# Naruse hook formula for linear
extensions of mobile posets

GaYee Park  Department of Mathematcs and Statistic, University of
Massachusetts, Amherst, MA<EMAIL_ADDRESS>
###### Abstract.

Linear extensions of posets are important objects in enumerative and algebraic
combinatorics that are difficult to count in general. Families of posets like
Young diagrams of straight shapes and $d$-complete posets have hook-length
product formulas to count linear extensions, whereas families like Young
diagrams of skew shapes have determinant or positive sum formulas like the
Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant
formulas to count linear extensions of a family of posets called mobile posets
that refine $d$-complete posets and border strip skew shapes. We give a Naruse
type hook-length formula to count linear extensions of such posets by proving
a major index $q$-analogue. We also a inversion index $q$-analogue of the
Naruse formula for mobile tree posets.

##  1\. Introduction

###  1.1. Hook-length formulas for linear extensions

Linear extensions of posets are fundamental objects in combinatorics. In
general, computing the number $e(P)$ linear extensions of any poset is a
difficult problem, it is $\\#P$-complete [2]. For certain posets like Young
diagrams, rooted trees, and more generally $d$-complete posets, there are
product formulas that compute the number of linear extensions efficiently,
such as the classical _hook-length formula_ (HLF) for the number of standard
Young tableaux (SYT) of shape $\lambda$.

######  Theorem 1.1 (Frame-Robinson-Thrall [4]).

Let $\lambda$ be a partition of $n$. We have

(HLF) $|\mathrm{SYT}(\lambda)|=n!\prod_{u\in[\lambda]}\frac{1}{h(u)}$

where $h(u)=\lambda_{i}+\lambda_{j}^{\prime}-i-j+1$ is the hook-length of the
square $u=(i,j)$.

For skew shapes, there is no known product formula, however, Naruse introduced
a generalization of the hook-length formula as a positive sums over excited
diagrams of products of hook-lengths. We call this the _Naruse hook-length
formula_ (NHLF).

######  Theorem 1.2 (Naruse [11]).

For a skew shape $\lambda/\mu$ of size $n$, we have

(NHLF)
$|\mathrm{SYT}(\lambda/\mu)|\,=\,n!\sum_{D\in\mathcal{E}(\lambda/\mu)}\prod_{u\in[\lambda]\setminus
D}\frac{1}{h(u)},$

where $\mathcal{E}(\lambda/\mu)$ is the set of excited diagrams of
$\lambda/\mu$.

The number of SYT of shape $\lambda/\mu$ can also be interpreted as the number
of linear extension of a poset induced by the Young diagram of $\lambda/\mu$.
In [13], Proctor defined the family of _d-complete_ posets, that include Young
diagrams of shape $\lambda$ and rooted trees and have a hook-length formula to
count the number of linear extensions.

######  Theorem 1.3 (Peterson-Proctor [13]).

The number of linear extensions of a $d$-complete poset $P$ with $n$ element
is

$e(P)=\frac{n!}{\prod_{x\in P}h_{P}(x)},$

where $h_{P}(x)$ is the hook-length of $x\in P$.

###  1.2. $q$-analogue of hook-length formulas

There are the following $q$-analogues of both the HLF and the NHLF for
semistandard Young tableaux. We state these results in terms of
$e_{q}^{\mathrm{maj}}(P,\omega):=\sum_{\sigma}q^{\mathrm{maj}(\sigma)}$ where
$(P,\omega)$ is a labeled poset, and $\sigma$ is a linear extension of it.
This polynomial also encodes the generating functions of $P$-partitions [16].

######  Theorem 1.4 (Stanley [15]).

For a shape $\lambda$ with associated poset $Q_{\lambda}$ of size $n$ with
$\omega$ Schur labeling, we have:

(1.1)
$\frac{e_{q}^{\mathrm{maj}}(Q_{\lambda},\omega)}{\prod^{n}_{i=1}(1-q^{i})}=q^{b(\lambda)}\prod_{u\in[\lambda]}\frac{1}{1-q^{h(u)}},$

where $b(\lambda)=\sum_{i}(i-1)\lambda_{i}$.

######  Theorem 1.5 (Morales-Pak-Panova [10]).

For a skew shape $\lambda/\mu$ with associated poset $Q_{\lambda/\mu}$ of size
$n$ with $\omega$ Schur labeling, we have:

(1.2)
$\frac{e_{q}^{\mathrm{maj}}(Q_{\lambda/\mu},\omega)}{\prod^{n}_{i=1}(1-q^{i})}=\sum_{D\in\mathcal{E}(\lambda/\mu)}q^{w(D)}\prod_{u\in[\lambda]\setminus
D}\frac{1}{1-q^{h(u)}},$

where $w(D)=\sum_{u\in Br(D)}h(u)$ is the sum of hook-lengths of the support
of broken diagonals.

For $d$-complete posets, we also have the following $q$-analogue in terms of
major index (see Section 2.5).

######  Theorem 1.6 (Peterson and Proctor [13]).

For a labeled $d$-complete poset $(P,\omega)$ of size $n$ with $\omega$ any
labeling, we have:

$e_{q}^{\mathrm{maj}}(P,\omega)=q^{\mathrm{maj}(P,\omega)}\frac{[n]_{q}!}{\prod_{x\in
P}[h_{P}(x)]_{q}},$

where $h_{P}(x)$ is the hook-length of $x\in P$ and
$\mathrm{maj}(P,\omega)=\sum_{x\in\mathrm{Des}(P,\omega)}h_{P}(x)$.

###  1.3. Hook formulas for mobile posets

A _border strips_ is a connected skew-shaped diagram with no $2\times 2$ box.
A _mobile poset_ is a recent common refinement of border strips and
$d$-complete posets introduced in [5] (see Figure 1, (a)). The authors found a
determinantal formula for the number of linear extensions of these posets,
similar to _Jacobi–Trudi_ formula and asked whether there was a Naruse-type
formula [5, Sec. 6.1] for this number. The first main result of this paper is
a $q$-analogue Naruse hook-length formula for mobile posets for the major
index, generalizing Theorem 1.5.

######  Theorem 1.7.

Let $P_{\lambda/\mu}(\mathbf{p})$ be a free-standing mobile poset of size $n$
with with underlying border strip $\lambda/\mu$ and
$\mathbf{p}=(p_{(r_{1},s_{1})},\dots,p_{(r_{k},s_{k})})$ the $d$-complete
posets hanging on $(r,s)$. For a labeled mobile poset
$(P_{\lambda/\mu}(\mathbf{p}),\omega)$ with $\omega$ reversed Schur labeling
on $[\lambda/\mu]$ and natural labeling on $d$-complete posets, we have:

$\frac{e_{q}^{\mathrm{maj}}(P_{\lambda/\mu},\omega)}{\prod^{n}_{i=1}(1-q^{i})}=\prod_{v\in\mathbf{p}}\frac{1}{1-q^{h(v)}}\sum_{D\in\mathcal{E}(\lambda/\mu)}q^{w^{\prime}(D)}\prod_{u\in[\lambda]\setminus
D}\frac{1}{1-q^{h^{\prime}(u)}},$

where $h(v)$ is the hook-length of the element $v$ in the $d$-complete posets
in $\mathbf{p}$,
$h^{\prime}(i,j)=\lambda_{i}-i+\lambda^{\prime}_{j}-j+1+\sum_{r\geq i,s\geq
j}|p_{({r,s})}|$ and $w^{\prime}(D)=\sum_{u\in\mathrm{Br}(D)}h^{\prime}(u)$ is
the sum of hook-lengths of the supports of broken diagonals.

This result is a common refinement of Theorem 1.5 and Theorem 1.6. Note that
Naruse-Okada [12] have a different $q$-analogue of
$e_{q}^{\mathrm{maj}}(P,\omega)$ for a family called _skew $d$-complete_
posets with _natural labelings_. See Section 7.4.

The proof of Theorem 1.7 is based on the method used in [8] to prove (NHLF)
for border strips. This involves the _Pieri–Chevalley formula_ (2.2) and a
recurrence of linear extensions for mobile posets. The proof of the latter is
combinatorial and uses a generalization of Stanley’s theory of
$(P,\omega)$-partitions [16].

By taking $q=1$, we obtain a Naruse hook-length formula for mobile posets as a
corollary.

######  Corollary 1.8 (NHLF for mobiles).

For a free-standing mobile poset $P_{\lambda/\mu}(\mathbf{p})$ of size $n$, we
have:

(1.3) $e(P_{\lambda/\mu}({\bf p}))=\frac{n!}{H({\bf
p})}\sum_{D\in\mathcal{E}(\lambda/\mu)}\prod_{(i,j)\in\gamma}\frac{1}{h^{\prime}(i,j)},$

where $H(\mathbf{p})$ is the product of hook-lengths of all elements in the
$d$-complete posets hanging from $\lambda/\mu$.

As an application of this corollary, we give bounds to generalizations of
Euler number defined in [5]. See Corollary 5.2 and Corollary 5.3.

Our second result is a $q$-analogue of NHLF for mobile tree posets (the
hanging $d$-complete posets are restricted to rooted trees) in terms of the
inversion statistic,
$e_{q}^{\mathrm{inv}}(P,\omega):=\sum_{\sigma}q^{\mathrm{inv}(\sigma)}$, where
$\sigma$ is a linear extension of $(P,\omega)$.

######  Theorem 1.9.

For a labeled mobile poset $(P_{\lambda/\mu},\omega)$ with $\omega$ reversed
Schur labeling on $[\lambda/\mu]$ and natural labeling on $d$-complete posets,

(1.4)
$\frac{e_{q}^{\mathrm{inv}}(P_{\lambda/\mu},\omega)}{\prod^{n}_{i=1}(1-q^{i})}=\prod_{v\in\mathbf{p}}\frac{1}{1-q^{h(v)}}\sum_{D\in\mathcal{E}(\lambda/\mu)}q^{w(D)+p_{D}}\prod_{u\in[\lambda]\setminus
D}\frac{1}{1-q^{h^{\prime}(u)}},$

where $w(D)=\sum_{u\in Br(D)}h(u)$ is the sum of hook-lengths of the supports
of broken diagonals and $p_{D}=\sum_{(i,j)\in[\mu]\setminus
D}\sum_{b=j}p_{r,s}$.

###  1.4. Paper outline

In Section 2, we give definitions and background results required for the
proof. In Section 3, we give results for $P$-partition with a fixed point. In
Section 4, we give an example and the proof of Theorem 1.7. In Section 5, we
show an application to the Corollary 1.8. In Section 6, we give examples and
the proof of the inversion index of the case of $q$-analogue. Lastly, we end
with final remarks in Section 7.

##  2\. Background and Preliminaries

###  2.1. Posets and linear extensions

A _partially-ordered set_ (poset) is a pair $(P,\leq_{P})$ where $P$ is a
finite set and $\leq_{P}$ is a binary relation that is reflexive, anti-
symmetric, and transitive. A _linear extension_ of an $n$-element poset $P$ is
a bijection $f:P\to[n]$ that is order-preserving. The number of SYT of a shape
$\lambda/\mu$ is equal to the number of _linear extensions_ of a poset of
shape $\lambda/\mu$. We denote the set of linear extensions of $P$ as
$\mathcal{L}(P)$, and $e(P)=|\mathcal{L}(P)|$.

###  2.2. Border strips and Mobile Posets

A _border strip_ is a connected skew shape $\lambda/\mu$ containing no
$2\times 2$ box. _d-complete posets_ are a large class of posets containing
rooted tree posets and posets arising from Young diagrams. Given a border
strip, we can convert it into a poset by letting the inner corners of the
diagram be the maximal points of the corresponding poset. Now we can construct
a mobile poset (See Figure 1 for an example).

######  Definition 2.1 (Garver-Grosser-Matherne-Morales [5]).

A _mobile 111What we call a mobile poset is called a free-standing mobile
poset in [5]. (tree) poset_ is a poset obtained from a border strip $Q$, by
allowing every element $x\in Q$ to cover the maximal element of a nonnegative
number of disjoint $d$-complete (rooted tree) posets.

Figure 1. The conversion of a mobile poset to a young diagram with
$d$-complete posets attached.

###  2.3. Excited diagrams and broken diagonals

Denote $[\lambda/\mu]$ as the skew shape Young diagram of a shape
$\lambda/\mu$. An _excited diagram_ of $\lambda/\mu$, denoted by $D$, is a
subset of $[\lambda]$ obtained from $\mu$ by applying a sequence of _excited
moves_ that we define next. Let $D\in\mathcal{E}(\lambda/\mu)$, then $(i,j)\in
D$ is an _active cell_ if $(i+1,j),(i,j+1)$, and $(i+1,j+1)$ are not in $D$.
We obtain a new excited diagram by replacing an active cell by $(i+1,i+j)$
(see Figure 1, (b)). Note that for border strips, the excited diagrams can
also be interpreted as the complement of its lattice paths $\gamma$ from
$(\lambda_{1}^{\prime},1)\to(1,\lambda_{1})$ that stay inside $[\lambda]$.
(see[8, Sec. 3]).

For each excited diagram $D\in\mathcal{E}(\lambda/\mu)$ we associate a set of
_broken diagonals_ $Br(D)\subset[\lambda]\setminus D$ as follows. Start with
$D=[\mu]$, then $\mathrm{Br}(D)=\\{(i,j)\in\lambda/\mu|i-j=\mu_{t}-t\\}$,
where $\mu_{t}=0$ if $\ell(\mu)<t\leq\ell(\lambda)$. For each active cell
$u=(i,j)$ and its excited move $\alpha_{u}:D\to D^{\prime}$, we have a
corresponding move for the broken diagonal where
$\mathrm{Br}(D^{\prime})=\mathrm{Br}(D)\setminus\\{(i+1,j+1)\\}\cup\\{(i+1,j)\\}\\}$.
See Figure 2, (a) and (b).

(a)

(b)

Figure 2. (a) Excited move (b) Excited diagrams and the corresponding broken
diagonals of $\lambda/\mu=(2,2,2,1)/(1,1)$

###  2.4. Multivariate function

For border strip $\lambda/\mu$, let

(2.1)
${}F_{\lambda/\mu}(\mathbf{x},\mathbf{y})=F_{\lambda/\mu}(x_{1},\dots,x_{\lambda_{1}^{\prime}},y_{1},\dots,y_{\lambda_{1}}):=\sum_{D\in\mathcal{E}(\lambda/\mu)}\prod_{(i,j)\in[\lambda]\setminus
D}\frac{1}{x_{i}-y_{j}}.$

Let $\lambda/\nu$ be the shape obtained by removing an inner corner of
$\lambda/\mu$. We denote this subtraction by $\mu\to\nu$. Note that
$[\lambda/\nu]$ is a disconnected skew shaped.

We need the following identity of $F_{\lambda/\mu}(\mathbf{x},\mathbf{y})$
from [8].

######  Lemma 2.2 (Pieri–Chevalley formula [8, Eq. (6.3)]).

(2.2) $\displaystyle F_{\lambda/\mu}(\mathbf{x},\mathbf{y})$
$\displaystyle=\frac{1}{x_{1}-y_{1}}\sum_{\mu\to\nu}F_{\lambda/\nu^{1}}({\bf
x},{\bf y})F_{\lambda/\nu^{2}}({\bf x},{\bf y}),$

where $\lambda/\nu^{1}$ and $\lambda/\nu^{2}$ are the two connected border
strips that form $\lambda/\nu$.

###  2.5. $q$-analogues of linear extensions

A _labeled poset_ $(P,\omega)$ is a poset $P$ with a labeling
$\omega:P\to[n]$. We call $\omega$ a _natural labeling_ if for any $x,y\in P$
with $x<_{P}y$, we have $\omega(x)<\omega(y)$ [16]. We call $\omega$ a
_reversed Schur labeling_ if the labeling decreases as it follows the path
from the bottom left to the top right of the tableau. For Theorem 1.9 and
Theorem 1.7, we use reversed Schur labeling on $[\lambda/\mu]$ and natural
labeling on the $d$-complete posets. In the case of the inversion statistic,
for each $\mu\to\nu$, we need $\omega(x_{1})>\omega(x_{2})$ for all
$x_{i}\in\lambda/\nu_{i}$ to satisfy the condition for Proposition 2.12

Given a linear extension $f:P\to[n]$, the permutation $\omega\circ
f^{-1}\in\mathfrak{S}_{n}$ is a _linear extension of the labeled poset_ ,
$\mathcal{L}(P,\omega)$. For the major index, we label the poset using the
_natural labeling_ on the $d$-complete posets and the _reversed Schur
labeling_ on the border-strip, where the labeling is increasing from right to
left and increasing from top to bottom of the diagram (See Figure 3 (b)). Such
labeling is derived from the Schur labeling of $\lambda/\mu$. We use the
reversed Schur labeling of instead of the Schur labeling because of the
orientation of the conversion from a Young diagram to a poset we have chosen
(see Figure 1).

Recall, $\mathrm{inv}(\sigma)=\\#\\{i\in[n-1]|\sigma_{i}>\sigma_{j}\mbox{
where }i<j\\}$ where
$\mathrm{Des}(\sigma):=\\{i\in[n-1]\,|\,\sigma_{i}>\sigma_{i+1}\\}$ and
$\mathrm{maj}(\sigma)=\sum_{i\in\mathrm{Des}(\sigma)}i$. The two common
statistics for $q$-analogues of the number of linear extensions for a labeled
poset $(P,\omega)$ are the major index and inversions. There is a version of
the major index and the inversion index for labeled $d$-complete posets.

######  Definition 2.3.

The major index of a labeled $d$-complete poset $(P,\omega)$ is

$\mathrm{maj}(P,\omega)=\sum_{x\in\mathrm{Des}(P,\omega)}h_{P}(x).$

######  Definition 2.4.

The inversion index of a labeled poset $(P,\omega)$ is

$\mathrm{inv}(P,\omega)=\\#\\{(x,y)\,|\,\omega(y)<\omega(x)\mbox{ and
}x<_{P}y\\}.$

Let $\mbox{stat}\in\\{\mathrm{maj},\mathrm{inv}\\}$, the _major index
(inversion) q-analogue_ of the number of linear extensions of a labeled poset
$(P,\omega)$ is

$e_{q}^{\mathrm{stat}}(P,\omega):=\sum_{\sigma\in\mathcal{L}(P,\omega)}q^{\mathrm{stat}(\sigma)}.$

Now we state the $q$-analogue of the hook-length formulas. We define the _q-
integer_ $[n]_{q}:=1+q+\cdots+q^{n-1}$ and the _q-factorial_
$[n]_{q}!:=[n]_{q}\cdots[2]_{q}[1]_{q}$. Peterson and Proctor [13] gave a
$\mathrm{maj}$ $q$-analogue formula for $d$-complete posets.

######  Theorem 2.5 (Peterson and Proctor [13]).

Let $(P,\omega)$ be a labeled $d$-complete poset of size $n$ with any
labeling. Then,

$e_{q}^{\mathrm{maj}}(P,\omega)=q^{\mathrm{maj}(P,\omega)}\frac{[n]_{q}!}{\prod_{x\in
P}[h_{P}(x)]_{q}}.$

Björner and Wachs [1] gave a $\mathrm{inv}$ $q$-analogue formula for rooted
tree posets.

######  Theorem 2.6 (Björner and Wachs [1]).

Let $(P,\omega)$ be a rooted tree poset with a natural labeling. Then

$e_{q}^{\mathrm{inv}}(P,\omega)=q^{\mathrm{inv}(P,\omega)}\frac{[n]_{q}!}{\prod_{x\in
P}[h_{P}(x)]_{q}}.$

###  2.6. Identities for $e_{q}^{\mathrm{maj}}$ from the theory of
$P$-partitions

In this section, we state the definition of $(P,\omega)$-partition and its
connection to $e_{q}^{\mathrm{maj}}(P,\omega)$.

######  Definition 2.7.

[16] A $(P,\omega)$-partition is a map $f:P\to\mathbb{N}$ satisfying the
conditions:

  1. (1)

If $s\leq t$ in P, then $f(s)\geq$ f(t).

  2. (2)

If $s\leq t$ and $\omega(s)>\omega(t)$, then $f(s)>f(t)$.

If $\sum_{t\in P}f(t)=n$, then we say $f$ is a $(P,\omega)$-partition of $n$.
We denote the set of all $(P,\omega)$-partitions as $\mathcal{A}(P,\omega)$.
First, we have the following definition.

######  Definition 2.8.

Let $w=w_{1}w_{2}\dots w_{n}\in\mathfrak{S}_{n}$. We say that the function
$f^{\prime}:[n]\to\mathbb{N}$ is $w$-compatible if the following two
conditions hold.

  1. (1)

$f^{\prime}(w_{1})\geq f^{\prime}(w_{2})\geq\cdots\geq f^{\prime}(w_{n})$.

  2. (2)

$f^{\prime}(w_{i})>f^{\prime}(w_{i+1})$ if $w_{i}>w_{i+1}$.

Define $f^{\prime}:[p]\to\mathbb{N}$ by $f^{\prime}(i)=f(\omega^{-1}(i))$, and
let $S_{w}$ be the set of all functions $f$ such that $f^{\prime}$ is
$w$-compatible. We have the following result called the fundamental lemma on
$(P,\omega)$-partitions.

######  Lemma 2.9 ([16], Lemma 3.15.3).

A function $f:P\to\mathbb{N}$ is a $(P,\omega)$-partition if and only if
$f^{\prime}$ is $w$-compatible with some $w\in\mathcal{L}(P,\omega)$. In other
words,

$\mathcal{A}(P,\omega)=\dot{\bigcup}_{w\in\mathcal{L}(P,\omega)}S_{w},$

where $\dot{\cup}$ denotes disjoint union.

Let $a_{P}(n)$ denote the number of $(P,\omega)$-partitions of $n$ and let
$G_{P,\omega}(x)=\sum a_{P}(n)x^{n}$ be generating function of these
partitions. Stanley gave the following specialization of the generating
function associated with $(P,\omega)$-partition:

######  Theorem 2.10 (Stanley [16]).

Let $(P,\omega)$ be a labeled poset of size $p$. Then we have,

$G_{P,\omega}(q)=\frac{e_{q}^{\mathrm{maj}}(P,\omega)}{\prod_{i=1}^{p}(1-q^{i})}.$

For any disjoint union of labeled posets $(P,\omega_{1})+(Q,\omega_{2})$, by
definition, we have since

$G_{P+Q,\omega_{1}+\omega_{2}}(q)=G_{P,\omega_{1}}(q)\cdot
G_{Q,\omega_{2}}(q).$

By applying Theorem 2.10 to above equation, we have the following corollary.

######  Corollary 2.11.

[16, Exercise 3.162(a)] Let $(P+Q,\omega)$ be a labeled disjoint sum of posets
with $|P+Q|=n$ and $|P|=n$. For any labeling $\omega$, we have

$e_{q}^{\mathrm{maj}}(P+Q,\omega)=\left[\begin{matrix}n\\\
p\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P,\omega_{1})\cdot
e_{q}^{\mathrm{maj}}(Q,\omega_{2}),$

where $\omega_{1}$ and $\omega_{2}$ is the labeling obtained by restricting
$\omega$ to $P$ and $Q$ respectively.

###  2.7. Identities for $e_{q}^{\mathrm{inv}}$ for disjoint union of posets

We also have the following disjoint union of poset identity for the case of
inversion index.

######  Proposition 2.12 (Björner-Wachs,[1]).

Let $(P+Q,\omega)$ be a labeled disjoint sum of posets with $|P+Q|=n$ and
$|P|=p$. Suppose that $\omega$ has the property that the label of every
element of $P$ is smaller than the label of every element of $Q$. We have

$e_{q}^{\mathrm{inv}}(P+Q,\omega)=\left[\begin{matrix}n\\\
p\end{matrix}\right]_{q}e_{q}^{\mathrm{inv}}(P,\omega_{1})\cdot
e_{q}^{\mathrm{inv}}(Q,\omega_{2}),$

where $\omega_{1}$ and $\omega_{2}$ are the labeling obtained by restricting
$\omega$ to $P$ and $Q$ respectively.

######  Remark 2.13.

Note that the disjoint poset identity for the inversion statistic has a more
specific condition on the poset labeling than the major index does. When
applying the Theorem 1.9, we need to label the mobile posets so that it
satisfies the condition of Proposition 2.12. More detail about the labeling of
the poset for the case of inversion index is stated in Section 6.

##  3\. $P$-partition with a fixed point

In this section, we discuss a variation of Theorem 2.10, where we fix the
position of an element in $P$. As a result, we have Corollary 3.5, which is
used to prove the recurrence in $e_{q}^{\mathrm{maj}}$.

Denote $\mathcal{L}(P,\omega;s)$ as the set of linear extensions of
$(P,\omega)$ that end with $\omega(s)$ for a fixed $s\in P$ and
$e_{q}^{\mathrm{maj}}(P,\omega;s)=\sum_{\sigma\in\mathcal{L}(P,\omega;s)}q^{\mathrm{maj}(\sigma)}$.
We omit $\omega$ and assume that the poset is labeled $(P,\omega)$ unless
specified. There is analogue of Corollary 2.11 for
$e_{q}^{\mathrm{maj}}(P,\omega;s)$.

We can restrict the conditions on $\mathcal{A}(P,\omega)$ so that it only
considers the linear extensions that ends with a fixed element $s\in P$. We
define the following restricted $(P,\omega)$-partition:

######  Definition 3.1.

A $(P,\omega;s)$-partition is a map $f:P\to\mathbb{N}$ satisfying the
following conditions:

  1. (1)

$f$ is a $(P,\omega)$-partition

  2. (2)

$f(s)\leq f(t)$ for all $t\in P$

  3. (3)

$f(s)=f(t)$ then $\omega(s)>\omega(t)$

Let $\mathcal{A}(P,\omega;s)$ denote the set of such partitions and let

$\mathcal{L}(P,\omega;\\{s\\}):=\\{\sigma\in\mathcal{L}(P,\omega)|\sigma\mbox{
ends with }\omega(s)\\}.$

######  Lemma 3.2.

A function $f:P\to\mathbb{N}$ is a $(P,\omega;s)$-partition if and only if
$f^{\prime}$ is $w$-compatible with some $w\in\mathcal{L}(P,\omega)$. In other
words,

$\mathcal{A}(P,\omega;s)=\dot{\bigcup}_{w\in\mathcal{L}(P,\omega;s)}S_{w}.$

###### Proof.

By the definition of a $(P,\omega;s)$-partition, the set
$\mathcal{A}(P,\omega;s)$ consists of the $(P,\omega)$-partitions such that
$f^{\prime}$ is $w$-compatible with some $w\in\mathcal{L}(P,\omega;s)$. Then
we can restrict the Lemma 2.9 to $\mathcal{L}(P,\omega;s)$. ∎

Let $a_{P}^{s}(n)$ be the number of $(P,\omega;s)$-partition of size $n$, and
let $G_{P;s}(x)=\sum a_{P}^{s}(n)x^{n}$ be the generating function. Then we
can restrict Theorem 2.10 to $\mathcal{L}(P,\omega;s)$. Denote
$e_{q}^{\mathrm{maj}}(P,\omega;\\{s\\}):=\sum_{\sigma\in\mathcal{L}(P_{\lambda/\mu},\omega;\\{s\\})}q^{\mathrm{maj}(\sigma)}$.

######  Theorem 3.3.

Let $(P,\omega)$ be a labeled poset of size $p$ and $s\in P$. Then we have,

$G_{P,\omega;s}(q)=\frac{e_{q}^{\mathrm{maj}}(P,\omega;s)}{\prod_{i=1}^{p}(1-q^{i})}.$

###### Proof.

This is a consequence of Lemma 3.2. ∎

We have the following lemma for the disjoint union of posets with the
restriction.

######  Lemma 3.4.

Let $(P+Q,\omega)$ be a labeled disjoint union of posets and fix $s\in P$ such
that $\omega(s)>\omega(t)$ for all $t\in Q$. Let $p=|P|$ and $n=|P+Q|$. Then
we have

$(1-q^{n})\cdot G_{P+Q;s}(q)=(1-q^{p})\cdot G_{P;s}(q)\cdot G_{Q}(q).$

###### Proof.

To prove the lemma, we first give a combinatorial interpretation of
$H_{P;s}(q):=(1-q^{p})\cdot G_{P;s}(q)$. Consider the difference
$G_{P;s}-q^{p}G_{P;s}$. We build an injection from the $(P;s)$-partition of
size $m-p$ to the $(P;s)$-partition of size $m$. The coefficients of $q^{m}$
from the generating function $q^{p}\cdot G_{P;s}(q)$ counts the number of
$(P;s)$-partitions of size $m-p$. Note that for any $(P;s)$-partition $f$ of
size $m-p$, we can obtain a $P$-partition of size $m$ by adding an element for
each part of $f$. Likewise, given any $(P;s)$-partition $g$ of size $m$ such
that $g(i)>0$ for all $i$, we can obtain a $(P;s)$-partition of size $m-p$ by
subtracting one from each part of $g$. Then the coefficients of $q^{m}$ of
$H_{P;s}(q)$ counts the number of $(P;s)$-partitions $g$ of size $m$ with
$g(s)$ is a minimum value, $\omega(s)>\omega(t)$ for all $t\in P$ where
$g(s)=g(t)$, and with at least one zero value. Thus it is necessary and
sufficient for $g(s)=0$. Similarly, $H_{P+Q;s}(q)$ is a generating function
for $(P+Q;s)$-partition $f$ of size $n$ such that (i) $f(s)=0$ and (ii)
$f(s)>f(t)$ for all $t\in P+Q$ such that $f(t)=0$.

Then consider the right hand side of the equation. The coefficients of $x^{m}$
counts size of a disjoint union of $(P;s)$-partition $f_{1}$ of size $k$
counted in $H_{P;s}(q)$ and $Q$-partition $f_{2}$ of size $n-k$ for some
$k=0,\ldots,n$. Such disjoint union is equivalent to the $(P+Q;s)$-partitions
counted in $H_{+Q;s}(q)$. To see this note that such $(P+Q;s)$-partition $f$
satisfies condition (i), namely $f(s)=f_{1}(s)=0$. Next, we verify that $f$
satisfies the condition (ii). If $t\in P$ with $f(t)=f_{1}(t)=0$ then
$\omega(s)>\omega(t)$ since $\sigma_{1}$ satisfies condition (ii). If $t\in Q$
with $f_{2}(t)=0$, then by assumption $\omega(s)>\omega(t)$. Thus, we have
that $H_{P+Q;s}(q)=H_{P;s}(q)\cdot G_{Q}(q)$ as desired. ∎

Then by applying Theorem 3.3 to Lemma 3.4, we have the following corollary
which we will use to prove our main result in the next section.

######  Corollary 3.5.

Let $(P+Q,\omega)$ be a labeled disjoint sum of posets and fix $s\in P$ such
that $\omega(s)>\omega(t)$ for all $t\in Q$. Let $|P+Q|=n$ and $|P|=p$. For
any labeling $\omega$, we have

$e_{q}^{\mathrm{maj}}(P+Q;s)=\left[\begin{matrix}n-1\\\
p-1\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P;s)e_{q}^{\mathrm{maj}}(Q).$

##  4\. A Major index $q$-analogue

In this section we give the proof of Theorem 1.7. The proof follows the proof
of the NHLF for border strips in [8]. We need to first define the hook-lengths
of mobile posets. Given a mobile poset $P_{\lambda/\mu}(\mathbf{p})$, define
the hook-length of $(i,j)\in[\lambda]$ as following:

(4.1) $h^{\prime}(i,j)=\lambda_{i}-i+\lambda_{j}^{\prime}-j+1+\sum_{a\geq
i,\\\ b\geq j}p_{a,b}.$

In other words, it is the usual hook-length of the cell in $\lambda$ plus the
size of the $d$-complete posets that are attached on the segment of the border
strip inside of the hook of $(i,j)$ (see Figure 3 (a)). We provide an example
of the of the theorem below.

(a)

(b)

(c)

Figure 3. (a): $h^{\prime}(u)$ is the usual hook-length plus the size of
$d$-complete posets in the shaded area. (b): mobile poset with reversed Schur
labeling. (c): mobile poset with hook-lengths.

######  Example 4.1.

Consider the poset $(P_{2221/11},\omega)$ from Figure 3 (b). By Theorem 1.7,
one can check that

$\displaystyle e_{q}^{\mathrm{maj}}(P)$
$\displaystyle=\frac{[13]!}{[1]^{2}[2]^{4}[3]^{2}}\left(\frac{q^{12}}{[5][7][1][6][7]}+\frac{q^{18}}{[5][7][12][6][7]}+\frac{q^{24}}{[5][7][12][13][7]}\right)$
$\displaystyle=q^{61}+2q^{60}+6q^{59}+11q^{58}+\cdots+6q^{14}+2q^{13}+q^{12}.$

We first introduce two lemmas required for the proof. We provide the proof of
each lemma in Section 4.1 and Section 4.2.

We have the following recurrence lemma for the the $q$-analogue of linear
extensions for the major index.

######  Lemma 4.2.

For a labeled mobile poset $(P_{\lambda/\mu}(\mathbf{p}),\omega)$, where
$\omega$ is a reversed Schur labeling,

(4.2)
$e_{q}^{\mathrm{maj}}(P_{\lambda/\mu},\omega)=\sum_{\mu\to\nu}q^{|P_{\lambda/\nu_{1}}|}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu},\omega_{\nu})$

where $P_{\lambda/\nu_{1}}$ is the left disconnected poset of
$P_{\lambda/\nu}$, and $\omega_{\nu}$ is the restricted labeling of $\omega$
onto $P_{\lambda/\nu}$.

Next, we have the following Pieri–Chevalley formula. Denote the RHS of (1.7)
as $H_{\lambda/\mu}(q)$.

$H_{\lambda/\mu}(q):=\prod_{v\in\mathbf{p}}\frac{1}{1-q^{h(v)}}\sum_{D\in\mathcal{E}(\lambda/\mu)}q^{w^{\prime}(D)}\prod_{u\in[\lambda]\setminus
D}\frac{1}{1-q^{h^{\prime}(u)}}$

######  Lemma 4.3.

(4.3) $(1-q^{n})\cdot
H_{\lambda/\mu}(q)=\sum_{\mu\to\nu}\frac{q^{|P_{\lambda/\nu^{1}}|}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}\cdot H_{\lambda/\nu^{1}}(q)\cdot
H_{\lambda/\nu^{2}}(q),$

where $T_{\nu}$ is the union of the $d$-complete posets that were hanging on
the removed inner corner $u$.

We are now ready to give the proof of Theorem 1.7.

###### Proof of Theorem 1.7.

We first evaluate the multivariate formula $F_{\lambda/\mu}$ at
$x=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}}$ and
$y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq j}p_{a,b}}$. We denote
$e_{q}^{\mathrm{maj}}(P,\omega)$ as $e_{q}(P)$ unless specified. We show that
$e_{q}^{\mathrm{maj}}(P_{\lambda/\mu}(\mathbf{p}))=\prod_{i=1}^{n}(1-q^{i})H_{\lambda/\mu}(q)$
by induction on $|\lambda/\mu|$ using Lemma 4.2. Note that $\lambda/\nu$ is
disconnected and

(4.4) $P_{\lambda/\nu}=P_{\lambda/\nu^{1}}+P_{\lambda/\nu^{2}}+T_{\nu},$

where $T_{\nu}$ is the union of the $d$-complete posets that were hanging on
the removed inner corner $u$. Denote $|P_{\lambda/\nu^{j}}|$ as $p_{j}$. By
induction, we have for $j=1,2$

$\frac{e_{q}(P_{\lambda/\nu^{j}})}{[p_{j}]_{q}!}=\prod_{i=1}^{p_{j}}(1-q^{i})\cdot{H}_{\lambda/\nu^{j}}\cdot\frac{(1-q)^{p_{j}}}{\prod_{i=1}^{p_{j}}(1-q^{i})}={(1-q)^{p_{j}}}\cdot{H}_{\lambda/\nu^{j}},$

and by Theorem 2.5, for each $T_{i}\subset T_{\nu}$ we have,

$\frac{e_{q}(T_{i})}{[t_{i}]_{q}!}=\frac{q^{\mathrm{maj}(T_{i})}}{\prod_{v\in
T_{i}}[h(v)]_{q}}=\frac{(1-q)^{t_{i}}}{\prod_{v\in T_{i}}(1-q^{h(v)})}.$

Note that $T_{i}$ are natural labeling, so $\mathrm{maj}(T_{i})=0$.

Using Corollary 2.11 and the equations above, we have

$e_{q}(P_{\lambda/\nu})=\frac{\prod_{i=1}^{n-1}(1-q^{i})}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}H_{\lambda/\nu^{1}}(q)\cdot H_{\lambda/\nu^{2}}(q).$

We now apply the equation to (4.2):

(4.5)
$e_{q}(P_{\lambda/\mu})=\prod_{i=1}^{n-1}(1-q^{i})\cdot\sum_{\mu\to\nu}\frac{q^{|\lambda/\nu_{1}|}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}\cdot H_{\lambda/\nu^{1}}(q)\cdot H_{\lambda/\nu^{2}}(q)$

Note that $\lambda_{1}^{\prime}-1+c(u)+p_{1}=|\lambda/\nu_{1}|$ when $\omega$
is a reversed Schur labeling. By (4.3), we can show the sum on the RHS of
(4.5) equals $(1-q^{n})\cdot H_{\lambda/\mu}(q)$, completing the proof. ∎

######  Remark 4.4.

We can generalize the labeling of the mobile poset by allowing non-natural
labeling on the $d$-complete posets. In such case, we we would have non-
trivial values for $q^{\mathrm{maj}(T_{i})}$ in our final formula.

###  4.1. Proof of Lemma 4.2

To prove Lemma 4.2, we need the following lemmas.

Let
$(P_{\lambda/\mu},\omega;U):=\\{\sigma\in\mathcal{L}(P_{\lambda/\mu},\omega)\,|\,\sigma\mbox{
ends with }u\in U\\}$ and

$e_{q}^{\mathrm{maj}}(P_{\lambda/\mu},\omega,U):=\sum_{\sigma\in(P_{\lambda/\mu},\omega;U)}q^{\mathrm{maj}(\sigma)}.$

Unless specified otherwise, we denote this as
$e_{q}^{\mathrm{maj}}(P_{\lambda/\mu},U)$.

######  Lemma 4.5.

For a labeled mobile poset $(P_{\lambda/\mu}(\mathbf{p}),\omega)$, where
$\omega$ is a reversed Schur labeling, let $\mu\to\nu$ be the removal of an
inner corner $u$, and $P_{\lambda/\nu_{1}}$ and $P_{\lambda/\nu_{2}}$ be the
two disconnected parts of $\lambda/\nu$. Then we have

(4.6)
$e_{q}^{\mathrm{maj}}(P_{\lambda/\mu};\\{u\\})=q^{n-1}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{1}])+e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{2}]).$

###### Proof.

The linear extension on the left is $\sigma\in\mathcal{L}(P_{\lambda/\mu})$
where $\sigma=\sigma_{1}\dots\sigma_{n-1}\omega(u)$, we have
$\sigma_{1}\dots\sigma_{n-1}\in\mathcal{L}(P_{\lambda/\nu})$ and
$\sigma_{n}=u$. Then $\sigma_{n-1}$ is either an element in
$Q_{\lambda/\nu_{1}}$ or $Q_{\lambda/\nu_{2}}$. If $\sigma_{n-1}\in
Q_{\lambda/\nu_{1}}$, since $\omega$ is a reversed Schur labeling,
$\omega(\sigma_{n-1})>\omega(\sigma_{n})$, so
$\mathrm{maj}(\sigma)=\mathrm{maj}(\sigma_{1}\dots\sigma_{n-1})+n-1$. If
$\sigma_{n-1}\in Q_{\lambda/\nu_{2}}$, then
$\omega(\sigma_{n-1})<\omega(\sigma_{n})$, so
$\mathrm{maj}(\sigma)=\mathrm{maj}(\sigma_{1}\dots\sigma_{n-1})$. ∎

Now we are ready to prove Lemma 4.2.

###### Proof of Lemma 4.2.

By Corollary 2.11 and a standard recurrence for $q$-binomial coefficients we
have

$\displaystyle e_{q}^{\mathrm{maj}}(P_{\lambda/\nu})$
$\displaystyle=\left(\left[\begin{matrix}n-2\\\
|\lambda/\nu_{1}|\end{matrix}\right]_{q}+q^{|\lambda/\nu_{1}|}\left[\begin{matrix}n-2\\\
|\lambda/\nu_{2}|-1\end{matrix}\right]_{q}\right)e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}})$
(4.7) $\displaystyle=\left[\begin{matrix}n-2\\\
|\lambda/\nu_{2}|\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}})+q^{|\lambda/\nu_{1}|}\left[\begin{matrix}n-2\\\
|\lambda/\nu_{2}|-1\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}}).$

Then the two parts of the sum can be interpreted in the following ways:

######  Proposition 4.6.

(4.8)
$e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{1}])=\left[\begin{matrix}n-2\\\
|\lambda/\nu_{1}|-1,|\lambda/\nu_{2}|\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}}),$

and

(4.9)
$e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{2}])=q^{|\lambda/\nu_{1}|}\left[\begin{matrix}n-2\\\
|\lambda/\nu_{2}|-1\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}}).$

###### Proof.

Let $\\{x_{1},\dots x_{k}\\}$ be the maximal elements of
$P_{\lambda/\nu_{1}}$. Note that

(4.10)
$e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{1}])=\sum_{i=1}^{k}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};x_{i}),$

and

(4.11)
$e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})=\sum_{i=1}^{k}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}};x_{i}).$

We know that $P_{\lambda/\nu_{1}}$ and $P_{\lambda/\nu_{2}}$ satisfy the
condition in Lemma 3.4. Then by Corollary 3.5, we have

(4.12) $e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};x_{i})=\left[\begin{matrix}n-2\\\
|\lambda/\nu_{1}|-1,|\lambda/\nu_{2}|\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}};x_{i})\cdot
e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}}).$

Applying (4.12) and (4.11) to (4.10), we have the desired result.

For the second equation, we know that
$e_{q}^{\mathrm{maj}}(P_{\lambda/\nu})=e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{1}])+e_{q}^{\mathrm{maj}}(P_{\lambda/\nu};[\lambda/\nu_{2}])$,
so subtracting (4.8) from $e_{q}^{\mathrm{maj}}(P_{\lambda/\nu})$, we have the
desired result as well. ∎

We now apply (4.8) and (4.9) to (4.6) to get the following equation. Let
$p_{1}=|P_{\lambda/\nu_{1}}|$ and $p_{2}=|P_{\lambda/\nu_{2}}|$.

$\displaystyle e_{q}^{\mathrm{maj}}(P_{\lambda/\mu};\\{u\\})$
$\displaystyle=q^{n-1}\left[\begin{matrix}n-2\\\
p_{2}\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})\cdot
e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}})+q^{p_{1}}\left[\begin{matrix}n-2\\\
p_{2}-1\end{matrix}\right]_{q}e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{1}})\cdot
e_{q}^{\mathrm{maj}}(P_{\lambda/\nu_{2}})$

After simplifying everything, we get

$e_{q}^{\mathrm{maj}}(P_{\lambda/\mu};\\{u\\})=q^{p_{1}}\cdot
e_{q}^{\mathrm{maj}}(P_{\lambda/\nu}).$

Such equation is true for all inner corners $u$ of $\mu\to\nu$, which
completes the proof of Lemma 4.2. ∎

###  4.2. Proof of Lemma 4.3

We first evaluate $x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}}$ and
$y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq j}p_{a,b}}$ in the
Pieri–Chevalley formula (2.2). The LHS of this formula becomes

(4.13)
$\left.F_{\lambda/\mu}(\mathbf{x},\mathbf{y})\right|_{\begin{subarray}{c}x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}},\\\
y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq
j}p_{a,b}}\end{subarray}}=(-1)^{n}\cdot\sum_{\begin{subarray}{c}\gamma:A\to
B,\\\
\gamma\subset\lambda\end{subarray}}\prod_{(i,j)\in\gamma}\frac{q^{\lambda_{j}^{\prime}-j+\sum_{b\geq
j}p_{a,b}}}{1-q^{h^{\prime}(i,j)}}$

By [10, Proposition 4.7] we have

$\displaystyle\sum_{(i,j)\in[\lambda]\setminus
D}\Big{(}(\lambda_{j}^{\prime}-j)+\sum_{b\geq j}p_{a,b}\Big{)}$
$\displaystyle=\sum_{(i,j)\in[\lambda]\setminus
D}\Big{(}(\lambda_{j}^{\prime}-i)+\sum_{b\geq
j}p_{a,b}\Big{)}-\sum_{(i,j)\in[\lambda]\setminus[\mu]}c(i,j),$

where in the last equality, we use [10, Proposition 7.16] to obtain,

(4.14) $\sum_{(i,j)\in[\lambda]\setminus
D}\Big{(}(\lambda_{j}^{\prime}-j)+\sum_{b\geq
j}p_{a,b}\Big{)}=w^{\prime}(\mathrm{Br}(D))+\Bigg{(}\sum_{(i,j)\in[\lambda]\setminus
D}\sum_{b\geq j}p_{a,b}-\sum_{\begin{subarray}{c}a\geq i,b\geq j\\\
(i,j)\in\mathrm{Br}(D)\end{subarray}}p_{a,b}\Bigg{)}-\sum_{(i,j)\in[\lambda]\setminus[\mu]}c(i,j)$

where
$w^{\prime}(\mathrm{Br}(D))=\sum_{(i,j)\in\mathrm{Br}(D)}h^{\prime}(i,j)$, and
the subtraction in the second sum is the $d$-complete posets that are included
in the new hook-lengths of the broken diagonals. We denote the quantity in
parenthesis on the RHS as $\widehat{p}_{D}$.

$\widehat{p}_{D}:=\sum_{(i,j)\in[\lambda]\setminus D}\sum_{b\geq
j}p_{a,b}-\sum_{\begin{subarray}{c}a\geq i,b\geq j\\\
(i,j)\in\mathrm{Br}(D)\end{subarray}}p_{a,b}$

We claim that $\widehat{p}_{D}$ is invariant among $D$.

######  Lemma 4.7.

The quantity $\widehat{p}_{D}$ is invariant among all
$D\in\mathcal{E}(\lambda/\mu)$.

###### Proof.

We prove by induction on excited moves $\beta:D\to D^{\prime}$. Denote
$a_{i}=(s_{i},t_{i})$ as the active cell and $b=(s+1,t+1),b^{\prime}=(s+1,t)$
be the old and new broken diagonal of an excited move $\beta:D\to D^{\prime}$,
Then for each excited move $\beta$, $[\lambda]\setminus
D^{\prime}=([\lambda]\setminus D)\setminus\\{b\\}\cup\\{a\\}$. Also,
$\mathrm{Br}(D^{\prime})=\mathrm{Br}(D)\setminus\\{b\\}\cup\\{b^{\prime}\\}$.
Thus

Figure 4. For each excited move $\beta_{i}$, we place the old broken diagonal
$b_{i}$ with the new broken diagonal $b_{i}^{\prime}$.

$\displaystyle\widehat{p}_{D^{\prime}}$
$\displaystyle=\left(\sum_{(i,j)\in[\lambda]\setminus D}\sum_{b\geq
j}p_{a,b}-\sum_{b\geq t_{i}+1}p_{a,b}+\sum_{b\geq
t_{i}}p_{a,b}\right)-\left(\sum_{\begin{subarray}{c}a\geq i,b\geq j\\\
(i,j)\in\mathrm{Br}(D)\end{subarray}}p_{a,b}-\sum_{\begin{subarray}{c}a\geq
s_{i}+1,\\\ b\geq t_{i}+1\end{subarray}}p_{a,b}+\sum_{\begin{subarray}{c}a\geq
s_{i}+1,\\\ b\geq t_{i}\end{subarray}}p_{a,b}\right)$
$\displaystyle=\left(\sum_{(i,j)\in[\lambda]\setminus D}\sum_{b\geq
j}p_{a,b}+\sum_{b=t_{i}}p_{a,b}\right)-\Bigg{(}\sum_{\begin{subarray}{c}a\geq
i,b\geq j\\\
(i,j)\in\mathrm{Br}(D)\end{subarray}}p_{a,b}+\sum_{\begin{subarray}{c}a\geq
s_{i}+1,\\\ b=t_{i}\end{subarray}}p_{a,b}\Bigg{)}$
$\displaystyle=\widehat{p}_{D}+\sum_{b=t_{i}}p_{a,b}-\sum_{\begin{subarray}{c}a\geq
s_{i}+1,\\\ b=t_{i}\end{subarray}}p_{a,b}$

For $\lambda/\mu$ border strip, there cannot be any $d$-complete posets
hanging above an active cell in the same column, so the difference on the RHS
of the equation above is zero and so
$\widehat{P}_{D^{\prime}}=\widehat{P}_{D}$. ∎

We then let $\widehat{p}_{D}=\widehat{p}$ for all
$D\in\mathcal{E}(\lambda/\mu)$. Putting $\widehat{p}$ and $c(i,j)$ outside of
the sum, we can rewrite (4.13) as:

(4.15)
$\left.F_{\lambda/\mu}(\mathbf{x}|\mathbf{y})\right|_{\begin{subarray}{c}x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}},\\\
y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq
j}p_{a,b}}\end{subarray}}=(-1)^{n}\cdot
q^{\widehat{p}_{\mu}-\operatorname{con}(\lambda/\mu)}\prod_{v\in\mathbf{p}}(1-q^{h(v)})\cdot
H_{\lambda/\mu},$

where $\operatorname{con}(\lambda/\mu):=\sum_{(i,j)\in[\lambda/\mu]}c(i,j)$.
Next we see what happens to the RHS of Pieri–Chevalley formula (2.2) when we
evaluate at $x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}}$ and
$y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq j}p_{a,b}}$, where the sizes
$p_{a,b}$ appearing in the sums are for the $d$-complete posets in
$P_{\lambda/\mu}(\mathbf{p})$. The linear factor on the RHS of Pieri–Chevalley
formula becomes

(4.16)
$\left.\frac{1}{x_{1}-y_{1}}\right|_{\begin{subarray}{c}x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}},\\\
y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq
j}p_{a,b}}\end{subarray}}\,=\,\frac{(-1)^{n-1}}{q^{\lambda_{1}-\sum_{a<1}p_{a,b}}-q^{1-\lambda^{\prime}_{1}-\sum_{b\geq
1}p_{a,b}}}=\frac{(-1)^{n}q^{\lambda_{1}^{\prime}-1+\sum_{(a,b)\in[\lambda/\mu]}p_{a,b}}}{(1-q^{n})}.$

Given an inner corner removed $\mu\to\nu$, denote $\mathbf{p}_{1}$ and
$\mathbf{p}_{2}$ be the set of $d$-complete poset hanging on
$P_{\lambda/\nu^{1}}$ and $P_{\lambda/\nu^{2}}$ respectively. Then for the
shapes $\lambda/\nu^{k}$ where $k=1,2$ we have

(4.17)
$\left.F_{\lambda/\nu^{k}}(\mathbf{x}|\mathbf{y})\right|_{\begin{subarray}{c}x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}},\\\
y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq
j}p_{a,b}}\end{subarray}}=(-1)^{|\lambda/\nu^{k}|}\cdot
q^{\widehat{p}_{\nu^{k}}-\operatorname{con}(\lambda/\nu^{k})}\prod_{v\in\mathbf{p}_{k}}(1-q^{h(v)})\cdot
H_{\lambda/\nu^{k}},$

$\frac{\prod_{v\in\mathbf{p}_{1}}(1-q^{h(v)})\cdot\prod_{v\in\mathbf{p}_{2}}(1-q^{h(v)})}{\prod_{v\in\mathbf{p}}(1-q^{h(v)})}=\frac{1}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}$

Thus, by (4.15), (4.16), and (4.17) the Pieri–Chevalley formula (2.2)
evaluated at such $x_{i}$ and $y_{j}$ becomes,

(4.18)
$q^{\widehat{p}_{\mu}-\operatorname{con}(\lambda/\mu)}H_{\lambda/\mu}=\frac{q^{\lambda_{1}^{\prime}-1+\sum_{(a,b)\in[\lambda/\mu]}p_{a,b}}}{(1-q^{n})}\sum_{\mu\to\nu}\frac{q^{\widehat{p}_{\nu_{1}}+\widehat{p}_{\nu_{2}}-\operatorname{con}(\lambda/\nu_{1})-\operatorname{con}(\lambda/\nu_{2})}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}H_{\lambda/\nu_{1}}H_{\lambda/\nu_{2}}.$

Note that for each inner corner $u:\mu\to\nu$, we have
$\operatorname{con}(\lambda/\mu)-\operatorname{con}(\lambda/\nu_{1})-\operatorname{con}(\lambda/\nu_{2})=c(u)$.
Thus the previous equation becomes

$(1-q^{n})H_{\lambda/\mu}=\sum_{\mu\to\nu}\frac{q^{\lambda^{\prime}_{1}-1+c(u)+\sum_{(a,b)\in[\lambda/\mu]}p_{a,b}+\widehat{p}_{\nu_{1}}+\widehat{p}_{\nu_{2}}-\widehat{p}_{\mu}}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}H_{\lambda/\nu_{1}}H_{\lambda/\nu_{2}}.$

######  Lemma 4.8.

For the equation defined above,

$\sum_{(a,b)\in[\lambda/\mu]}p_{a,b}+\widehat{p}_{\nu_{1}}+\widehat{p}_{\nu_{2}}-\widehat{p}_{\mu}=|\mathbf{p}_{1}|.$

###### Proof.

Consider $\sum_{b\geq
1}p_{a,b}+\widehat{p}_{\nu_{1}}+\widehat{p}_{\nu_{2}}-\widehat{p}_{\mu}$. We
know that $\widehat{p}$ is invariant among the excited diagrams, so without
loss of generality, assume $D=[\mu]$.

We have
$\mathrm{Br}([\nu_{1}])\cup\mathrm{Br}([\nu_{2}])\cup\\{u_{0}\\}=\mathrm{Br}([\lambda/\mu]),$
where $u_{0}=(u_{1}+1,u_{2})$ is a broken diagonal of $[\mu]$ below
$u=(u_{1},u_{2})$. Denote $p_{i}$ as the size of the $d$-complete posets
hanging on $\lambda/\nu_{i}$. Then,

Figure 5. $[\lambda/\mu]=[\lambda/\nu_{1}]+[\lambda/\nu_{2}]+u$. Note that in
$\lambda/\nu_{1}$, we are missing a broken diagonal underneath $u$.

(4.19) $\sum_{b\geq
1}p_{a,b}+\widehat{p}_{\nu_{1}}+\widehat{p}_{\nu_{2}}-\widehat{p}_{\mu}=\sum_{b\geq
1}p_{a,b}+\Bigg{(}\sum_{(i,j)\in[\lambda/\nu_{1}]}\sum_{b\geq
j}p_{a,b}+\sum_{(i,j)\in[\lambda/\nu_{2}]}\sum_{b\geq
j}p_{a,b}-\sum_{(i,j)\in[\lambda/\mu]}\sum_{b\geq j}p_{a,b}\Bigg{)}\\\ \hskip
10.0pt-\Bigg{(}\sum_{\begin{subarray}{c}a\geq i,b\geq j\\\
(i,j)\in\mathrm{Br}([\nu_{1}])\end{subarray}}p_{a,b}+\sum_{\begin{subarray}{c}a\geq
i,b\geq j\\\
(i,j)\in\mathrm{Br}([\nu_{2}])\end{subarray}}p_{a,b}-\sum_{\begin{subarray}{c}a\geq
i,b\geq j\\\ (i,j)\in\mathrm{Br}([\mu])\end{subarray}}p_{a,b}\Bigg{)}$

Let $S_{1}$ and $S_{2}$ be the sums in parenthesis on the RHS above. Since
$\lambda/\nu_{1}$ and $\lambda/\nu_{2}$ do not contain the inner corner $u$
(see Figure 5) then $S_{1}$ and $S_{2}$ simplify to

(4.20) $S_{1}=\sum_{b\geq u_{2}}p_{a,b},\qquad S_{2}=\sum_{a\geq u_{1}+1,b\geq
u_{2}}p_{a,b}.$

Thus Equation (4.19) becomes

$\displaystyle\sum_{b\geq
1}p_{a,b}+\widehat{p}_{\nu_{1}}+\widehat{p}_{\nu_{2}}-\widehat{p}_{\mu}$
$\displaystyle=\sum_{b\geq 1}p_{a,b}-\Bigg{(}\sum_{b\geq
u_{2}}p_{a,b}-\sum_{a\geq u_{1}+1,b\geq u_{2}}p_{a,b}\Bigg{)}.$

Note that the equation in the parenthesis counts the number of $d$-complete
posets hanging on and to the the right of $u$. These are exactly the
$d$-complete posets on $P_{\lambda/\nu^{1}}$. ∎

Finally, note that
$\lambda_{1}^{\prime}-1+c(u)+|\mathbf{p}_{1}|=|P_{\lambda/\nu_{1}}|$. Then we
can simplify (4.2) to obtain the desired formula.

##  5\. Application: bounds for the number of linear extensions

In this section, we provide a short proof of Corollary 1.8 along with an
example. We also discuss an application of the formula to the bounds of
generalized Euler numbers.

###  5.1. The case of $q=1$

###### proof of Corollary 1.8.

Taking $q=1$ for Theorem 1.7, we get the proof of Corollary 1.8. One can also
prove Corollary 1.8 directly by evaluating the multivariate formula
$F_{\lambda/\mu}$ at $x_{i}=\lambda_{i}-i+1-\sum_{a<i}p_{a,b}$ and
$y_{j}=j-\lambda_{j}^{\prime}-\sum_{b\geq j}p_{a,b}$ ∎

We give the example of the theorem below.

######  Example 5.1.

Consider the mobile poset $P_{2221/11}$ from Figure 3 (b). Then by Corollary
1.8 we have

(5.1) $\displaystyle e(P)=\frac{13!}{2^{\cdot}4^{2}}\left(\frac{1}{5\cdot
6\cdot 7^{2}}+\frac{1}{5\cdot 6\cdot 7^{2}\cdot 12}+\frac{1}{5\cdot 7^{2}\cdot
12\cdot 13}\right)=33000.$

(a)

(b)

Figure 6. (a) the $\omega_{\mathrm{inv}}$ labeled mobile tree poset, (b)
illustration of the posets $\mathcal{C}_{p}(k)$ (top) and $\mathcal{A}_{p}(k)$
(bottom).

###  5.2. Bounds to generalizations of Euler numbers

As an application to Corollary 1.8 gives bounds to
$e(P_{\lambda/\mu}(\mathbf{p}))$ just as in [9].

######  Corollary 5.2.

For any mobile poset $e(P_{\lambda/\mu}(\mathbf{p}))$ of size $n$,

$\frac{n!}{H(\mathbf{p})\prod_{u\in[\lambda/\mu]}h^{\prime}(u)}\leq
e(P_{\lambda/\mu}(\mathbf{p}))\leq|\mathcal{E}(\lambda/\mu)|\cdot\frac{n!}{H(\mathbf{p})\prod_{u\in[\lambda/\mu]}h^{\prime}(u)}$

where $[\lambda/\mu]$ is the border strip of the mobile poset.

###### Proof.

For any skew shape $\lambda/\mu$, we have $[\mu]\in\mathcal{E}(\lambda/\mu)$,
so the lower bound is given by Corollary 1.8. For the upper bound, note that
under the excited move, the product $\prod_{v\in[\lambda/\mu]}h^{\prime}(u)$
increases. Then this product is minimal when the excited diagram is $[\mu]$
and the upper bound follows from Corollary 1.8. ∎

For more detail about asymptotic of linear extensions of skew shaped tableaux,
see [9].

One application of the formula is that it provides bounds to generalizations
of _Euler numbers_ defined in [5]. The authors give two generalizations of
Euler number using two different families of posets, up-down posets with $k-1$
downs and chains (or anti-chains) of size $p$ hanging on every minimal
element,denoted as $\mathcal{C}_{p}(k)$ and $\mathcal{A}_{p}(k)$ (see Figure 3
(b)). See [14, A332471] and [14, A332568] for examples of these sequences.

######  Corollary 5.3.

$\frac{(2k+kp)!}{(p+1)!^{k}(2p+3)^{k-1}(p+2)}\leq
e(\mathcal{C}_{p}(k))\leq\mathrm{Cat}(k)\cdot\frac{(2k+kp)!}{(p+1)!^{k}(2p+3)^{k-1}(p+2)}$

$\frac{(2k+kp)!}{(p+1)^{k}(2p+3)^{k-1}(p+2)}\leq
e(\mathcal{A}_{p}(k))\leq\mathrm{Cat}(k)\cdot\frac{(2k+kp)!}{(p+1)^{k}(2p+3)^{k-1}(p+2)}$

where $\mathcal{Z}$ is the up-down border strip with $k-1$ many down steps.

###### Proof.

The result follows from Corollary 5.2, a routine calculations of hooks, and
the fact that the excited diagrams of up-down posets are given by the Catalan
numbers [8, Corollary 8.1]

∎

##  6\. An Inversion index $q$-analogue

In this section we give an example and the proof of Theorem 1.4. Unless
specified otherwise, $(P_{\lambda/\mu}(\mathbf{p}),\omega)$ is a labeled
mobile tree poset.

###  6.1. Labeling of the poset for the case of inversion index

The mobile tree poset must satisfy a very specific labeling for the case of
the inversion statistic. One of the reasons why is because of the condition
stated in Proposition 2.12. Another reason is so that the labeling needs to
satisfy Lemma 6.3. To satisfy both conditions, we must label the poset in the
following way: let $\\{u_{1},\dots,u_{k}\\}$ be the list of inner corners of
$P_{\lambda/\mu}$ from $(1,\lambda_{1})$ to $(\lambda^{\prime}_{1},1)$, and
$u_{0}=(1,\lambda_{1})$ and $u_{k+1}=(\lambda_{1}^{\prime},1)$. Partition the
mobile posets into $P_{1},\dots,P_{k+1}$ such that for each $P_{i}$, it
contains all elements $(s,t)\in P_{\lambda/\mu}$ such that $u_{i}<t\leq
u_{i-1}$ and all the elements of rooted trees hanging on such $(s,t)$. If
$u_{k+1}=(\lambda_{1}^{\prime},1)$, then $P_{k+1}=u_{k+1}$. Starting from
$P_{1}$, we label each $P_{i}$ such that all the hanging rooted trees are
naturally labeled and the elements in the border strip have reversed Schur
labeling. See Figure 7 for an example. We denote such labeling as
$\omega_{\mathrm{inv}}$.

Figure 7. A mobile tree poset with labeling $\omega_{\mathrm{inv}}$. On the
left we show the partitions of the poset into $P_{1},\dots,P_{4}$. We label
each $P_{i}$ so that the $d$-complete posets are naturally labeled and the
elements in the border-strip have reversed Schur labeling.

######  Example 6.1.

Consider labeled the mobile poset $(P_{2221/11},\omega_{\mathrm{inv}})$ from
Figure 6 (a). Then by Theorem 1.9 we have

$\displaystyle e_{q}^{\mathrm{inv}}(P)$
$\displaystyle=\frac{[11]!}{[1]^{4}[3]^{2}}\left(\frac{q^{4}}{[4][6][1][5][6]}+\frac{q^{9}}{[4][6][10][5][6]}+\frac{q^{14}}{[4][6][10][11][6]}\right)$
$\displaystyle=q^{38}+4q^{37}+9q^{36}+17q^{35}+\cdots+9q^{6}+4q^{5}+q^{4}.$

######  Remark 6.2.

Note that Theorem 1.9 is only for mobile trees, where the $d$-complete posets
are restricted to rooted trees. This is because there is no known hook-length
formula for $e_{q}^{\mathrm{inv}}(P)$ when $P$ is a general $d$-complete
poset.

We need the following recursion for the inversion index $q$-analogue.

######  Lemma 6.3.

(6.1)
$e_{q}^{\mathrm{inv}}(P_{\lambda/\mu},\omega)=\sum_{\mu\to\nu}q^{n-\omega(u)}e_{q}^{\mathrm{inv}}(P_{\lambda/\nu},\omega_{\nu}),$

where $\omega$ is a reversed Schur labeling and $\omega(u)$ is the label of
the inner corner $u$ from $\mu\to\nu$.

We also need the following Pieri–Chevalley formula for the inversion index.
Denote the the RHS of (1.4) as $\widetilde{H}_{\lambda/\mu}(q)$:

$\widetilde{H}_{\lambda/\mu}(q):=\prod_{v\in\mathbf{p}}\frac{1}{1-q^{h(v)}}\sum_{D\in\mathcal{E}(\lambda/\mu)}q^{w(D)+p_{D}}\prod_{u\in[\lambda]\setminus
D}\frac{1}{1-q^{h^{\prime}(u)}},$

######  Lemma 6.4.

(6.2)
$(1-q^{n})\cdot\widetilde{H}_{\lambda/\mu}=\sum_{\mu\to\nu}\frac{q^{q^{n-\omega_{\mathrm{inv}}(u)}}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}\cdot\widetilde{H}_{\lambda/\nu^{1}}\cdot\widetilde{H}_{\lambda/\nu^{2}},$

where $\omega_{\mathrm{inv}}(u)$ is the label of $u$, the inner corner from
$\mu\to\nu$

We provide the proof of the lemmas in Section 6.2.

###### Proof.

Recall that for a fixed tree mobile $P_{\lambda/\mu}$, a linear extension of
$\sigma\in\mathcal{L}(P_{\lambda/\mu})$ consists of an inner corner of
$\lambda/\mu$ followed by a linear extension of the remaining poset of shape
$\lambda/\nu$, where $\mu\to\nu$. Conversely, given a linear extension
$\sigma^{\prime}\in\mathcal{L}(P_{\lambda/\nu})$, by inserting the new element
in the beginning we obtain a linear extension of $P_{\lambda/\mu}$. Note that
$\mathrm{inv}(\sigma)=\mathrm{inv}(\sigma^{\prime})+n-\omega$, where
$n-\omega$ is the number of inversion caused by the inner corner. The result
follows from this correspondence ∎

We are now ready to give the proof of Theorem 1.9.

###### Proof of Theorem 1.4.

Similarly as in the case of major index, we show that
$e_{q}^{\mathrm{inv}}(P_{\lambda/\mu}(\mathbf{p}))={\prod_{i=1}^{n}(1-q^{i})}\cdot\widetilde{H}_{\lambda/\mu}(q)$
by induction on $|\lambda/\mu|$ using Lemma 6.3. Recall $P_{\lambda/\nu}$ can
be expressed as (4.4). By induction and Theorem 2.5, we have

$\frac{e_{q}(P_{\lambda/\nu^{j}})}{[p_{j}]_{q}!}=\prod_{i=1}^{p_{j}}(1-q^{i})\cdot\widetilde{H}_{\lambda/\nu^{j}}\cdot\frac{(1-q)^{p_{j}}}{\prod_{i=1}^{p_{j}}(1-q^{i})}={(1-q)^{p_{j}}}\cdot\widetilde{H}_{\lambda/\nu^{j}},$

where $p_{j}=|P_{\lambda/\nu^{j}}|$. Also, for each $T_{i}\subset T_{\nu}$,

(6.3)
$\frac{e_{q}^{\mathrm{inv}}(T_{i})}{[t_{i}]_{q}!}=\frac{q^{\mathrm{inv}(T_{i})}}{\prod_{v\in
T_{i}}[h(v)]_{q}}=\frac{(1-q)^{t_{i}}}{\prod_{v\in T_{i}}(1-q^{h(v)})}.$

Note that $T_{i}$ are naturally labeled, so $\mathrm{inv}(T_{i})=0$. Using
Proposition 2.12 and (6.3), we have

$e_{q}(P_{\lambda/\nu})=\frac{\prod_{i=1}^{n-1}(1-q^{i})}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}\widetilde{H}_{\lambda/\nu^{1}}\cdot\widetilde{H}_{\lambda/\nu^{2}}.$

By this equation and Lemma 6.3.

(6.4)
$e_{q}(P_{\lambda/\mu})={\prod_{i=1}^{n-1}(1-q^{i})}\sum_{\mu\to\nu}\frac{q^{n-\omega_{\mathrm{inv}}(u)}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}\cdot\widetilde{H}_{\lambda/\nu^{1}}\cdot\widetilde{H}_{\lambda/\nu^{2}}.$

By (6.2), we can show the sum on the RHS of (6.4) equals
$(1-q^{n})\cdot\widetilde{H}_{\lambda/\mu}$, completing the proof. ∎

###  6.2. Proof of Lemma 6.4

We first evaluate $F_{\lambda/\mu}(\mathbf{x},\mathbf{y})$ at
$x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}}$ and
$y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq j}p_{a,b}}$.

(6.5)
$\left.F_{\lambda/\mu}(\mathbf{x},\mathbf{y})\right.\mid_{\begin{subarray}{c}x_{i}=q^{\lambda_{i}-i+1-\sum_{a<i}p_{a,b}},\\\
y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq
j}p_{a,b}}\end{subarray}}=(-1)^{n}\sum_{\begin{subarray}{c}\gamma:A\to B,\\\
\gamma\subset\lambda\end{subarray}}\prod_{(i,j)\in\gamma}\frac{q^{\lambda_{j}^{\prime}-j+\sum_{b\geq
j}p_{a,b}}}{1-q^{h^{\prime}(i,j)}}$

By [10, Prop 4.7] and [10, Lemma 7.17], we have

$\displaystyle\sum_{(i,j)\in\lambda\setminus
D}\Big{(}(\lambda_{j}^{\prime}-j)+\sum_{b\geq j}p_{a,b}\Big{)}$
$\displaystyle=\sum_{(i,j)\in[\lambda]\setminus
D}\Big{(}(\lambda_{j}^{\prime}-i)+\sum_{b\geq
j}p_{a,b}\Big{)}-\sum_{(i,j)\in[\lambda]\setminus[\mu]}c(i,j)$ (6.6)
$\displaystyle=w(\mathrm{Br}(D))+\sum_{(i,j)\in[\lambda]\setminus
D}\sum_{b\geq j}p_{a,b}-\sum_{(i,j)\in[\lambda]\setminus[\mu]}c(i,j),$

where $c(i,j)=j-i$ and $w(\mathrm{Br}(D))=\sum_{(i,j)\in Br(D)}h(i,j)$. Note
that unlike the case of major index, we do not include the size of the rooted
trees in to $w(\mathrm{Br}(D))$ (see (4.14)) .

Denote $\tilde{p}_{\lambda/\mu}:=\sum_{(i,j)\in[\lambda]/[\mu]}\sum_{b\geq
j}p_{a,b}$. For each $D\in\mathcal{E}(\lambda/\mu)$, we have

$\sum_{(i,j)\in[\lambda]\setminus D}\sum_{b\geq
j}p_{a,b}-\tilde{p}_{\lambda/\mu}=\sum_{(i,j)\in[\mu]/D}\sum_{b=j}p_{a,b}=p_{D}.$

Then $\tilde{p}_{\lambda/\mu}$ and $c(i,j)$ from (6.6) do not depend on $D$,
so we can take them outside of the sum to rewrite (6.5) as:

(6.7)
$\left.F_{\lambda/\mu}(\mathbf{x}|\mathbf{y})\right.\mid_{\begin{subarray}{c}x_{i}=q^{\lambda^{\prime}_{1}-i+1-\sum_{a<i}p_{a,b}},\\\
y_{j}=q^{j-\lambda^{\prime}_{j}-\sum_{b\geq
j}p_{a,b}}\end{subarray}}=(-1)^{n}\cdot
q^{\tilde{p}_{\lambda/\mu}-\operatorname{con}(\lambda/\mu)}\prod_{v\in\mathbf{p}}(1-q^{h(v)})\cdot\widetilde{H}_{\lambda/\mu}$

Now as done for the case of major index in Section 4, we evaluate the
Pieri–Chevalley formula at such $x_{i}$ and $y_{j}$. Then applying (6.7) to
(2.2), and simplifying everything as we did in the case of major index, we
have

(6.8)
$(1-q^{n})\widetilde{H}_{\lambda/\mu}=\sum_{\mu\to\nu}\frac{q^{\lambda_{1}-1+c(u)+\sum_{b\geq
1}p_{a,b}+\tilde{p}_{\lambda/\nu_{1}}+\tilde{p}_{\lambda/\nu_{2}}-\tilde{p}_{\lambda/\mu}}}{\prod_{v\in
T_{\nu}}(1-q^{h(v)})}\widetilde{H}_{\lambda/\nu_{1}}\widetilde{H}_{\lambda/\nu_{2}}.$

Note that from (4.20), this is equivalent to

$\sum_{b\geq
1}p_{a,b}+\tilde{p}_{\lambda/\nu_{1}}+\tilde{p}_{\lambda/\nu_{2}}-\tilde{p}_{\lambda/\mu}=\sum_{b\geq
1}p_{a,b}-\sum_{b\geq u_{2}}p_{a,b},$

where $u_{2}$ is the column of the the inner corner $u=(u_{1},u_{2})$. It is
left to show the following lemma to complete the proof.

######  Lemma 6.5.

Let $(P_{\lambda/\mu}(\mathbf{p}),\omega_{\mathrm{inv}})$ be a mobile tree
poset of size $n$ with a labeling $\omega_{\mathrm{inv}}$ and $u$ be the inner
corner for $\mu\to\nu$. Then we have,

$n-\omega_{\mathrm{inv}}(u)=\lambda^{\prime}_{1}-1+c(u)+\sum_{b\geq
1}p_{a,b}-\sum_{b\geq u_{2}}p_{a,b}.$

###### Proof.

First, we show that for a border strip $(P_{\lambda/\mu},\omega)$ of size
$n_{0}$ with a reversed Schur labeling we have
$n_{0}-\omega(x)=\lambda^{\prime}_{1}-1+c(x)$ for all $x\in P_{\lambda/\mu}$.
Note that in a border strip, there is only one element per content. Also, for
Schur labeling of a border strip, we have $\omega(\lambda_{1}^{\prime},1)=1$
and $\omega(1,\lambda_{1})=n_{0}$. The element $(\lambda_{1}^{\prime},1)$
satisfies the equation. Then as you follow the border strip, the content
decreases by one while the label increases by one, so the rest of the elements
satisfy the equation $n_{0}-\omega(x)=\lambda^{\prime}_{1}-1+c(x)$.

Now for a labeled mobile tree poset
$(P_{\lambda/\mu}(\mathbf{p}),\omega_{\mathrm{inv}})$ of size $n$, for any
$x\in P_{\lambda/\mu}(\mathbf{p})$, $\omega_{\mathrm{inv}}(x)$ gets shifted by
$\sum_{b\geq x_{2}}p_{a,b}$. Then
$\omega_{\mathrm{inv}}(x)=\omega(x)+\sum_{b\geq x_{2}}p_{a,b}$. Also, we have
that $n=n_{0}+\sum_{b\geq 1}p_{a,b}$. Then applying such shifts to the
equation obtained from a border strip, we have the desired equation. ∎

Then we can simplify (6.8), completing the proof of Lemma 6.4.

##  7\. Final remarks

###  7.1. Theorem 1.8 for border-strips

In [10] Morales, Pak, and Panova gave a proof of Theorem 1.5 using factorial
Schur functions. In [8] the same authors gave another proof of Theorem 1.5
reducing it to the case of border strips. The latter proof included an
analogue of Lemma 4.3 to border strips, but there was no explicit analogue of
Lemma 4.2. Instead they relied on an identity [8, Lemma 7.2] proved using
factorial Schur functions. Our Lemma 4.2 can be reduced to the case of border
strips as follows.

######  Corollary 7.1.

For a labeled border-strip poset $(Q_{\lambda/\mu},\omega)$, where $\omega$ is
a reversed Schur labeling,

$e_{q}^{\mathrm{maj}}(Q_{\lambda/\mu},\omega)=\sum_{\mu\to\nu}q^{|Q_{\lambda/\nu_{1}}|}e_{q}^{\mathrm{maj}}(Q_{\lambda/\nu},\omega_{\nu}),$

where $Q_{\lambda/\nu_{1}}$ is the left disconnected poset of
$Q_{\lambda/\nu}$, and $\omega_{\nu}$ is the restricted labeling of $\omega$
onto $Q_{\lambda/\nu}$.

Then using the Pieri–Chevalley formula ((2.2) proved in [8]) and Corollary
7.1, we obtain a proof of Theorem 1.5 for border strips without using
factorial Schur functions.

###  7.2. Bijective proof between maj and inv index for border strips

The inversion statistic analogue of Lemma 4.2 is Lemma 6.3. In the case of
border strips $Q_{\lambda/\mu}$, since $n-\omega(u)=|Q_{\lambda/\nu_{1}}|$, we
obtain the same recurrence as $e_{q}^{\mathrm{maj}}(Q_{\lambda/\mu})$ as in
Corollary 7.1. Thus, we obtain the following equation of the $q$-analogues for
border strips.

######  Corollary 7.2.

For a border strip $Q_{\lambda/\mu}$,

$e_{q}^{\mathrm{inv}}(Q_{\lambda/\mu},\omega)=e_{q}^{\mathrm{maj}}(Q_{\lambda/\mu},\omega),$

where $\omega$ is a reversed Schur labeling

This identity can also be proved bijectively using Foata’s classical bijection
on permutations, denoted by $\varphi$, defined as follows (see [16, Sec.
1.4]). Let $w=w_{1}\cdots w_{n}\in\mathfrak{S}_{n}$, and we define
$\gamma_{1},\dots\gamma_{n}$, where $\gamma_{k}$ is a permutation of
$\\{w_{1},\dots,w_{k}\\}$. Let $\gamma_{1}=w_{1}$. For each $k\geq 1$, if the
last letter of $\gamma_{k}$ is greater (respectively smaller) than $w_{k+1}$,
then split $\gamma_{k}$ after each letter greater (respectively smaller) than
$w_{k+1}$. To obtain $\gamma_{k+1}$, cyclically shift each compartment of
$\gamma_{k}$ to the right, then place $w_{k+1}$ at the end. We set
$\varphi(w)=\gamma_{n}$. We have the following theorem.

######  Theorem 7.3 (Foata [3]).

Let $\varphi$ be the Foata bijection. For all $w\in\mathfrak{S}_{n}$,

$\mathrm{Des}(w^{-1})=\mathrm{Des}(\varphi(w)^{-1}).$

Because Foata’s bijection preserves descent sets, we have the following
bijection between the major and inversion index.

######  Lemma 7.4.

Given a $\sigma\in\mathcal{L}(P,\omega)$, where $\omega$ is a (reversed) Schur
labeling, $\varphi(\sigma)$ is also in $\mathcal{L}(P,\omega)$, and
$\mathrm{maj}(\sigma)=\mathrm{inv}(\varphi(\sigma))$

More detailed information about the equidistribution of major and inversion
index in trees can be found in [1]. The situation for mobile posets is more
subtle since the $q$-analogues do not agree (see Example 4.1 and Example 6.1).

###  7.3. Mobiles of general skew shapes

The formula (NHLF) holds true for all posets coming from skew shapes, but the
combinatorial proof of the formula is restricted to the case of border strips.
Recall that a mobile is obtained by hanging $d$-complete posets from a border-
strip. It would be interesting to see if Theorem 1.7 holds for posets where
the border strip is replaced by general skew shape. Calculations suggest that
the Naruse formula (1.7) would need some adjustments.

On the other hand, we use a version of Pieri–Chevalley formula and the
recurrence for our proof. There is a version of the Pieri–Chevalley formula
for general skew shapes, shown algebraically by Ikeda and Naruse [6], and
combinatorically by Konvalinka [7].

###  7.4. Relation with Naruse-Okada hook-length formula

Naruse-Okada [12] have a different $q$-analogue of
$e_{q}^{\mathrm{maj}}(P,\omega)$ for a family called _skew d-complete_ posets,
which intersects with the family of mobile posets [5, Section 6.1].

######  Definition 7.5.

[12] A _skew $d$-complete poset_ is a $d$-complete poset $P$ with an order
filter $I$ removed. We denote such a poset by $P\setminus I$.

The Naruse-Okada formula for counting linear extensions of skew $d$-complete
posets uses the hook-length of _excited peaks_ (see [10, Section 6] and
[12])instead of broken diagonals.

######  Theorem 7.6 (Naruse-Okada [11]).

Let $P\setminus I$ be a skew $d$-complete poset with $n$ elements. Then

$e_{q}^{\mathrm{maj}}(P\setminus
I)=\prod_{i=1}^{n}(1-q^{i})\sum_{D\in\mathcal{E}(P\setminus
I)}\frac{\prod_{v\in B(D)}q^{h(v)}}{\prod_{v\in P\setminus D}(1-q^{h(v)})},$

where $h(v)$ is the hook length of element $v$ in $P\setminus I$ and $B(D)$ is
a set of excited peaks of $D$.

For posets that are both mobiles and skew $d$-complete, the notion of hook-
lengths are the same (see Figure 8). Then for such posets Theorem 7.6 at $q=1$
and Corollary 1.8 agree.

However, the $q$-analogues in Theorem 1.8 and Theorem 7.6 are different (see
Example 7.7). This is because the NHLF formula for skew $d$-complete posets
uses the natural labeling of the poset as opposed to the reversed Schur
labeling. For the case of skew shapes, their $q$-analogue agrees with the
reverse plane partition $q$-analogue of the Naruse formula in (see [10]
Corollary 6.17) instead of SSYT $q$-analogue, which uses the Schur labeling
(see Figure 8).

(a)

(b)

(c)

Figure 8. A skew $d$-complete poset $P\setminus I$ that is also a mobile
labeled using (a) the Schur labeling $\omega^{\prime}$ and (b) the natural
labeling $\omega$. (c) The hook lengths of the elements in the poset.

######  Example 7.7.

Consider the poset $Q=P\setminus I$ in Figure 8 that is both a mobile poset
and a skew $d$-complete poset [5, Ex. 6.3]. If we label it using the reversed
Schur labeling on the border strip and natural labeling on the $d$-complete
posets, then by Theorem 1.7, we have

$\displaystyle e_{q}^{\mathrm{maj}}(Q,\omega^{\prime})$
$\displaystyle=q^{11}+2q^{10}+3q^{9}+3q^{8}+3q^{7}+2q^{6}+1q^{5}+q^{4}$
$\displaystyle=[6]!\left(\frac{q^{4}}{[1][1][2][2][3][5]}+\frac{q^{7}}{[1][1][2][3][5][6]}\right).$

Now, label the same skew $d$-complete poset $Q=P\setminus I$ using the natural
labeling. Then, by Naruse–Okada formula (Theorem 7.6), we have

$\displaystyle e_{q}^{\mathrm{maj}}(Q,\omega)$
$\displaystyle=q^{9}+q^{8}+2q^{7}+2q^{6}+2q^{5}+2q^{4}+2q^{3}+2q^{3}+2q^{2}+q+1$
$\displaystyle=[6]!\left(\frac{q^{0}}{[1][1][2][2][3][5]}+\frac{q^{6}}{[1][1][2][3][5][6]}\right).$

It would be interesting to see if one can give a proof of Theorem 7.6 using
the technique from [8]. This would involve proving a variation of Lemma 4.2
where $\omega$ is a natural labeling instead of reversed Schur labeling.

## Acknowledgement

I would like to thank Alejandro Morales for introducing this problem and for
all of his guidance throughout this project. I would also like to thank Stefan
Grosser, Jacob Matherne, and Soichi Okada for helpful comments.

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# The Orbital Structure and Selection Effects of the Galactic Center S-Star
Cluster

A. Burkert  University Observatory Munich (USM)
Scheinerstrasse 1, 81679 Munich, Germany  Max-Planck-Institut für
extraterrestrische Physik (MPE)
Giessenbachstr. 1, 85748 Garching, Germany  S. Gillessen  Max-Planck-Institut
für extraterrestrische Physik (MPE)
Giessenbachstr. 1, 85748 Garching, Germany  D. N. C. Lin  Department of
Astronomy and Astrophysics, University of California
Santa Cruz, USA  Institute for Advanced Studies, Tsinghua University
Beijing, China  X. Zheng  Beijing Planetarium, Beijing Academy of Science and
Technology
Beijing, China  P. Schoeller  ORIGINS Excellence Cluster
Boltzmannstr. 2, 85748 Garching, Germany  F. Eisenhauer  Max-Planck-Institut
für extraterrestrische Physik (MPE)
Giessenbachstr. 1, 85748 Garching, Germany  R. Genzel  Max-Planck-Institut für
extraterrestrische Physik (MPE)
Giessenbachstr. 1, 85748 Garching, Germany A. Burkert<EMAIL_ADDRESS>
###### Abstract

The orbital distribution of the S-star cluster surrounding the supermassive
black hole in the center of the Milky Way is analyzed. A tight, roughly
exponential dependence of the pericenter distance rp on orbital eccentricity
e⋆ is found, $\log($r${}_{p})\sim$(1-e⋆), which cannot be explained simply by
a random distribution of semi-major axis and eccentricities. No stars are
found in the region with high e⋆ and large log rp or in the region with low e⋆
and small log rp. G-clouds follow the same correlation. The likelihood P(log
rp,(1-e⋆)) to determine the orbital parameters of S-stars is determined. P is
very small for stars with large e⋆ and large log rp. S-stars might exist in
this region. To determine their orbital parameters, one however needs
observations over a longer time period. On the other hand, if stars would
exist in the region of low log rp and small e⋆, their orbital parameters
should by now have been determined. That this region is unpopulated therefore
indicates that no S-stars exist with these orbital characteristics, providing
constraints for their formation. We call this region, defined by $\log$
(rp/AU) $<$ 1.57+2.6(1-e${}_{\star})$, the zone of avoidance. Finally, it is
shown that the observed frequency of eccentricities and pericenter distances
is consistent with a random sampling of log rp and e⋆. However, only if one
takes into account that no stars exist in the zone of avoidance and that
orbital parameters cannot yet be determined for stars with large rp and large
e⋆.

Galaxy: center – Galaxy: nucleus – Galaxy: kinematics and dynamics – Stars:
massive

##  1 Introduction

The galactic center hosts one of the most mysterious structures of the Milky
Way, the S-star cluster (SSC; see Genzel et al. (2010) for a review). This
spheroidal ensemble of at least 50 young, preferentially B-type stars with
masses in the range of 8-20 M⊙ and ages of less than 15 Myrs (Gillessen et
al., 2017; Habibi et al., 2017) is confined to a region of 0.04 pc from Sgr
A∗(Genzel et al., 2003; Ghez et al., 2003a; Gillessen et al., 2009). Their
eccentric to nearly parabolic orbits have been reconstructed based on proper
motion measurements (Genzel et al., 1997, 2010; Ghez et al., 1998, 2003b,
2008; Gillessen et al., 2009; Schödel et al., 2009; Boehle et al., 2016). This
kinematic data not only confirms the existence of a super-massive black hole
(SMBH) with a mass $M_{\rm SMBH}=4.297\times 10^{6}M_{\odot}$ at the Galactic
Center (Genzel et al., 1997; Ghez et al., 1998; Schödel et al., 2002; Ghez et
al., 2005; Eisenhauer et al., 2005). It also provides valuable clues on the
dynamical evolution of these recently formed stars in its proximity (Levin &
Beloborodov, 2003).

The origin of the SSC is puzzling as single stars typically condense out of
molecular cloud cores with radii of order 0.1 pc (Hester et al., 1996; Mellema
et al., 2006; Forbes et al., 2021). This is larger than the whole SSC.
Therefore, the S-stars formed under very exotic conditions (Goodman, 2003;
Thompson et al., 2005; Nayakshin et al., 2007; Levin, 2007) or were
transported into this tiny, hostile environment (Syer et al., 1991; Artymowicz
et al., 1993; Davies & Lin, 2020) of the central SMBH. If the S-stars formed
in situ, some violent star formation event must have happened very close to
the SMBH a few of Myrs ago (Morris, 1993; Ghez et al., 2003a). Currently, the
S-stars are embedded in a 106K hot, diffuse gas bubble (Xu et al., 2006;
Murchikova et al., 2019) which is not a favorite condition for star formation.
One should, however, note that not all of the gas is hot. A cluster of cold
G-clouds with temperatures of order 104K has been discovered in the same
region as the SSC (Gillessen et al., 2012, 2013; Pfuhl et al., 2015; Plewa et
al., 2017). These gas clumps have very small masses, of order several Earth
masses. Their nature and origin are still a matter of debate (Burkert et al.,
2012; Murray-Clay & Loeb, 2012; Scoville & Burkert, 2013; Ballone et al.,
2013; Owen & Lin, 2023) and at the moment, it is not clear whether they are
related to the origin of the S-stars.

Today, nearly 200 massive stars have been detected in the Sgr A∗ proximity. In
addition to the SSC, $\sim 20\%$ of the known stars belong to a subgroup of
O/WR stars with top-heavy luminosity function and estimated age $\sim 6\pm 2$
Myr (Ghez et al., 2003a). They reside in a possibly-warped disk with clock-
wise rotation, a semi-major axis of $0.04{\rm pc}\lesssim a_{\star}\lesssim
0.5$ pc and an eccentricity of $e_{\star}\sim 0.2-0.4$, commonly referred to
as the clockwise disk (CWSs, Levin & Beloborodov (2003); Paumard et al.
(2006); Lu et al. (2006, 2009); Löckmann & Baumgardt (2009); Bartko et al.
(2009, 2010); Yelda et al. (2014)). The disk stars have an inner semi-major-
axis hole of 0.04 pc, corresponding to the sharp outer edge in the semi-major
axis distribution of the SSC. This coexistence and the very similar ages
provide evidence that the SSC and its surrounding disk population might have a
common origin. The remaining fraction of the young stars are the “off-disk”
stars (ODSs) with much less well-determined, probably-large eccentricities and
inclinations, as well as a non-isotropic distribution, surrounding the disk-
star population. Several over-dense concentrations in their orbital angular-
momentum orientation distribution have been identified, including the
controversial identification of a second disk, inclined by $\sim 110^{\circ}$
with respect to the clockwise disk (Paumard et al., 2006; Lu et al., 2009;
Yelda et al., 2014; Ali et al., 2020). The luminosity function of the “off-
disk” stars is similar to that of the S stars and less top-heavy than the disk
stars (von Fellenberg et al., 2022). There are suggestions that two disks of
massive stars formed 6 Myrs ago, orbiting the central SMBH from one or more
infalling and colliding clouds (Bonnell & Rice, 2008; Hobbs & Nayakshin, 2009;
Alig et al., 2013). It is also possible that secular perturbation by an
intermediate-mass companion (Zheng et al., 2020, 2021; GRAVITY Collaboration
et al., 2023) and dynamical relaxation (Rauch & Tremaine, 1996; Freitag et
al., 2006; Alexander et al., 2007; Madigan et al., 2011; Kocsis & Tremaine,
2011, 2015) heated a common-natal disk, generating all the diverse kinematic
properties in each subgroup, including the spheroidal SSC. We will discuss
this scenario in greater detail in a companion paper (Zheng et al. in
preparation).

Another, maybe complementary scenario for the origin of the S-stars is Hill’s
mechanism, whereby massive stellar binaries, generated in the larger
environment of the SMBH were deflected by some process onto an orbit that
brought them so close to the SMBH that they broke up (Hills, 1975, 1988; Gould
& Quillen, 2003; Yu & Tremaine, 2003; Ginsburg & Loeb, 2006; Amaro-Seoane et
al., 2012). Typically the more massive star will lose orbital energy and go
into a tight, bound orbit around the SMBH while the other star gains energy
and might even be ejected as a high-velocity star or a hyper-velocity star
that is unbound to the Milky Way. Stars with such high velocities have been
detected in the galactic halo. Observations indicate that these stars have
their origin in the galactic center, consistent with this theory. One caveat
of the Hills mechanism is, however, the fact that it alone can only explain
the current orbits of the most bound S-stars. One would still need secular
evolution scatter processes to account for the S-stars at larger distances. In
addition, this mechanism does not provide an answer to the question of why
almost all S-stars are B-stars, whereas Hill’s mechanism should also work
equally well for O-stars.

The size and geometry of the SSC provide insight into its origin. More
detailed information is, however, available from measurements of the orbital
distribution of its stars. By now, we have up to 25 yrs of uninterrupted
monitoring of stellar orbits, and the number of stars with known orbits
(Schödel et al., 2002; Ghez et al., 2003a) has increased to $\sim$ 50
(Eisenhauer et al., 2005; Ghez et al., 2005; Gillessen et al., 2009, 2017).
One of the most prominent members of the SSC is S2 which passed its first
observed pericenter in 2002 and again in 2018, providing the most precise
measurement of the black hole mass and allowing for the first time to detect
gravitational redshift (GRAVITY Collaboration et al., 2018; Do et al., 2019)
and Schwarzschild precession (GRAVITY Collaboration et al., 2020), in
excellent agreement with the predictions by general relativity. In order to
understand the formation of the SSC, one however needs to investigate the
physical property of the whole cluster, which requires a detailed
understanding of biasing and selection effects.

In this paper, we focus on the distribution of the orbital parameters of the
SSC. In section 2, we demonstrate that the S-stars follow a tight correlation
between their orbital eccentricity e⋆ and their pericenter distance rp.
Interestingly the G-clouds follow the same distribution, which might indicate
a common origin. In section 3, we show that this correlation differs strongly
from a thermal distribution which is usually adopted to characterize the SSC
or other theoretical models that explain the SSC as the result of secular
heating of a previous disk of stars. We also demonstrate that the lack of
stars with high e⋆ and large rp can be understood as a result of selection
effects. Stars could indeed exist in this regime. One would, however, need
observations on a longer timeline to determine their orbital parameters. The
zone of avoidance, on the other hand, is real. It is defined by a sharp edge
of rp versus e⋆, below which no S-stars exist. A summary follows in section 4.

##  2 The correlation between orbital eccentricity and pericenter distance

Figure 1: Pericenter distances rp of S-stars (red points) and G-clouds (cyan
triangles) versus their orbital eccentricities e. More circular orbits,
corresponding to larger values of (1-e), have larger pericenter distances.
Dashed lines show the expected dependence of rp on (1-e) for semi-major axi of
$2\times 10^{3}$AU and $2\times 10^{4}$AU, respectively. Stars and clouds with
larger rp, on average, have larger values of a than objects with smaller rp. A
random distribution of semi-major axi, independent of e, therefore cannot
explain the data.

The red points with error bars in figure 1 show the logarithm of rp versus e⋆
for the S-stars, compiled by Gillessen et al. (2017). The pericenter increases
roughly exponentially with increasing (1-e⋆). In addition to the stars, cyan
triangles mark the currently known G-gas clouds (Gillessen et al., 2012;
Eckart et al., 2013; Plewa et al., 2017; Ciurlo et al., 2020; Peißker et al.,
2020) which follow the same correlation closely. This might indicate a common
origin of S-stars and G-clouds or a common secular redistribution process that
generated the currently observed configuration. Note however that scattering
processes should be more disruptive for the G-clouds if they are diffuse,
pressure-confined gas clumps, with negligible self-gravity (Burkert et al.,
2012).

One might expect a dependence of rp on e⋆ if all the stars have a similar
semi-major axis, and a spread in e⋆, as for given a⋆ the pericenter distance
rp = a⋆ $\times$ (1-e⋆) decreases with increasing e⋆. In this case, rp should
however increase linearly with (1-e⋆), which is different from the
approximately exponential correlation, seen in figure 1. The two dashed black
lines depict the expected correlation for a semi-major axis of a⋆=$2\times
10^{3}$ AU and a⋆=$2\times 10^{4}$ AU, respectively. One can see that the more
circular the orbits, the larger the semi-major axis. This is demonstrated more
clearly in figure 2, which shows the eccentricity dependence of a⋆ (left
panel) and of the apocenter distance ra=a⋆ $\times$ (1+e⋆). Both quantities
are not randomly distributed but depend on (1-e⋆). We, therefore, can conclude
that the dependence of pericenter distance on eccentricity is not just a
result of a universal semi-major axis, independent of eccentricity.

Figure 2: Semi-major axis a⋆ (left panel) and apocenter distance ra of S-stars
versus their orbital eccentricities e⋆. More circular orbits have larger semi-
major axis and larger apocenter distances.

##  3 Selection effects

One possible explanation of the correlations seen in figure 1 and figure 2 is
selection effects. It might, for example, be harder to determine the complete
orbital parameters of stars in the upper left and lower right corners of
figure 1. In order to test this conjecture, we have performed a detailed Monte
Carlo study. We start by focusing on a pair of rp and (1-e⋆). Stars are then
generated with this set of orbital parameters on randomly oriented orbital
planes. Each star is placed somewhere on its orbit with the starting point at
time t0 chosen randomly between zero and the orbital period. The star is then
advanced by 20 years which corresponds to the average observational period and
its projected position (xi,yi) at time ti=t0+(i-1)$\Delta$t in the x-y plane
is recorded with $\Delta$t = 1 yr. Here we assume that the z-axis represents
the line of sight to the observer. Next, a Gaussian random error of 10AU is
added to each position which is the expected mean observational error. From
this, we determine the time dependence of the distance to the central black
hole ri(ti)=$\sqrt{x_{i}^{2}+y_{i}^{2}}$. A second-order regression line
r(t)=a+b$\times$t+c$\times$t2 is now fitted through the 21 data points ri(ti).
Repeating this procedure a large number of times allows us to determine the
mean value $\langle$c$\rangle$ and its error $\delta c$. It is this curvature
$\langle$c$\rangle$ in the orbit that is required in order to determine the
full orbital parameters of a star. We now follow the selection criterion of
Gillessen et al. (2017) and assume that the orbital parameters are known if
$\delta c$/$\langle$c$\rangle\leq 10^{-3}$. The fraction of cases that fulfill
this requirement then defines the probability P to determine the orbital
parameters for a given of rp and (1-e⋆). For P=1, orbital parameters can be
determined for all configurations. The smaller P, the less frequently can
these parameters be specified. Regions with small values of P should not be
populated in figure 1, even if stars exist on these orbits, because the
curvature c and by this, the orbital parameters rp and e⋆ cannot be determined
reliably for a large fraction of orbital configurations.

We find that, to good approximation, the probability P to measure the orbital
parameters of a star with pericenter distance rp in units of AU and orbital
eccentricity e⋆ is given by the following fit formula

$logP=-2+\left(\frac{-q_{2}+\sqrt{q_{2}^{2}-4q_{1}q_{3}}}{2q_{1}}\right)$ (1)

with

$\displaystyle q_{1}=0.095\times x$ (2) $\displaystyle q_{2}=1.266-0.95\times
x-0.075\times x\times\ln x$ $\displaystyle
q_{3}=\ln(r_{p}/AU)-11.4+0.855\times x+0.075\times x\times\ln x-\ln x$

and x=(1-e⋆).

Figure 3: Dotted lines show contours of constant probability P (see labels) to
determine the orbits of S-stars within 20 years of observations. It should be
possible to determine the orbital parameters for all stars with pericenter
distances below the solid black line, labeled 1.0. Red points with error bars
show those S-stars for which orbital parameters have been determined. Cyan
triangles show G-clouds. The dashed black line marks the upper boundary of the
zone of avoidance within which no stars with known orbital parameters are
found, despite the fact that it should be easy to determine their orbital
parameters. For reference, the solid lines in the zone of avoidance show
positions of constant semi-major axi a⋆.

The dotted lines in figure 3 show contours of constant P. Above a given
contour, the probability is smaller than the corresponding value. Red points
show the S-stars with known orbital parameters. One can see that the
unpopulated upper left region with high eccentricities and high rp is
characterized by very small probabilities of a few percent or lower. Indeed,
almost no stars lie in this region. And those few stars that are found there
have large error bars. We, therefore, cannot rule out that this region is
populated with S-stars. However, determining their orbital parameters will
require a longer epoch of observations and/or higher observational precision.
On the other hand, the orbital parameters of S-stars in the empty lower right
triangular region below the dashed line should by now have been measured. We
call this region the zone of avoidance. It is characterized by

$\log(r_{p}/AU)\leq 1.57+2.6(1-e_{\star})$ (3)

with the dashed line showing its upper boundary. That this zone of avoidance
is not populated by stars therefore indicates that no S-stars exist with these
combinations of orbital parameters. The zone of avoidance overlaps with the
P=1 line (solid black line) close to e⋆=0. The orbits of stars on circular
orbits above the zone are, therefore, harder to determine as the probability
to measure their parameters within on average 20 years is small. This might
account for the fact that not many stars with eccentricities close to unity
have been reliably detected yet.

Figure 4: Number of S-stars with known orbital parameters and with (1-e⋆) less
or equal to a given value. The red points show the observations. The cyan
points with error bars correspond to a homogeneous distribution in log rp and
e⋆. Yellow points depict a Rayleigh distribution with $\sigma$=0.6 (see
equation 4). An isothermal distribution with P(e⋆)$\sim$e⋆ is shown by the
open points. In all cases, stars were only taken if they lie above the zone of
avoidance. In addition, the stars were selected considering the probability of
determining their orbits.

To more quantitatively evaluate the effect of the detection probability on the
distribution of observed orbital parameters, figure 4 shows a Monte-Carlo test
where we randomly draw values of e⋆ in the regime of 0 and 1 and values of
log(r${}_{p}/$AU) within a range of 1 and 5. We then discard this set of
parameters if it lies in the zone of avoidance (equation 3). Otherwise, we
accept the set with a probability as given by equation 2. Note that this
procedure makes the result insensitive to the exact values of the upper and
lower boundary of the adopted log(rp) as long as the lower boundary lies
within the zone of avoidance and the upper boundary lies in the region where
the probability of determining the orbital parameters is small. As soon as 41
stars have been selected this way, we determine their cumulative eccentricity
distribution. We repeat this procedure 1000 times. The cyan points in figure 4
show the mean of the cumulative e⋆ distribution with the error bars depicting
the 1$\sigma$ mean square variation. The distribution within the error bars is
very similar to the observed distribution of red points. This also indicates
that a lot of stars could hide in the upper left corner at high pericenter
distances and large eccentricities. We have repeated this test, adopting a
Rayleigh distribution which is characterized by the probability

$P(e_{\star})=\left(\frac{e_{\star}}{\sigma}\right)\times\exp\left[0.5\times(1-\left(\frac{e_{\star}}{\sigma}\right)^{2})\right]$
(4)

The best fit in this case corresponds to $\sigma=0.6$ (yellow points). Despite
this additional free parameter it cannot fit the data as well as the random
distribution. Finally, the open points show an isothermal distribution
P(e⋆)$\sim$e⋆ which cannot fit the data at all.

##  4 Summary

We demonstrate that the Galactic center S-star cluster shows a narrow,
exponential dependence of pericenter distance rp on (1-e⋆), with e⋆ as the
orbital eccentricity. The G-clouds follow the same correlation, indicating
that they might be of similar origin or have experienced a similar secular
evolution. If scattering mechanisms were important, the fact that the G-clouds
were not disrupted would indicate that they might have an invisible, compact
central source (Scoville & Burkert, 2013; Peißker et al., 2021). We then
derive the likelihood to determine orbital parameters for S-stars
observationally and show that small values of (1-e⋆), combined with large
pericenter distances rp require a larger observational time span than what is
available at the moment. This explains why no stars have yet been found in
this region. It might be populated with yet undetected S-stars. On the other
hand, the orbital parameters of stars characterized by $\log$ (rp/AU) $<$
1.57+2.6(1-e⋆) should be easy to determine. That no stars exist in this zone
of avoidance indicates that whatever dynamical processes shaped the S-star
cluster kinematics did not favor this region. Considering the observational
biases and the zone of avoidance, we find that the S-stars follow a random
sampling of $\log$ rp with (1-e⋆). Our results could provide valuable
constraints for theoretical models of the origin of the S-star cluster and its
kinematics. For example, we will show in an accompanying paper that this
correlation arises naturally from the perturbation of an intermediate-mass
companion (Zheng et al. in preparation) without invoking any tidal-disruption-
of-binary hypothesis (Hills, 1988; Chu et al., 2023).

This work was supported by the Excellence Cluster ORIGINS which is funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - EXC-2094 - 390783311.

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