_{r}}_{b_{r}-b_{r-1}\text{
times}}\right).$

Hence $\ell(\lambda)=b_{r}$ and $|\lambda|=\sum_{i=1}^{r}a_{i}b_{i}n_{i}$. We
then observe that $\lambda_{b_{i}}-\lambda_{b_{i}+1}=a_{i}n_{i}$ for each $i$,
and thus $n_{i}=(\lambda_{b_{i}}-\lambda_{b_{i}+1})/a_{i}$. Notice also that
$\lambda_{b_{i}}=\lambda_{b_{i}-1}=\cdots=\lambda_{b_{i-1}+1}$ for all $i>1$,
and we have that

  1. (1)

$\lambda_{b_{i}}\equiv\lambda_{b_{i}+1}\equiv\dots\equiv\lambda_{b_{i+1}}\pmod{a_{i}}$,
and

  2. (2)

$\lambda_{b_{r}}\equiv 0\pmod{a_{r}}.$

We can see that $\mathcal{S}_{B}(A)$ is a generalization of $\mathcal{S}$,
since $\mathcal{S}=\mathcal{S}_{\mathbb{N}}((1,2,3,\dots))$. Throughout this
section, if $B$ is omitted, then we consider $B=\mathbb{N}$, and if $A$ is
omitted, then we consider $A=(1,2,3,\dots)$. We also sometimes refer to $A$ as
a subset of $\mathbb{N}$ instead of a sequence of natural numbers when
appropriate.

Schneider and Schneider conjectured [12] that there are bijective maps between
$\mathcal{P}_{A}$, the set of partitions of $n$ with parts from $A$ (in this
context, $A\subseteq\mathbb{N}$), and $\mathcal{S}_{B}(A)$ that are analogous
to the bijections $\pi$ and $\sigma$ between $\mathcal{P}$ and $\mathcal{S}$.
Recall that $\pi:\mathcal{P}(n)\to\mathcal{S}_{\operatorname{lg}=n}$ and
$\sigma:\mathcal{S}_{\operatorname{lg}=n}\to\mathcal{P}(n)$. Furthermore,
recall that $\sigma\circ\pi$ is equivalent to conjugation, and
$\pi\circ\sigma$ gives an analogue of conjugation specific to sequentially
congruent partitions, as defined in Section 3. We define generalizations of
$\pi$ and $\sigma$ which relate size and largest part, and we also define
different generalizations whose compositions give conjugation and the “squish-
flip-stretch” analogue of conjugation for $\mathcal{S}_{B}(A)$. It turns out
that there is no generalization that simultaneously has both properties,
except in very special cases.

Let $\mathcal{S}_{B}(A)_{\operatorname{lg}=n}$ be the set of partitions in
$\mathcal{S}_{B}(A)$ with largest part $n$, and let $\mathcal{P}_{A}(n)$ be
the set of partitions in $\mathcal{P}_{A}$ of size $n$. If we wish to preserve
the exact relationship between size and largest part as in the original maps
$\pi$ and $\sigma$, we can define
$\sigma_{AB}:\mathcal{S}_{B}(A)_{\operatorname{lg}=n}\to\mathcal{P}_{A}(n)$ by

$\sigma_{AB}([n_{1},\dots,n_{r}]_{A,B}):=\left<a_{1}^{n_{1}},\dots,a_{r}^{n_{r}}\right>.$

In order for this map to be a bijection, however, the terms of $A$ must be
distinct, or else $\sigma_{AB}$ is not injective. If $A$ is not increasing,
then the terms in the frequency notation will not be in increasing order, but
by reordering the terms, the bijection holds. For a sequentially congruent
partition $\phi=[n_{1},\dots,n_{r}]_{A,B}$, we see that
$|\sigma_{AB}(\phi)|=\phi_{1}$, since $\phi_{1}=\sum_{i=1}^{r}a_{i}n_{i}$.

Unfortunately, the domain of the corresponding generalization of $\pi$ is
different from the range of $\sigma_{AB}$. Notice that $\sigma_{AB}$ maps into
partitions whose parts are from $A$, whereas the generalization of $\pi$
defined below maps from partitions whose columns in the Young diagram are
elements of $A$, i.e., the set of conjugates of partitions in
$\mathcal{P}_{A}$. This difference mirrors the key difference mentioned in
Section 3 between $\sigma$ and $\pi$. The set of conjugates of partitions in
$\mathcal{P}_{A}$ can be written $\mathcal{S}_{A}((1,1,1,\dots))$, and we let
$\mathcal{S}_{B}(A,n)$ denote the set of partitions in $\mathcal{S}_{B}(A)$ of
size $n$, for any $B\subseteq\mathbb{N}$ and any sequence $A$ of natural
numbers. We define
$\pi_{AB}:\mathcal{S}_{A}((1,1,1,\dots),n)\to\mathcal{S}_{B}(A)_{\operatorname{lg}=n}$
to transform the $a_{i}$ column in the Young diagram into a rectangle with
width $a_{i}$ and height $b_{i}$. Explicitly, we define the map $\pi_{AB}$ by

$\pi_{AB}((\lambda_{1},\dots,\lambda_{a_{r}})):=[n_{1},\dots,n_{r}]_{A,B},$

where $n_{i}=\lambda_{a_{i}}-\lambda_{a_{i}+1}$ for $1\leq i<r$ and
$n_{r}=\lambda_{a_{r}}$. This is also a bijection if and only if the terms of
$A$ are distinct, and again, we might have to reorder if $A$ is not
increasing.

As mentioned in Section 3, we can easily replicate the graphical
transformations of $\pi$ and $\sigma$ by simply replacing the $c$s in
Propositions 3.1 and 3.2 with $n$s as defined above. Unfortunately, in doing
so, the relationship between size and largest part is no longer preserved in
general. However, conjugation and the “squish-flip-stretch” analogue of
conjugation still hold under composition. Thus we define
$\pi^{\prime}_{AB}:\mathcal{P}\rightarrow{}\mathcal{S}_{B}(A)$ by

$\pi^{\prime}_{AB}((\lambda_{1},\dots,\lambda_{r})):=[n_{1},\dots,n_{r}]_{A,B},$

where $n_{i}=\lambda_{i}-\lambda_{i+1}$ for $1\leq i<r$ and
$n_{r}=\lambda_{r}$. In standard notation, we have

(5)
$\pi^{\prime}_{AB}((\lambda_{1},\dots,\lambda_{r}))=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots,\lambda_{b_{r}}^{\prime}),$

where
$\lambda_{i}^{\prime}=a_{i}\lambda_{i}+\sum_{j=i+1}^{r}(a_{j}-a_{j-1})\lambda_{j}$
for $b_{i-1}<i\leq b_{i}$, for all $1\leq i\leq r$, noting that we consider
$b_{0}=0$ for any $B\subseteq\mathbb{N}$. We define
$\sigma^{\prime}_{AB}:\mathcal{S}_{B}(A)\rightarrow{}\mathcal{P}$ by

$\sigma^{\prime}_{AB}([n_{1},\dots,n_{r}]_{A,B}):=\left<1^{n_{1}},2^{n_{2}},\dots,r^{n_{r}}\right>,$

so that for any partition
$\phi=(\phi_{1},\phi_{2},\dots,\phi_{r})\in\mathcal{S}_{B}(A)$, we have

(6)
$\sigma^{\prime}_{AB}((\phi_{1},\phi_{2},\dots,\phi_{r}))=\big{<}1^{(\phi_{b_{1}}-\phi_{b_{2}})/a_{1}},2^{(\phi_{b_{2}}-\phi_{b_{3}})/a_{2}},\dots,r^{\phi_{b_{r}}/a_{r}}\big{>}.$

By (5) and (6), it is clear that

$(\sigma^{\prime}_{AB}\circ\pi^{\prime}_{AB})(\lambda)=\big{\langle}1^{\lambda_{1}-\lambda_{2}},2^{\lambda_{2}-\lambda_{3}},\dots,(r-1)^{\lambda_{r-1}-\lambda_{r}},r^{\lambda_{r}}\big{\rangle},$

so this composition is still equivalent to conjugation by Proposition 1.5.
Additionally, $\sigma^{\prime}_{AB}$ and $\pi^{\prime}_{AB}$ act in a similar
manner (“squish-flip” and “stretch”, respectively) on the Young diagram of a
partition. Thus the composition $\pi^{\prime}_{AB}\circ\sigma^{\prime}_{AB}$
is still equivalent to “squish-flip-stretch,” but with $a_{i}\times b_{i}$
rectangles instead of $i\times i$ squares.

Although the relation between size and largest part is not usually preserved
by $\pi^{\prime}_{AB}$ and $\sigma^{\prime}_{AB}$, there is a special case in
which size is mapped to a multiple of the largest part and vice versa.

######  Proposition 4.2.

Suppose the sequence $A$ is of the form $A=(a,2a,3a,\dots)$. Then we have that
$\pi^{\prime}_{AB}:\mathcal{P}(n)\rightarrow{}\mathcal{S}_{B}(A)_{\operatorname{lg}=an}$
and
$\sigma^{\prime}_{AB}:\mathcal{S}_{B}(A)_{\operatorname{lg}=an}\rightarrow{}\mathcal{P}(n)$.

###### Proof.

Note that the largest part of $[n_{1},\dots,n_{r}]_{A,B}\in\mathcal{S}_{B}(A)$
is $\sum_{i=1}^{r}a_{i}n_{i}$ and the size of the output of
$\sigma^{\prime}_{AB}$ is $n_{1}+2n_{2}+\dots+rn_{r}=\sum_{i=1}^{r}in_{i}$.
Let $\lambda$ be any partition of $n$, and consider
$\pi^{\prime}_{AB}(\lambda)=[n_{1},\dots,n_{r}]_{A,B}$. We have that

$\displaystyle|\lambda|$
$\displaystyle=\lambda_{1}+\lambda_{2}+\dots+\lambda_{r}$
$\displaystyle=\lambda_{1}+(-\lambda_{2}+\lambda_{2})+\lambda_{2}+2(-\lambda_{3}+\lambda_{3})+\dots+(r-1)(-\lambda_{r}+\lambda_{r})+\lambda_{r}$
$\displaystyle=(\lambda_{1}-\lambda_{2})+2(\lambda_{2}-\lambda_{3})+3(\lambda_{3}-\lambda_{4})+\dots+r\lambda_{r}$
$\displaystyle=n_{1}+2n_{2}+\dots+rn_{r}$
$\displaystyle=\sum_{i=1}^{r}in_{i}.$

Thus $\pi^{\prime}_{AB}$ maps a partition of size $\sum_{i=1}^{r}in_{i}$ to a
partition with largest part $\sum_{i=1}^{r}a_{i}n_{i}$, and similarly
$\sigma^{\prime}_{AB}$ maps a partition with largest part
$\sum_{i=1}^{r}a_{i}n_{i}$ to a partition of size $\sum_{i=1}^{r}in_{i}$.

Now, suppose $f:\mathbb{N}\to\mathbb{N}$ is a map so that

$f\left(\sum_{i=1}^{\infty}in_{i}\right)=\sum_{i=1}^{\infty}a_{i}n_{i}$

for all partitions $[n_{1},n_{2},\dots]_{A,B}$. Then $f$ describes a
relationship between largest part and size. Since this equality holds for all
partitions, consider the partition $[k]$, where $n_{1}=k\in\mathbb{N}$ and
$n_{i}=0$ for all $i\geq 2$. Then we have that
$f(n_{1}+2n_{2}+\cdots)=a_{1}n_{1}+a_{2}n_{2}+\cdots$, so that $f(k)=a_{1}k$.
Similarly, if we consider the partition where $n_{j}=1$ for some
$j\in\mathbb{N}$, and all other $n_{i}=0$, we see that $f(j)=a_{j}$, meaning
that $a_{j}=a_{1}j$. Therefore, if the largest part of a partition
$\phi\in\mathcal{S}_{B}(A)$ is related to the size of
$\sigma^{\prime}_{AB}(\phi)$, or the size of a partition
$\lambda\in\mathcal{P}$ is related to the largest part of
$\pi^{\prime}_{AB}(\lambda)$, then this relationship must be scalar by a
factor of $a_{1}$, and the sequence $A$ must satisfy $a_{j}=a_{1}j$ for all
$j\in\mathbb{N}$.

Note that if $A$ satisfies $a_{j}=a_{1}j$ for all $j\in\mathbb{N}$, then

$\sum_{i=1}^{\infty}a_{i}n_{i}=\sum_{i=1}^{\infty}a_{1}in_{i}=a_{1}\sum_{i=1}^{\infty}in_{i}.$

Thus when $A$ is defined as above, the relationship holds. ∎

###  4.2. Partitions into $k$th powers

In [13], Schneider, Sellers, and Wagner defined $S(j,k)$ to be the set of
partitions $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ that satisfy

  1. (1)

$\lambda_{i}-\lambda_{i+1}=ji^{k}$ for $1\leq i\leq r-1$, and

  2. (2)

$\lambda_{r}=ji^{r}$.

They found that partitions in $S(j,k)$ of size $n$ are in bijection with
partitions of $n$ into $(k+1)$th powers where each part occurs exactly $j$
times. As a generalization, we define $S(k)$ to be the subset of partitions
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ that satisfy

  1. (1)

$\lambda_{i}\equiv\lambda_{i+1}\pmod{i^{k}}$, and

  2. (2)

$\lambda_{r}\equiv 0\pmod{r^{k}}$.

Note that the set of sequentially congruent partitions is the specialization
$S(1)$, and in fact all $S(k)$ are actually just
$S_{\mathbb{N}}(\mathbb{N}^{k})$ where
$\mathbb{N}^{k}=\\{1^{k},2^{k},3^{k},\dots\\}$.

We define two bijections,
$\sigma_{k}:\mathcal{S}_{B}(\mathbb{N}^{k})_{\operatorname{lg}=n}\to\mathcal{P}_{\mathbb{N}^{k}}(n)$
and
$\psi_{k}:\mathcal{S}(\mathbb{N}^{k},n)\to\mathcal{P}_{\mathbb{N}^{k+1}}(n)$,
by

$\sigma_{k}([n_{1},\dots,n_{r}]):=\big{<}{(1^{k})}^{n_{1}},{(2^{k})}^{n_{2}},\dots,{(r^{k})}^{n_{r}}\big{>}$

and

$\psi_{k}([n_{1},\dots,n_{r}]):=\big{<}{(1^{k+1})}^{n_{1}},{(2^{k+1})}^{n_{2}},\dots,{(r^{k+1})}^{n_{r}}\big{>}.$

Note that $\sigma_{1}$ is identical to the map $\sigma$ defined by
Schneider–Schneider in [12], and $\psi_{1}$ is identical to the bijection
between $\mathcal{S}(n)$ and $\mathcal{P}_{\mathbb{N}^{2}}(n)$ given by
Schneider–Sellers–Wagner in [13].

Notice that we can compose $\sigma_{k+1}$ and $\psi_{k}^{-1}$, meaning that
$\mathcal{S}_{B}(\mathbb{N}^{k+1})_{\operatorname{lg}=n}$ is in bijection with
$\mathcal{S}(\mathbb{N}^{k},n)$. This becomes clear when we consider the Young
diagrams of the partitions in each set. The rectangles that comprise the Young
diagrams of the partitions in
$\mathcal{S}_{B}(\mathbb{N}^{k+1})_{\operatorname{lg}=n}$ have height $i$ and
width $i^{k+1}$. Since the largest part comes from the sum of all the
rectangles’ widths, we simply transform the top row of each rectangle into a
rectangle of height $i$ and width $i^{k}$ to get the corresponding partition
in $\mathcal{S}(\mathbb{N}^{k},n)$. This gives the bijection
$\sigma_{k+1}\circ\psi_{k}^{-1}:\mathcal{S}_{B}(\mathbb{N}^{k+1})_{\operatorname{lg}=n}\to\mathcal{S}(\mathbb{N}^{k},n)$
defined by

$\sigma_{k+1}\circ\psi_{k}^{-1}\left([n_{1},\dots,n_{r}]_{\mathbb{N}^{k+1},B}\right)=[n_{1},\dots,n_{r}]_{\mathbb{N}^{k},\mathbb{N}}.$

As an extension of this idea, we can define
$\eta_{k,p}:\mathcal{S}_{B}(\mathbb{N}^{k})_{\operatorname{lg}=n}\to\mathcal{S}_{\mathbb{N}^{p}}(\mathbb{N}^{k-p},n)$
for $1\leq p\leq k$ by

$\eta_{k,p}\left([n_{1},\dots,n_{r}]_{\mathbb{N}^{k},B}\right):=[n_{1},\dots,n_{r}]_{\mathbb{N}^{k-p},\mathbb{N}^{p}}.$

The Young diagrams of partitions in
$\mathcal{S}_{B}(\mathbb{N}^{k})_{\operatorname{lg}=n}$ consist of rectangles
of width $i^{k}$, and so $\eta$ maps the top row of each rectangle to a
rectangle with area $i^{k}$. Therefore, $\eta$ maps partitions with largest
part $n$ to partitions of size $n$.

We can define a similar extension
$\tau_{k,p,q}:\mathcal{S}_{\mathbb{N}^{p}}(\mathbb{N}^{k-p},n)\to\mathcal{S}_{\mathbb{N}^{q}}(\mathbb{N}^{k-q},n)$
for integers $1\leq p,q\leq k$ by

$\tau_{k,p,q}([n_{1},\dots,n_{r}]_{\mathbb{N}^{k-p},\mathbb{N}^{p}}):=[n_{1},\dots,n_{r}]_{\mathbb{N}^{k-q},\mathbb{N}^{q}}.$

This $\tau$ maps rectangles of area $i^{k}$ to different rectangles with the
same area, thus preserving size.

##  5\. Partition Ideals

In this section, we change gears and investigate a different question of
Schneider and Schneider regarding partition ideals. Partition ideals were
defined by Andrews in the 1970s [1, 2] and have been studied more recently by
Andrews, Chern, and Li [4, 5, 6]. In 2019, Schneider and Schneider [12]
speculated that frequency congruent partitions, the conjugates of sequentially
congruent partitions, form a type of “quasi-ideal”. Although we do not expand
on the notion of “quasi-ideal” with regard to frequency congruent partitions,
this speculation motivated our investigation into the connection between
sequentially congruent partitions and partition ideals. First, we recall the
definition of a partition ideal. For the remainder of the paper, parts with
frequency zero are included in the frequency notation representation of all
partitions.

######  Definition 5.1.

A subset $\mathscr{I}$ of $\mathcal{P}$ is a partition ideal if for any
$\lambda\in\mathscr{I}$, removing any number of parts from $\lambda$ yields a
new partition $\gamma\in\mathscr{I}$.

In order to find a partition ideal composed of sequentially congruent
partitions, we define the sequence

$\mathcal{A}:=(1,2,6,\dots,\operatorname{lcm}(1,\dots,i),\dots)$

and consider $\mathcal{S}(\mathcal{A})$, the set of all partitions
$\lambda=(\lambda_{1},\lambda_{2},\dots)$ such that
$\lambda_{i}\equiv\lambda_{i+1}\pmod{\operatorname{lcm}(1,\dots,i)}$ for all
$i$. We will use the following lemma to prove that $\mathcal{S}(\mathcal{A})$
is the largest possible subset of sequentially congruent partitions which
forms a partition ideal.

######  Lemma 5.2.

A partition $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ is in
$\mathcal{S}(\mathcal{A})$ if and only if each part $\lambda_{i}$ is divisible
by every positive integer less than or equal to $i$.

###### Proof.

Let
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})\in\mathcal{S}(\mathcal{A})$.
We know that $\lambda_{r}\equiv 0\pmod{\operatorname{lcm}(1,\dots,r)}$ by
definition. We proceed by induction, assuming that
$\operatorname{lcm}(1,\dots,i)\mid\lambda_{i}$ for some $1<i\leq r$. Since
$\operatorname{lcm}(1,\dots,i-1)\mid\operatorname{lcm}(1\dots,i)$, we have
$\operatorname{lcm}(1,\dots,i-1)\mid\lambda_{i}$. Furthermore, since
$\lambda_{i-1}\equiv\lambda_{i}\pmod{\operatorname{lcm}(1,\dots,i-1)}$ by
definition, we also have $\operatorname{lcm}(1,\dots,i-1)\mid\lambda_{i-1}$.
Thus $\operatorname{lcm}(1,\dots,i)\mid\lambda_{i}$ for all $i$.

Now, let $\lambda$ be a partition so that
$\operatorname{lcm}(1,\dots,i)\mid\lambda_{i}$ for all $i$. Then for any $i$,
$\lambda_{i}-\lambda_{i+1}$ is also divisible by
$\operatorname{lcm}(1,\dots,i)$, and we have
$\lambda_{i}\equiv\lambda_{i+1}\pmod{\operatorname{lcm}(1,\dots,i)}$. Hence
$\lambda\in\mathcal{S}(\mathcal{A})$. ∎

######  Proposition 5.3.

The set $\mathcal{S}(\mathcal{A})$ is a maximal partition ideal of
sequentially congruent partitions.

###### Proof.

Let $\lambda\in\mathcal{S}(\mathcal{A})$. By Lemma 5.2, we have that
$\lambda_{i}\equiv 0\pmod{j}$ for all $j\leq i$. Thus, for all $k$ we have
$\lambda_{k}\equiv\lambda_{k+1}\equiv 0\pmod{k}$, and hence $\lambda$ is
sequentially congruent and $\mathcal{S}(\mathcal{A})\subseteq\mathcal{S}$.

Now consider removing any one part from $\lambda$. Each remaining part is
either going to stay at the same index or its index will decrease by one.
Since each part is divisible by every index less than or equal to its initial
index, each part will still be congruent to $0$ modulo its own index and every
smaller index. In other words, the resulting partition is still in
$\mathcal{S}(\mathcal{A})$.

Now we need to show that $\mathcal{S}(\mathcal{A})$ contains all possible
partition ideals of sequentially congruent partitions. Assume by way of
contradiction that there exists some $\lambda\notin\mathcal{S}(\mathcal{A})$
which is in a partition ideal of sequentially congruent partitions. Because
$\lambda\notin\mathcal{S}(\mathcal{A})$, there is at least one part
$\lambda_{i}$ such that there exists some index $k\leq i$ which does not
divide $\lambda_{i}$. Because $\lambda$ is in a partition ideal of
sequentially congruent partitions, we can remove all of the parts after
$\lambda_{i}$, and the resulting partition will still be in this partition
ideal. We can also remove the first $i-k$ parts of the partition, so that
$\lambda_{i}$ ends up at index $k$, and this new partition is still in the
partition ideal. However, we know that $k\nmid\lambda_{i}$, so
$\lambda_{i}\not\equiv 0\pmod{k}$, and therefore $\lambda$ is not a
sequentially congruent partition. ∎

In the literature, interest in partition ideals has typically been focused on
either order 1 partition ideals or linked partition ideals. These types of
ideals permit the application of some extremely useful generating function
results due to Andrews (see [2] for a summary of these results). We will
categorize all possible order 1 partition subideals of
$\mathcal{S}(\mathcal{A})$ and show that no subideals of
$\mathcal{S}(\mathcal{A})$ are linked. We will conclude this section with a
discussion of a new class of partition ideals which have infinite order and
their potential interest as mathematical objects. We now recall the definition
of the order of a partition ideal.

######  Definition 5.4.

A partition ideal $\mathscr{I}$ has _order_ $k$ if $k$ is the least positive
integer such that for all $\lambda=\langle
1^{f_{1}},2^{f_{2}},3^{f_{3}},\dots\rangle\notin\mathscr{I}$, there exists $m$
such that
$\lambda^{\prime}=\big{\langle}1^{f_{1}^{\prime}},2^{f_{2}^{\prime}},3^{f_{3}^{\prime}},\dots\big{\rangle}\notin\mathscr{I}$,
where

$f_{i}^{\prime}=\begin{dcases}f_{i}&\text{for }i=m,m+1,\dots,m+k-1,\\\
0&\text{otherwise}.\end{dcases}$

We can think of the order of a partition ideal as the maximum number of
consecutive integers’ frequencies we must consider to see that a given
partition is not an element of $\mathscr{I}$.

To determine the order of $\mathcal{S}(\mathcal{A})$, observe that
$\lambda=(m,1)\notin\mathcal{S}(\mathcal{A})$ while
$(m)\in\mathcal{S}(\mathcal{A})$ and $(1)\in\mathcal{S}(\mathcal{A})$ for all
$m$. Letting $m$ become arbitrarily large, we have to consider an arbitrarily
large number of frequencies of $\lambda$ to obtain
$\lambda^{\prime}\notin\mathcal{S}(\mathcal{A})$. Therefore
$\mathcal{S}(\mathcal{A})$ has no well defined order, and so we say that
$\mathcal{S}(\mathcal{A})$ has _infinite order_. We can, however, obtain order
1 subideals of $\mathcal{S}(\mathcal{A})$, to which we can apply Andrews’
results to get generating functions for each such subideal. Note that each
order 1 subideal $C\subseteq\mathcal{S}(\mathcal{A})$ is determined by a
single partition $\lambda\in C$, and all other partitions in $C$ can be
obtained by removing one or more parts from $\lambda$. Any such partition
ideal is necessarily finite and therefore of limited interest, since the
corresponding generating functions devolve into products of finite geometric
series. While $\mathcal{S}(\mathcal{A})$ has order 1 subideals, albeit finite
ones, we will show that there are no subideals of $\mathcal{S}(\mathcal{A})$
that are linked. Nevertheless, the discussion surrounding
$\mathcal{S}(\mathcal{A})$ and linked partition ideals is a somewhat more
interesting one.

We now recall some definitions and examples required to understand the concept
of a linked partition ideal. As in [1, 2, 6], we define the bijection
$\varphi$ by
$\varphi((\lambda_{1},\lambda_{2},\dots,\lambda_{r}))=(\lambda_{1}+1,\lambda_{2}+1,\dots,\lambda_{r}+1)$.
Next, for a partition ideal $\mathscr{I}$, define
$\mathscr{I}^{(m)}=\\{\lambda=\langle
1^{f_{1}},2^{f_{2}},\dots\rangle\in\mathscr{I}\mid
f_{1}=f_{2}=\dots=f_{m}=0\\}$.

######  Definition 5.5.

A partition ideal $\mathscr{I}$ has _modulus_ $m$ if $m$ is a positive integer
such that $\varphi^{m}\mathscr{I}=\mathscr{I}^{(m)}$.

Note that $\mathscr{I}$ can have more than one modulus. In fact, if
$\mathscr{I}$ has modulus $m$, then any multiple of $m$ is also a modulus of
$\mathscr{I}$. In order for a partition ideal to be linked, it must have a
modulus. We see immediately that $\mathcal{S}(\mathcal{A})$ has no modulus,
since for any finite $m$, there is a partition
$\lambda\in\mathcal{S}(\mathcal{A})$ of length $m+1$ such that adding $m$ to
the last part makes that part no longer a multiple of $m+1$. For example, let
$m=4$ and consider $\lambda=(60,60,60,60,60)\in\mathcal{S}(\mathcal{A})$. Note
that $(64,64,64,64,64)\notin\mathcal{S}(\mathcal{A})$ since $5\nmid 64$.
Additionally, any order 1 subideal of $\mathcal{S}(\mathcal{A})$ does not have
a modulus, since it has a largest allowable part. If $k$ is the largest
allowable part of an order 1 partition ideal
$I\subseteq\mathcal{S}(\mathcal{A})$, then for any $m$, note that $(k+m)\notin
I$, and thus $m$ is not a modulus for $I$.

We can define a subideal of $\mathcal{S}(\mathcal{A})$ with a modulus by
limiting the maximum length of the partitions in that subideal. We define
$C=\\{\lambda\in\mathcal{S}(\mathcal{A})\mid\ell(\lambda)\leq r\\}$ so that
$C$ is an infinite set. The order of $C$ is again infinite by the same
reasoning we used to show that $\mathcal{S}(\mathcal{A})$ has infinite order.
To see that $C$ has a modulus, note that $\varphi^{m}C$ maps surjectively into
$C^{(m)}$ when $m$ is a multiple of $\operatorname{lcm}(1,\dots,r)$, meaning
$C$ has modulus $m=\operatorname{lcm}(1,\dots,r)$. In fact, a subideal
$C\subseteq\mathcal{S}(\mathcal{A})$ has a modulus if and only if the maximum
length of the partitions in $C$ is bounded.

We now define a critically important set in any partition ideal with modulus
$m$, and then we state Lemma 8.9 from Andrews [2], which is required to define
linked partition ideals. Following the lemma, we give one more useful
definition from Chern and Li [6], and then we will be sufficiently equipped to
define linked partition ideals.

######  Definition 5.6.

For each partition ideal $\mathscr{I}$ with modulus $m$, define

$L_{\mathscr{I}}:=\big{\\{}\lambda=\big{\langle}1^{f_{1}},2^{f_{2}},\dots\big{\rangle}\in\mathscr{I}\mid
f_{i}=0\text{ for }i>m\big{\\}}.$

For two partitions $\lambda,\gamma\in\mathscr{I}$, define
$\lambda\oplus\gamma$ to be the partition of length
$\ell(\lambda)+\ell(\gamma)$ formed by combining all parts of the two
partitions in weakly decreasing order. Note that $\oplus$ is a different
additive operation than $\star$, which we defined in Section 1.

######  Lemma 5.7 (Andrews, Lemma 8.9).

Let $\mathscr{I}$ be a partition ideal with modulus $m$. For each
$\lambda\in\mathscr{I}$, we uniquely have

$\lambda=\pi_{1}\oplus\big{(}\varphi^{m}\pi_{2}\big{)}\oplus\big{(}\varphi^{2m}\pi_{3}\big{)}\oplus\dots,$

where $\pi_{i}\in L_{\mathscr{I}}$.

######  Definition 5.8.

For any partition $\lambda\in\mathcal{P}$, its _$m$ -tail_
$\operatorname{Tail}_{m}(\lambda)$ is the collection of parts of $\lambda$
which are at most $m$.

For example,

$\operatorname{Tail}_{2}((3,3,2,1,1,1))=(2,1,1,1).$

Finally, we define a linked partition ideal.

######  Definition 5.9.

A partition ideal $\mathscr{I}$ is a _linked partition ideal_ if

  1. (1)

$\mathscr{I}$ has a modulus, $m$;

  2. (2)

the set $L_{\mathscr{I}}$ corresponding to $m$ is finite;

  3. (3)

for each $\pi\in L_{\mathscr{I}}$ there corresponds a minimal subset
$\mathscr{L}_{\mathscr{I}}(\pi)\subseteq L_{\mathscr{I}}$ and a positive
integer $l(\pi)$ such that for any $\lambda\in\mathcal{P}$, we have that
$\lambda\in\mathscr{I}$ with $\operatorname{Tail}_{m}(\lambda)=\pi$ if and
only if we can find a partition $\tilde{\pi}$ with
$\operatorname{Tail}_{m}(\tilde{\pi})\in\mathscr{L}_{\mathscr{I}}(\pi)$ such
that

$\lambda=\pi\oplus\big{(}\varphi^{l(\pi)m}\tilde{\pi}\big{)}.$

The subset $\mathscr{L}_{\mathscr{I}}(\pi)$ is called the _linking set of
$\pi$_, and the integer $l(\pi)$ is called the _span of $\pi$_.

Two important observations are that the empty partition (which we will refer
to as $\pi_{0}$) is in the linking set of every element of $L_{\mathscr{I}}$,
and that the linking set of $\pi_{0}$ is
$\mathscr{L}_{\mathscr{I}}(\pi_{0})=L_{\mathscr{I}}$.

The following two examples can also be found in [6].

######  Example 5.10.

Let $\mathscr{D}$ denote the set of partitions into distinct parts. Note that
$\mathscr{D}$ is a partition ideal with modulus 1,
$L_{\mathscr{D}}=\\{\pi_{0},(1)\\}$, and

$\displaystyle\mathscr{L}_{\mathscr{D}}(\pi_{0})$
$\displaystyle=L_{\mathscr{D}},\;\;\;\;$ $\displaystyle l(\pi_{0})=1,$
$\displaystyle\mathscr{L}_{\mathscr{D}}((1))$
$\displaystyle=L_{\mathscr{D}},\;\;\;\;$ $\displaystyle l((1))=1.$

######  Example 5.11.

Consider $\mathscr{R}$, the set of Rogers–Ramanujan partitions, where adjacent
parts differ by at least two. Note that $\mathscr{R}$ is a partition ideal
with modulus 1 and that $L_{\mathscr{R}}=\\{\pi_{0},(1)\\}.$ Then we have

$\displaystyle\mathscr{L}_{\mathscr{R}}(\pi_{0})$
$\displaystyle=L_{\mathscr{R}},\;\;\;\;$ $\displaystyle l(\pi_{0})=1,$
$\displaystyle\mathscr{L}_{\mathscr{R}}((1))$
$\displaystyle=L_{\mathscr{R}},\;\;\;\;$ $\displaystyle l((1))=2.$

Alternatively, we may consider $\mathscr{R}$ to be a partition ideal with
modulus 2 and $L_{\mathscr{R}}=\\{\pi_{0},(1),(2)\\}.$ In this case we have

$\displaystyle\mathscr{L}_{\mathscr{R}}(\pi_{0})$
$\displaystyle=L_{\mathscr{R}},\;\;\;\;$ $\displaystyle l(\pi_{0})=1,$
$\displaystyle\mathscr{L}_{\mathscr{R}}((1))$
$\displaystyle=L_{\mathscr{R}},\;\;\;\;$ $\displaystyle l((1))=1,$ and
$\displaystyle\mathscr{L}_{\mathscr{R}}((2))$
$\displaystyle=\\{\pi_{0},(2)\\},\;\;\;\;$ $\displaystyle l((2))=1.$

We have shown that we can construct ideals
$C\subseteq\mathcal{S}(\mathcal{A})$ with modulus $m$ where
$m=\operatorname{lcm}(1,\dots,r)$ by choosing all elements of
$\mathcal{S}(\mathcal{A})$ whose length is no longer than $r$ so that
condition (1) from Definition 5.9 is satisfied. Observe that such a subset $C$
also satisfies condition (2), since the length of the partitions in $C$ must
be bounded in order for $C$ to have a modulus. However, we will show that
condition (3) does not hold for any such subset $C$.

######  Proposition 5.12.

Any subideal $C\subseteq\mathcal{S}(\mathcal{A})$ is not a linked partition
ideal.

###### Proof.

Let $C\subseteq\mathcal{S}(\mathcal{A})$ be a subideal, and suppose the length
of the partitions in $C$ is bounded by $r$ for some $r\geq 2$, so that C has
modulus $m=\operatorname{lcm}(1,\dots,r)$. Since
$2\leq\operatorname{lcm}(1,\dots,r)$, we have $(2)\in L_{C}$. Note that
$(\operatorname{lcm}(1,\dots,r)+2,2)=(m+2,2)\in C$, and thus we have
$(2)\in\mathscr{L}_{C}((2))$ and $l((2))=1$. Therefore $(2m+2,m+2,2)\in C$,
but $2\not\equiv 0\pmod{3}$, which contradicts the fact that $C$ is a subset
of $\mathcal{S}(\mathcal{A})\subseteq\mathcal{S}$. Thus no such subideal $C$
is linked. ∎

Since $\mathcal{S}(\mathcal{A})$ and its subideals are not linked, we cannot
apply the generating function results of Andrews to sequentially congruent
partitions. However, there has been no other research so far on partition
ideals of infinite order, so here we describe several more partition ideals of
infinite order and ask whether they are linked.

######  Example 5.13.

Let $\mathscr{R}^{\prime}$ denote the set of all partitions with no parts
below their Durfee square. A similar argument to the one for
$\mathcal{S}(\mathcal{A})$ above verifies that $\mathscr{R}^{\prime}$ is an
infinite order partition ideal. Additionally, $\mathscr{R}^{\prime}$ is not
linked by a similar argument to that in the proof of Proposition 5.12. By a
simple Young diagram transformation, we know that $\mathscr{R}^{\prime}$ is in
bijection with the set of Rogers–Ramanujan partitions $\mathscr{R}$ from
Example 5.11. It is straightforward to confirm that $\mathscr{R}$ has order 2,
which leads to the interesting observation that infinite order partition
ideals can be in bijection with finite order partition ideals. Furthermore,
note that
$\mathcal{S}(\mathcal{A})\subseteq\mathcal{S}\subseteq\mathscr{R}^{\prime}$,
since for any partition
$\phi=(\phi_{1},\phi_{2},\dots,\phi_{r})\in\mathcal{S}$, we have that
$\phi_{r}=kr$ for some $k\geq 1$. Thus $\phi_{r}\geq\ell(\phi)$ and therefore
$\phi\in\mathscr{R}^{\prime}$.

######  Example 5.14.

Let $\mathscr{A}$ be the set of all partitions
$(\lambda_{1},\dots,\lambda_{r})$ that satisfy
$\lambda_{r-i}-\lambda_{r-i+1}\geq i$ for all $1\leq i\leq r-1$. Consider
$\lambda=(k+1,k,1)$. Note that $\lambda_{1}-\lambda_{2}=1\not\geq 2$ and thus
$\lambda\notin\mathscr{A}$. Letting $k$ be arbitrarily large, we see that
$\mathscr{A}$ has no finite order. Notice that $\mathscr{A}$ has modulus 1,
and $L_{\mathscr{A}}=\\{\pi_{0},(1)\\}$. However, the subpartition $(1)$ does
not have a consistent span, since $(2,1)\in\mathscr{A}$ but
$(3,2,1)\notin\mathscr{A}$. Therefore $\mathscr{A}$ is not linked.

######  Example 5.15.

Let $\mathscr{N}$ be the set of all partitions with length at most $n$. This
is an infinite order partition ideal with modulus 1, and the set
$L_{\mathscr{N}}$ is finite with cardinality $n+1$. However, $\mathscr{N}$ is
not linked, since we can construct partitions with length greater than $n$
using the representation described in condition (3) of Definition 5.9.

######  Example 5.16.

Let $\mathscr{P}$ denote the set of all partitions whose parts have the same
parity. Notice that $\mathscr{P}$ is also an infinite order partition ideal
with modulus 1. This ideal is not linked because the set $L_{\mathscr{P}}$ is
not finite. These parity partitions can be generalized to sets of partitions
whose parts are all the same modulo $k$ for some $k\geq 2$. Each such set is a
partition ideal with infinite order which is not linked.

After finding that none of these infinite order partition ideals are linked,
we prove the following.

######  Theorem 5.17.

Any linked partition ideal has finite order.

###### Proof.

Suppose $C$ is a linked partition ideal and we have a partition $\lambda\notin
C$. Then there is no representation of $\lambda$ as in Lemma 5.7 that
satisfies the requirements of Definition 5.9. If
$\operatorname{Tail}_{m}(\lambda)\notin L_{C}$, then we choose

$\lambda^{\prime}=\operatorname{Tail}_{m}(\lambda)\oplus\varphi^{m}\pi_{0}\oplus\varphi^{2m}\pi_{0}\oplus\cdots\notin
C$

so that the frequencies of the first $m$ natural numbers in $\lambda^{\prime}$
match those of $\lambda$ and the rest are 0. If
$\operatorname{Tail}_{m}(\lambda)\in L_{C}$, then we have a representation of
the form

$\lambda=\pi_{1}\oplus\varphi^{m}\pi_{2}\oplus\cdots\oplus\varphi^{m(b-1)}\pi_{b},$

where each $\varphi^{m(i-1)}\pi_{i}$ satisfies the required linking set
properties up until $\varphi^{m(b-1)}\pi_{b}$. Notice that
$\varphi^{m(b-1)}\operatorname{Tail}_{m}\pi_{b}$ corresponds to $m$
frequencies of $\lambda$. If $\operatorname{Tail}_{m}\pi_{b}\notin L_{C}$,
then we choose

$\lambda^{\prime}=\pi_{0}\oplus\varphi^{m}\pi_{0}\oplus\cdots\oplus\varphi^{m(b-2)}\pi_{0}\oplus\varphi^{m(b-1)}\operatorname{Tail}_{m}\pi_{b}\oplus\varphi^{mb}\pi_{0}\oplus\cdots\notin
C,$

where again $m$ consecutive frequencies of $\lambda^{\prime}$ match those of
$\lambda$ and the rest are 0. Otherwise, it must be that $\pi_{b}$ is the
first $\pi_{i}$ where $\operatorname{Tail}_{m}(\pi_{b})\neq\pi_{0}$ after some
nonempty partition $\pi_{a}$. If
$\operatorname{Tail}_{m}\pi_{b}\in\mathscr{L}_{C}(\pi_{a})$, then

$\lambda=\pi_{1}\oplus\varphi^{m}\pi_{2}\oplus\cdots\oplus\varphi^{m(a-1)}\pi_{a}\oplus\underbrace{\varphi^{ma}\pi_{0}\oplus\cdots\oplus\varphi^{m(b-2)}\pi_{0}}_{<l(\pi_{a})-1\text{
times}}\oplus\varphi^{m(b-1)}\pi_{b},$

and we can choose

$\displaystyle\lambda^{\prime}$
$\displaystyle=\pi_{0}\oplus\cdots\oplus\varphi^{m(a-2)}\pi_{0}\oplus\varphi^{m(a-1)}\pi_{a}\oplus$
$\displaystyle\hskip
42.67912pt\underbrace{\varphi^{ma}\pi_{0}\oplus\cdots\oplus\varphi^{m(b-2)}\pi_{0}}_{<l(\pi_{a})-1\text{
times}}\oplus\varphi^{m(b-1)}\operatorname{Tail}_{m}\pi_{b}\oplus\varphi^{mb}\pi_{0}\oplus\cdots\notin
C,$

which corresponds to less than $(l(\pi_{a})+1)m$ consecutive frequencies
matching those of $\lambda$ and the rest equal to 0. Otherwise, we have that
$\operatorname{Tail}_{m}\pi_{b}\notin\mathscr{L}_{C}(\pi_{a})$, and we can
choose

$\displaystyle\lambda^{\prime}$
$\displaystyle=\pi_{0}\oplus\cdots\oplus\varphi^{m(a-2)}\pi_{0}\oplus\varphi^{m(a-1)}\pi_{a}\oplus$
$\displaystyle\hskip
42.67912pt\underbrace{\varphi^{ma}\pi_{0}\oplus\cdots\oplus\varphi^{m(b-2)}\pi_{0}}_{\leq
l(\pi_{a})-1\text{
times}}\oplus\varphi^{m(b-1)}\operatorname{Tail}_{m}\pi_{b}\oplus\varphi^{mb}\pi_{0}\oplus\cdots\notin
C,$

which corresponds to at most $(l(\pi_{a})+1)m$ consecutive frequencies
matching those of $\lambda$ and the rest 0. Since
$l(\pi_{a})\leq\max(\\{l(\pi_{i}):\pi_{i}\in L_{C}\\})$, we have that $C$ has
order at most $(\max\\{l(\pi_{i}):\pi_{i}\in L_{C}\\}+1)m$, and therefore $C$
has finite order. ∎

Notice that the partition ideals in Example 5.16, and the suggested
generalizations thereof, can be thought of as unions of order 1 partition
ideals whose intersections contain only the empty partition. For this reason,
finding generating functions for such partition ideals is simple, even without
Andrews’ generating function results. For example, the generating function for
$\mathscr{P}$ as defined in Example 5.16 is simply

$\sum_{n=0}^{\infty}p(\mathscr{P},n)q^{n}=\prod_{i=1}^{\infty}\frac{1}{1-q^{2n-1}}+\prod_{i=1}^{\infty}\frac{1}{1-q^{2n}}-1,$

where we subtract 1 to avoid double counting the empty partition. It does not
appear that Examples 5.13, 5.14, and 5.15 are similarly composed of order 1
partition ideals.

Tweaking the definition of the order of a partition ideal provides further
evidence that Examples 5.13, 5.14, 5.15, and 5.16 fall into different classes.
For example, suppose we say that a partition ideal $\mathscr{I}$ has “weak
order” $k$ if $k$ is the least positive integer such that for all
$\lambda=\Big{\langle}\lambda_{1}^{f_{\lambda_{1}}},\lambda_{2}^{f_{\lambda_{2}}},\lambda_{3}^{f_{\lambda_{3}}},\dots\Big{\rangle}\notin\mathscr{I}$,
there exists $m$ such that
$\lambda^{\prime}=\Big{\langle}\lambda_{1}^{f_{\lambda_{1}}^{\prime}},\lambda_{2}^{f_{\lambda_{2}}^{\prime}},\lambda_{3}^{f_{\lambda_{3}}^{\prime}},\dots\Big{\rangle}\notin\mathscr{I}$,
where

$f_{\lambda_{i}}^{\prime}=\begin{dcases}f_{\lambda_{i}}&\text{for
}i=m,m+1,\dots,m+k-1,\\\ 0&\text{otherwise}.\end{dcases}$

In other words, weak order omits parts whose frequencies are zero so that the
$k$ consecutive parts need not be consecutive positive integers. Under this
new definition, the ideals $\mathscr{R}^{\prime}$ and $\mathscr{A}$ from
Examples 5.13 and 5.14 still have infinite weak order, while $\mathscr{N}$ and
$\mathscr{P}$ from Examples 5.15 and 5.16 have weak order $n+1$ and $2$,
respectively.

To provide even further evidence of distinctions among these partition ideals,
we can tweak the definition of a linked partition ideal to allow for the span
of $\pi_{0}$ to be 1 before a nonempty partition appears in the representation
given in Lemma 5.7, and to change after such a partition appears in the
representation. If we call partitions satisfying this definition “quasi-
linked”, note that the ideals discussed in Examples 5.13, 5.14, and 5.15 are
not quasi-linked. Furthermore, we see that $\mathscr{P}$ from Example 5.16 and
its suggested generalizations are also not quasi-linked, as $L_{\mathscr{P}}$
is not finite. However, this issue can be remedied by taking
$\mathscr{P}^{\prime}\subseteq\mathscr{P}$ to be the set of all partitions in
$\mathscr{P}$ with distinct parts (or parts occurring no more than $q$ times
for some $q\in\mathbb{N}$). In this case, we do have that
$\mathscr{P}^{\prime}$ and its generalizations are quasi-linked, since
$l(\pi_{0})$ can be defined as 1 before a nonempty partition occurs in the
Lemma 5.7 representation of a partition in $\mathscr{P}^{\prime}$, and 2 (or
$k$ in the generalizations) after a nonempty partition occurs in the
representation.

These observations, paired with the difficulty of defining properties
intrinsic to infinite order partition ideals, have led us to think of infinite
order as the lack of known exploitable properties relating to partition
ideals, rather than a property worthy of research in its own right. However,
the various characteristics of the examples given above do suggest the
potential for further categorization of partition ideals. Such categorization
may yield new results, such as ways to find generating functions for partition
ideals that are unions of order 1 partition ideals, or opportunities to adjust
certain key definitions relevant to partition ideals so as to be more broadly
applicable.

## Acknowledgments

The authors were supported by a National Science Foundation grant
(DMS-2149921).

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