Gamma positivity of the Excedance based Eulerian polynomial in positive elements of Classical Weyl Groups

TL;DR

This paper investigates gamma positivity properties of Eulerian polynomials associated with positive elements in classical Weyl groups, revealing conditions under which these polynomials are gamma positive and expressing some as sums of gamma positive polynomials.

Contribution

It establishes gamma positivity for positive Eulerian polynomials in Weyl groups, including type D, and for sums over derangements, extending known results to new algebraic structures.

Findings

Gamma positivity holds for certain n and parity conditions.

Positive Eulerian polynomials can be expressed as sums of gamma positive polynomials.

Gamma positivity is shown for polynomials summed over conjugacy classes and derangements.

Abstract

The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that $\mathsf{AExc^{+}}_n(t)$ is gamma positive iff $n \geq 5$ and $n \equiv 1$ (mod 2). When $n \geq 4$, and $n \equiv 0$ (mod 2) we show that $\mathsf{AExc^{+}}_n(t)$ can be written as a sum of two gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials. Finally, we show gamma positivity results when we sum excedances over derangements with positive and negative sign. Our main resuls is that the polynomial obtained by summing excedance over a conjugacy class indexed by $\lambda$ is gamma positive.

Equations

$${\mathsf{AExc}}_n(t)$$

$${\mathsf{AExc}}_n^{+}(t)$$

$${\mathsf{AExc}}_n^{-}(t)$$

$$A_n(p,q,t)=\sum _{\pi \in {\mathfrak{S}}_n}{p}^{2-13(\pi )}{q}^{31-2(\pi )}{t}^{\mathsf{des}(\pi )}=\sum _{i=0}^{\lfloor n/2\rfloor }{a}_{n,i}(p,q)t^i(1+t{)}^{n-2i}$$

$${\mathsf{AExc}}_n(p,q,t)=\sum _{\pi \in {\mathfrak{S}}_n}{p}^{{\mathsf{des}}^{\ast }(\pi )}{q}^{\mathsf{maj}(\pi )-\mathsf{exc}(\pi )}{t}^{\mathsf{exc}(\pi )}=\sum _{i=0}^{\lfloor n/2\rfloor }{\gamma }_{n,i}(p,q)t^i(1+t{)}^{n-2i}$$

$${\mathsf{ADerExc}}_n(t)$$

$${\mathsf{ADerExc}}_n^{+}(t)$$

$${\mathsf{ADerExc}}_n^{-}(t)$$

$$\sum _{\pi \in {\mathfrak{S}}_n^1}{t}^{\mathsf{des}(\pi )}{s}^{\mathsf{asc}(\pi )}=\sum _{\pi \in {\mathfrak{S}}_n^{n-1}}{t}^{\mathsf{exc}(\pi )}{s}^{\mathsf{nexc}(\pi )-1}$$

$$\sum _{\pi \in {\mathcal{A}}_n^r}{t}^{\mathsf{exc}(\pi )}{s}^{\mathsf{nexc}(\pi )-1}=\sum _{\pi \in {\mathfrak{S}}_n^r-{\mathcal{A}}_n^r}{t}^{\mathsf{exc}(\pi )}{s}^{\mathsf{nexc}(\pi )-1}=\frac{1}{2}\sum _{\pi \in {\mathfrak{S}}_n^r}{t}^{\mathsf{exc}(\pi )}{s}^{\mathsf{nexc}(\pi )-1}$$

$${\mathsf{AExc}}_n^{+}(s,t)$$

$${\mathsf{AExc}}_n^{-}(s,t)$$

$$\sum _{\pi \in {\mathfrak{S}}_n}{t}^{\mathsf{des}(\pi )}{s}^{\mathsf{asc}(\pi )}$$

$$\sum _{\pi \in {\mathfrak{S}}_n^n}{t}^{\mathsf{des}(\pi )}{s}^{\mathsf{asc}(\pi )}$$

$$\sum _{r=2}^{n-1}\sum _{\pi \in {\mathfrak{S}}_n^r}{t}^{\mathsf{des}(\pi )}{s}^{\mathsf{asc}(\pi )}=\sum _{r=1}^{n-2}\sum _{\pi \in {\mathfrak{S}}_n^r}{t}^{\mathsf{exc}(\pi )}{s}^{\mathsf{nexc}(\pi )-1}$$

$${\mathsf{AExc}}_n^{+}(s,t)$$

$$\sum _{\pi \in {\mathfrak{S}}_n}(-1{)}^{\mathsf{inv}(\pi )}{t}^{\mathsf{exc}(\pi )}{s}^{\mathsf{nexc}(\pi )-1}$$

$$a_{n,k}^{+}=k{a}_{n-1,k}^{-}+(n-k)a_{n-1,k-1}^{-}+a_{n-1,k}^{+}\text{mbox}and{a}_{n,k}^{-}=k{a}_{n-1,k}^{+}+(n-k)a_{n-1,k-1}^{+}+a_{n-1,k}^{-}.$$

$${\mathsf{AExc}}_n^{+}(s,t)=\frac{1}{2}\left(A_n(s,t)+(s-t{)}^{n-1}\right)\text{mbox}and{\mathsf{AExc}}_n^{-}(s,t)=\frac{1}{2}\left(A_n(s,t)-(s-t{)}^{n-1}\right).$$

$$D{\mathsf{AExc}}_n^{+}(s,t)$$

$${\mathsf{AExc}}_{n+4}^{+}(s,t)$$

$${\mathsf{AExc}}_{n+4}^{-}(s,t)$$

$${\mathsf{AExc}}_5^{+}(s,t)$$

$${\mathsf{AExc}}_5^{-}(s,t)$$

$${\mathsf{AExc}}_7^{+}(s,t)$$

$${\mathsf{AExc}}_7^{-}(s,t)$$

$${\mathsf{AExc}}_{2m+2}^{+}(s,t)$$

$${\mathsf{AExc}}_{2m+2}^{+}(t)$$

$$=$$

$${\mathsf{BExc}}_n(t)$$

$$B_n(t)$$

$$B_n^{+}(s,t)$$

$${\mathsf{BExc}}_n^{+}(t)$$

$${\mathsf{BExc}}_n^{-}(t)$$

$${\mathsf{SgnBExc}}_n(s,t)$$

$${\mathsf{SgnBExc}}_n(t)$$

$${\mathrm{SgnB}}_{\{a_1,a_2,\dots ,a_n\}}(s,t,u)=\sum _{\pi \in {\mathfrak{B}}_{\{a_1,a_2,\dots ,a_n\}}}(-1{)}^{{\mathsf{inv}}_B(\pi )}{t}^{{\mathsf{des}}_B(\pi )}{s}^{{\mathsf{asc}}_B(\pi )}{u}^{{\mathsf{pos}}_{a_n}(\pi )}$$

$$\sum _{\pi \in {\mathfrak{B}}_{\{a_1,a_2,\dots ,a_n\}},{\mathsf{pos}}_{a_n}(\pi )\ne n}(-1{)}^{{\mathsf{inv}}_B(\pi )}{t}^{{\mathsf{des}}_B(\pi )}{s}^{{\mathsf{asc}}_B(\pi )}{u}^{{\mathsf{pos}}_{a_n}(\pi )}=0$$

Full text

Gamma positivity of the Excedance based Eulerian

polynomial in positive elements of Classical Weyl Groups

Hiranya Kishore Dey

Department of Mathematics

Indian Institute of Technology, Bombay

Mumbai 400 076, India.

email: hkdey@math.iitb.ac.in

  

Sivaramakrishnan Sivasubramanian

Department of Mathematics

Indian Institute of Technology, Bombay

Mumbai 400 076, India.

email: krishnan@math.iitb.ac.in

1 Introduction

Let $f(t)\in\mathbb{Q}[t]$ be a degree $n$ univariate polynomial with $f(t)=\sum_{i=0}^{n}a_{i}t^{i}$ where $a_{i}\in\mathbb{Q}$ with $a_{n}\not=0$. Let $r$ be the least non-negative integer such that $a_{r}\not=0$. Define $\mathsf{len}(f)=n-r$. $f(t)$ is said to be palindromic if $a_{r+i}=a_{n-i}$ for $0\leq i\leq(n-r)/2$. Define the center of symmetry of $f(t)$ to be $\mathrm{\operatorname{c-o-s}}(f(t))=(n+r)/2$. Note that for a palindromic polynomial $f(t)$, its center of symmetry $\mathrm{\operatorname{c-o-s}}(f(t))$ could be half integral.

Let $\mathrm{PalindPoly}_{(n+r)/2}(t)$ denote the vector space of palindromic univariate polynomials $f(t)=\sum_{i=0}^{n}a_{i}t^{i}$ with $r$ being the least non-negative integer such that $a_{r}>0$ and with $\mathrm{\operatorname{c-o-s}}(f(t))=(n+r)/2$. Clearly, $\displaystyle\Gamma=\{t^{r+i}(1+t)^{n-r-2i}:0\leq i\leq\lfloor(n-r)/2\rfloor\}$ is a basis of $\mathrm{PalindPoly}_{(n+r)/2}(t)$. Thus, if $f(t)\in\mathrm{PalindPoly}_{(n+r)/2}(t)$, then we can write $f(t)=\sum_{i=0}^{\lfloor(n-r)/2\rfloor}\gamma_{n,i}t^{r+i}(1+t)^{n-2i}$. $f(t)$ is said to be “gamma positive” if $\gamma_{n,i}\geq 0$ for all $i$ (that is, if $f(t)$ has positive coefficients when expressed in the basis $\Gamma$).

For a positive integer $n$, let $[n]=\{1,2,\ldots,n\}$ and let $\mathfrak{S}_{n}$ be the set of permutations on $[n]$. For $\pi=\pi_{1},\pi_{2},\ldots,\pi_{n}\in\mathfrak{S}_{n}$, define its excedance set as $\mathsf{ExcSet}(\pi)=\{i\in[n]:\pi_{i}>i\}$ and its number of excedances as $\mathsf{exc}(\pi)=|\mathsf{ExcSet}(\pi)|$. Define its number of non-excedances as $\mathsf{nexc}(\pi)=|\{i\in[n]:\pi_{i}\leq i\}|$. For $\pi\in\mathfrak{S}_{n}$, define its number of inversions as $\mathsf{inv}(\pi)=|\{1\leq i<j\leq n:\pi_{i}>\pi_{j}\}|$. Let $\mathsf{Des\_Set}(\pi)=\{i\in[n-1]:\pi_{i}>\pi_{i+1}\}$ and $\mathsf{Asc\_Set}(\pi)=\{i\in[n-1]:\pi_{i}<\pi_{i+1}\}$ be its set of descents and ascents respectively. Let $\mathsf{des}(\pi)=|\mathsf{Des\_Set}(\pi)|$ be its number of descents and $\mathsf{asc}(\pi)=|\mathsf{Asc\_Set}(\pi)|$ be its number of ascents. The polynomials $A_{n}(t)=\sum_{\pi\in\mathfrak{S}_{n}}t^{\mathsf{des}(\pi)}$ are the classical Eulerian polynomials. Let $\mathcal{A}_{n}\subseteq\mathfrak{S}_{n}$ be the subset of even permutations. Define

[TABLE]

It is a well known result of MacMahon [9] that both descents and excedances are equidistributed over $\mathfrak{S}_{n}$. That is, for all non negative integers $n$, $A_{n}(t)=\mathsf{AExc}_{n}(t)$. It is well known that the Eulerian polynomials $\mathsf{AExc}_{n}(t)$ are palindromic (see Graham, Knuth and Patashnik [7]). Gamma positivity of $\mathsf{AExc}_{n}(t)$ was first proved by Foata and Schützenberger (see [5]). Foata and Strehl [6] later used a group action based proof which has been termed as “valley hopping” by Shapiro, Woan and Getu [14]. This approach gives a combinatorial interpretation for the gamma coefficients. Several refinements of the gamma positivity of $\mathsf{AExc}_{n}(t)$ are known when enumeration is done with respect to both descents $\mathsf{des}()$ and with respect to excedance $\mathsf{exc}()$. For two statistics $\operatorname{2-13},\operatorname{31-2}:\mathfrak{S}_{n}\rightarrow\mathbb{Z}_{\geq 0}$, Brändén [1] and Shin-and-Zeng [17, 18] have shown a $p,q$-refinement by showing that the polynomial

[TABLE]

where the $a_{n,i}(p,q)$’s are polynomials with positive coefficients. Similarly, Shareshian and Wachs [16] have shown for statistics $\mathsf{des}^{*},\mathsf{maj}:\mathfrak{S}_{n}\mapsto\mathbb{Z}_{\geq 0}$ that the polynomial

[TABLE]

where the $\gamma_{n,i}(p,q)$’s are polynomials with positive coefficients. Dey and Sivasubramanian had recently given gamma positivity results (see [4]) when one sums descents over $\mathcal{A}_{n}$. In this paper, we consider the case when we sum excedances over $\mathcal{A}_{n}$. Two of our main results in this paper are the following.

Theorem 1

For all positive integers $n\geq 5$ with $n\equiv 1$ (mod 2), $\mathsf{AExc}_{n}^{+}(s,t)$ and $\mathsf{AExc}_{n}^{-}(s,t)$ are gamma positive, with both polynomials having the same center of symmetry $(n-1)/2$.

Theorem 2

For all even positive integers $n=2m$ with $n\geq 4$, $\mathsf{AExc}_{n}^{+}(t)$ and $\mathsf{AExc}_{n}^{-}(t)$ can be written as a sum of two gamma positive polynomials whose centers of symmetry differ by one.

Similar results are known when we sum excedances over derangements. Let $\mathfrak{SD}_{n}=\{\pi\in\mathfrak{S}_{n}:\pi_{i}\not=i\mbox{ for all }i\}$ denote the set of derangements in $\mathfrak{S}_{n}$. Let $\mathfrak{SD}_{n}^{+}=\mathfrak{SD}_{n}\cap\mathcal{A}_{n}$ and $\mathfrak{SD}_{n}^{-}=\mathfrak{SD}_{n}\cap(\mathfrak{S}_{n}-\mathcal{A}_{n})$. Define

[TABLE]

The polynomial $\mathsf{ADerExc}_{n}(t)$ is known to be real rooted (see Zhang [22]) and easily seen to be palindromic. Hence it is gamma positive (see Petersens book [13, Page 82]). Shin and Zeng in [17, 18] have proved the following refinement with respect to the number of inversions $\mathsf{inv}(\pi)$, number of cycles $\mathsf{cyc}(\pi)$ and the nesting number $\mathsf{nest}(\pi)$.

Theorem 3** (Shin and Zeng)**

For all positive integers $n$, and for all choices $\mathsf{stat}(\pi)\in\{\mathsf{cyc}(\pi),\mathsf{inv}(\pi),\mathsf{nest}(\pi)\}$, the polynomial $\sum_{\pi\in\mathfrak{SD}_{n}}q^{\mathsf{stat}(\pi)}t^{\mathsf{exc}(\pi)}=\sum_{i=0}^{\lfloor n/2\rfloor}b_{n,i}(q)t^{i}(1+t)^{n-2i}$ where $b_{n,i}(q)$ is a polynomial with positive integral coefficients.

We prove the following refinement of Theorem 3 in this paper.

Theorem 4

For all positive integers $n$, and for all choices $\mathsf{stat}(\pi)\in\{\mathsf{cyc}(\pi),\mathsf{inv}(\pi)\}$, the polynomials $\sum_{\pi\in\mathfrak{SD}_{n}^{+}}q^{\mathsf{stat}(\pi)}t^{\mathsf{exc}(\pi)}=\sum_{i=0}^{\lfloor n/2\rfloor}b_{n,i}^{+}(q)t^{i}(1+t)^{n-2i}$ and $\sum_{\pi\in\mathfrak{SD}_{n}^{-}}q^{\mathsf{stat}(\pi)}t^{\mathsf{exc}(\pi)}=\sum_{i=0}^{\lfloor n/2\rfloor}b_{n,i}^{-}(q)t^{i}(1+t)^{n-2i}$ where both $b_{n,i}^{+}(q)$ and $b_{n,i}^{-}(q)$ are polynomials with positive integral coefficients.

We generalize our results to the case when excedances are summed up over the elements with positive sign in classical Weyl groups. Let $\mathfrak{B}_{n}$ denote the group of signed permutations on $T_{n}=\{-n,-(n-1),\ldots,-1,1,2,\ldots,n\}$, that is $\sigma\in\mathfrak{B}_{n}$ consists of all permutations of $T_{n}$ that satisfy $\sigma(-i)=-\sigma(i)$ for all $i\in[n]$. Similarly, let $\mathfrak{D}_{n}\subseteq\mathfrak{B}_{n}$ denote the subset consisting of those elements of $\mathfrak{B}_{n}$ which have an even number of negative entries. As both $\mathfrak{B}_{n}$ and $\mathfrak{D}_{n}$ are Coxeter groups, they have a natural notion of descent associated to them. Similar to the classical Eulerian polynomials, we thus have the Eulerian polynomials of type-B and type-D. There is a natural notion of length in these groups and we get results when restricted to elements with even length. Our results for Type-B Weyl groups are Theorems 24 and 25. Similarly for Type-D Weyl groups, our main result is Theorem 34.

2 Preliminaries on Gamma positive polynomials

Most of the gamma positive polynomials in this work have homogeneous bivariate counterparts with the following slightly more general definition of gamma positivity. Let $f(s,t)=\sum_{i=0}^{n}a_{i}s^{n-i}t^{i}$ be a homogenous bivariate polynomial with $a_{n}\not=0$ and let $r$ be the smallest non-negative integer such that $a_{r}\not=0$. As before, define$f(s,t)$ to be palindromic if $a_{r+i}=a_{n-i}$ for all $i$ and define $\mathrm{\operatorname{c-o-s}}(f(s,t))=(n+r)/2$.

Let $f(s,t)$ have $\mathrm{\operatorname{c-o-s}}(f(s,t))=n/2$. $f(s,t)$ is said to be bivariate gamma positive if $f(s,t)=\sum_{i=0}^{n/2}\gamma_{n,i}(st)^{i}(s+t)^{n-2i}$ with $\gamma_{n,i}\geq 0$ for all $i$. Clearly, if $f(s,t)$ is a bivariate gamma positive polynomial, then $f(t)=f(s,t)|_{s=1}$ is clearly a univariate gamma positive palindromic polynomial with the same center of symmetry. We need the following lemmas. Since all of them are from [4]), we omit proofs.

Lemma 5

Let $f_{1}(s,t)$ and $f_{2}(s,t)$ be two bivariate gamma positive polynomials with respective centers of symmetry $\mathrm{\operatorname{c-o-s}}(f_{1}(s,t))=r_{1}+(n-r_{1})/2$ and $\mathrm{\operatorname{c-o-s}}(f_{2}(s,t))=r_{2}+(n-r_{2})/2$. Then $f_{1}(s,t)f_{2}(s,t)$ is gamma positive with $\mathrm{\operatorname{c-o-s}}(f_{1}(s,t)f_{2}(s,t))=(r_{1}+r_{2})+(2n-r_{1}-r_{2})/2$.

Let $D$ be the operator $\displaystyle\left(\frac{d}{ds}+\frac{d}{dt}\right)$ in $\mathbb{Q}[s,t]$.

Lemma 6

Let $f(s,t)$ be a bivariate gamma positive polynomial with $\mathrm{\operatorname{c-o-s}}(f(s,t))=n/2$. Then, $g(s,t)=Df(s,t)$ is gamma positive with $\mathrm{\operatorname{c-o-s}}(g(s,t))=(n-1)/2$.

The following are easy corollaries.

Corollary 7

Let $f(s,t)$ be a bivariate gamma positive polynomial with $\mathrm{\operatorname{c-o-s}}(f(s,t))=n/2$.

Then, for a natural number $\ell$, $\displaystyle g(s,t)=D^{\ell}f(s,t)$ is gamma positive with $\mathrm{\operatorname{c-o-s}}(g(s,t))=(n-\ell)/2$. 2. 2.

Then, $h(s,t)=(st)^{i}f(s,t)$ is gamma positive with $\mathrm{\operatorname{c-o-s}}(h(s,t))=i+n/2$ 3. 3.

Then, $h(s,t)=(s+t)f(s,t)$ is gamma positive with $\mathrm{\operatorname{c-o-s}}(h(s,t))=(n+1)/2$

Lemma 8

Let $f(t)$ be gamma positive with $\mathrm{\operatorname{c-o-s}}(f(t))=n/2$ and with $\mathsf{len}(f(t))$ being odd. Then, $f(t)$ is the sum of two gamma positive sequences $p_{1}(t)$ and $p_{2}(t)$ with centers of symmetry $(n-1)/2$ and $(n+1)/2$ respectively. Further, both $p_{1}(t)$ and $p_{2}(t)$ have even length.

3 Type-A Coxeter Groups

Define $\mathsf{pos\_n}(\pi)$=index $i$ such that $\pi(i)=n$ and define $\pi^{\prime}$ to be $\pi$ restricted to $[n-1]$. Let $\mathfrak{S}_{n}^{i}=\{\pi\in\mathfrak{S}_{n}:\mathsf{pos\_n}(\pi)=i\}$, $\mathcal{A}_{n}^{i}=\{\pi\in\mathcal{A}_{n}:\mathsf{pos\_n}(\pi)=i\}$, and $\mathfrak{S}_{n}^{i}-\mathcal{A}_{n}^{i}=\{\pi\in\mathfrak{S}_{n}-\mathcal{A}_{n}:\mathsf{pos\_n}(\pi)=i\}$. We need Foata’s First Fundamental Transformation which is a bijection that maps descents to excedances (See Lothaire [8, Section 10.2]).

Theorem 9** (Foata’s First Fundamental Transformation)**

For all $n\geq 2$, there exist a bijection $\mathsf{FFT}_{n}:\mathfrak{S}_{n}\mapsto\mathfrak{S}_{n}$ such that $\mathsf{des}(\mathsf{FFT}_{n}(\pi))=\mathsf{exc}(\pi)$. Hence, the statistics $\mathsf{des}()$ and $\mathsf{exc}()$ are equidistributed over $\mathfrak{S}_{n}$ for all positive integers $n$.

We use the bijection $\mathsf{FFT}_{n}$ of Theorem 9 to prove the following.

Lemma 10

For all integers $n\geq 2$, the following holds:

[TABLE]

Proof:   We give a bijection $f_{n}:\mathfrak{S}_{n}^{n-1}\mapsto\mathfrak{S}_{n}^{1}$ as follows. For any $\pi=\pi_{1},\pi_{2},\ldots,\pi_{n-2},n,\pi_{n}\in\mathfrak{S}_{n}^{n-1}$, let $\mathsf{FFT}_{n-1}(\pi^{\prime})=a_{1},a_{2},\ldots,a_{n-1}$. Then, let $f_{n}(\pi)=n,a_{1},a_{2},\ldots,a_{n-1}$. Clearly, $f_{n}$ is well-defined and is a bijection. Further, for $\pi\in\mathfrak{S}_{n}^{n-1}$, it is easy to see that $\mathsf{exc}(\pi)=\mathsf{des}(f_{n}(\pi))$ and hence that $\mathsf{nexc}(\pi)=\mathsf{asc}(f_{n}(\pi))+1$.

Lemma 11

For all positive integers $n$ and $r$ with $1\leq r\leq n-2$, the following holds:

[TABLE]

Proof:   Define $g:\mathcal{A}_{n}^{r}\mapsto\mathfrak{S}_{n}^{r}-\mathcal{A}_{n}^{r}$ by $g(\pi_{1},\pi_{2},\ldots,\pi_{n-1},\pi_{n})=\pi_{1},\pi_{2},\ldots,\pi_{n},\pi_{n-1}$. Clearly, $\mathsf{inv}(\pi)\not\equiv\mathsf{inv}(g(\pi))$ (mod 2) and further $g^{2}=\mathsf{id}$. As $\mathsf{pos\_n}(\pi)\leq n-2$, we also have $\mathsf{exc}(\pi)=\mathsf{exc}(g(\pi))$, completing the proof.

Lemma 12

For all positive integers $n$, the polynomials $\mathsf{AExc}_{n}^{+}(s,t)$ and $\mathsf{AExc}_{n}^{-}(s,t)$ satisfy the following recurrence.

[TABLE]

Proof:   Foata’s First Fundamental bijection, applied on $\mathfrak{S}_{n}$ and $\mathfrak{S}_{n-1}$ gives us:

[TABLE]

Subtracting equations (7) and (11) from (10), we get

[TABLE]

We know

[TABLE]

The second line follows by using Lemma 11. The third line follows using (12). The last line follows using the fact that $stDA_{n-1}(s,t)$ is the contribution of all the permutations where $n$ is not in the initial or the final position. The proof of (9) is identical and is hence omitted.

A simple corollary of Lemma 12 is the following Theorem of Mantaci (see [11]).

Corollary 13

For all positive integers $n\geq 2$, $\sum_{\pi\in{\mathfrak{S}_{n}}}(-1)^{\mathsf{inv}(\pi)}t^{\mathsf{exc}(\pi)}s^{\mathsf{nexc}(\pi)-1}=(s-t)^{n-1}.$

Proof:   We induct on $n$ with the base case being $n=2$. When $n=2$, it is simple to note that $\sum_{\pi\in{\mathfrak{S}_{2}}}(-1)^{\mathsf{inv}(\pi)}t^{\mathsf{exc}(\pi)}s^{\mathsf{nexc}(\pi)-1}=s-t$. Subtracting (9) from (8) we get

[TABLE]

Let $\mathsf{AExc}_{n}^{+}(t)=\sum_{k=0}^{n-1}a_{n,k}^{+}t^{k}$ and let $\mathsf{AExc}_{n}^{-}(t)=\sum_{k=0}^{n-1}a_{n,k}^{-}t^{k}$. The following recurrences for the numbers $a_{n,k}^{+}$ and $a_{n,k}^{-}$ were proved by Mantaci in [11, 10]. It is easy to derive them from Lemma 12

Corollary 14

For all positive integers $n$ with $n\geq 2$, we have

[TABLE]

We move on to palindromicity of the polynomials $\mathsf{AExc}_{n}^{+}(s,t)$ and $\mathsf{AExc}_{n}^{-}(s,t)$.

Lemma 15

For all natural numbers $n$, $\mathsf{AExc}_{n}^{+}(s,t)$ and $\mathsf{AExc}_{n}^{-}(s,t)$ are palindromic if and only if $n\equiv 1\mod 2$.

Proof:   By Corollary 13,

[TABLE]

Clearly, $A_{n}(s,t)$ is palindromic for all $n$ while $(s-t)^{n-1}$ is palindromic if and only if $n$ is odd. Hence, $\mathsf{AExc}_{n}^{+}(s,t)$ and $\mathsf{AExc}_{n}^{-}(s,t)$ are palindromic iff $n\equiv 1\mod 2$.

Lemma 16

For all positive integers $n\geq 2$, $D\mathsf{AExc}_{n}^{+}(s,t)=D\mathsf{AExc}_{n}^{-}(s,t)=\frac{1}{2}DA_{n}(s,t).$

Proof:   We induct on $n$. Clearly, when $n=2$, $\mathsf{AExc}_{2}^{+}(s,t)=s$, $\mathsf{AExc}_{2}^{-}(s,t)=t$ while $A_{2}(s,t)=s+t$. Thus, $D\mathsf{AExc}_{2}^{+}(s,t)=D\mathsf{AExc}_{2}^{-}(s,t)=1=\frac{1}{2}DA_{2}(s,t.$ Assume, $D\mathsf{AExc}_{n-1}^{+}(s,t)=D\mathsf{AExc}_{n-1}^{-}(s,t)=\frac{1}{2}DA_{n-1}(s,t).$ Further,

[TABLE]

The proof is complete.

We use the recurrences (8) and (9) four times to get the following. Since this is the main idea of the proof, we omit some details.

Lemma 17

For all positive integers $n$, we have

[TABLE]

where the $L_{i}(s,t)$ with their centers of symmetry are as follows.

\begin{array}[]{l|r}f(s,t)&\mathrm{\operatorname{c-o-s}}(f(s,t))\\ \hline\cr L_{1}(s,t)=(s+t)^{4}+7st(s+t)^{2}+16(st)^{2}&2\\ \hline\cr L_{2}(s,t)=15st(s+t)^{2}&2\\ \hline\cr L_{3}(s,t)=3(5s^{2}+30st+5t^{2})st(s+t)&5/2\\ \hline\cr L_{4}(s,t)=25(st)^{2}(s+t)^{2}+20(st)^{3}&3\\ \hline\cr L_{5}(s,t)=10(st)^{3}(s+t)&7/2\\ \hline\cr L_{6}(s,t)=(st)^{4}&4\end{array}**

Proof:   (Of Theorem 1) When $n=5$ and $n=7$ one can check that

[TABLE]

Let $n$ be odd with $n>7$ and let $m=n-4$. By induction, both $\mathsf{AExc}_{m}^{+}(s,t)$ and $\mathsf{AExc}_{m}^{-}(s,t)$ are gamma positive with centers of symmetry $\frac{1}{2}(m-1)$. It is easy to check that all $L_{i}(s,t)$’s for all $1\leq i\leq 6$ that appear in Lemma 17 are gamma positive. Further, each of the six terms in (13) has the same center of symmetry $\frac{1}{2}(m+3)$. Thus by Lemma 17, $\mathsf{AExc}_{n}^{+}(s,t)$ is gamma positive. In an identical manner, one can prove that $\mathsf{AExc}_{n}^{-}(s,t)$ is gamma positive with center of symmetry $\frac{1}{2}(n-1)$, completing the proof.

3.1 When $n\equiv 0$ (mod 2)

When $n=2m$ for a natural number $m$, by Lemma 15, the polynomials $\mathsf{AExc}_{n}^{+}(t)$ and $\mathsf{AExc}_{n}^{-}(t)$ are not even palindromic.

Proof:   (Of Theorem 2) We induct on $n$. Our base case is when $n=2m=4$, one can check that $\mathsf{AExc}_{4}^{+}(t)=1+4t+7t^{2}=(1+4t+t^{2})+6t^{2}$ and $\mathsf{AExc}_{4}^{-}(t)=t^{3}+4t^{2}+7t=t(1+4t+t^{2})+6t$. By Lemma 12, we have

[TABLE]

The polynomial $p(s,t)=stD\mathsf{AExc}_{2m+1}(s,t)$ is gamma positive with $\mathrm{\operatorname{c-o-s}}(p(s,t))=m-1/2+1$. Hence, $p(t)=p(s,t)|_{s=1}$ is univariate gamma positive polynomial with $\mathrm{\operatorname{c-o-s}}(p(t))=m+1/2$. Further, it is simple to see that $p(t)$ has odd length and thus by Lemma 8, it can be written as $p(t)=p_{1}(t)+p_{2}(t)$ with $\mathrm{\operatorname{c-o-s}}(p_{1}(t))=m$ and $\mathrm{\operatorname{c-o-s}}(p_{2}(t))=m+1$. Thus, (15) becomes

[TABLE]

where $w_{1}(t)=\mathsf{AExc}_{2m+1}^{+}(t)+p_{1}(t)$ has $\mathrm{\operatorname{c-o-s}}(w_{1}(t))=m$ and $w_{2}(t)=t\mathsf{AExc}_{2m+1}^{-}(t)+p_{2}(t)$ has $\mathrm{\operatorname{c-o-s}}(w_{2}(t))=m+1$. The proof is complete.

4 Type B Coxeter Group

Let $\mathfrak{B}_{n}$ be the set of permutations $\pi$ of $\{-n,-(n-1),\ldots,-1,1,2,\ldots n\}$ satisfying $\pi(-i)=-\pi(i)$. We denote $-k$ as $\overline{k}$ as well. $\mathfrak{B}_{n}$ is referred to as the hyperoctahedral group or the group of signed permutations on $[n]$ and $|\mathfrak{B}_{n}|=2^{n}n!$. For $\pi\in\mathfrak{B}_{n}$ we alternatively denote $\pi(i)$ as $\pi_{i}$. For $\pi\in\mathfrak{B}_{n}$, define $\mathsf{Negs}(\pi)=\{\pi_{i}:i>0,\pi_{i}<0\}$ be the set of elements which occur with a negative sign. Define $\mathsf{inv}_{B}(\pi)=|\{1\leq i<j\leq n:\pi_{i}>\pi_{j}\}|+\sum_{i\in\mathsf{Negs}(\pi)}i$ and let $\mathfrak{B}^{+}_{n}\subseteq\mathfrak{B}_{n}$ denote the subset of elements having even $\mathsf{inv}_{B}()$ value and let $\mathfrak{B}_{n}^{-}=\mathfrak{B}_{n}-\mathfrak{B}_{n}^{+}$.

We follow Brenti’s[2] definition of excedance: $\mathsf{exc}_{B}(\pi)=|\{i\in[n]:\pi_{|\pi(i)|}>\pi_{i}\}|+|\{i\in[n]:\pi_{i}=-i\}|$ and $\mathsf{nexc}_{B}(\pi)=n-\mathsf{exc}_{B}(\pi)$. For all the permutations $\pi\in\mathfrak{B}_{n}$, let $\pi_{0}=0$. Define $\mathsf{des}_{B}(\pi)=|\{i\in[0,1,2\ldots,n-1]:\pi_{i}>\pi_{i+1}\}|$ and $\mathsf{asc}_{B}(\pi)=|\{i\in[0,1,2\ldots,n-1]:\pi_{i}<\pi_{i+1}\}|$. Define

[TABLE]

$B_{n}(s,t)$ is called the type-B Eulerian polynomial and Brenti in [2] proved that $B_{n}(s,t)=\mathsf{BExc}_{n}(s,t)$.

Theorem 18** (Brenti)**

For all positive integers $n$, there exists a bijection $h_{n}:\mathfrak{B}_{n}\mapsto\mathfrak{B}_{n}$ such that $\mathsf{asc}_{B}(h_{n}(\pi))=\mathsf{wkexc}_{B}(\pi)$ and $\mathsf{Negs}((h(_{n}\pi))=\mathsf{Negs}(\pi)$.

Gamma positivity of the type-B Eulerian polynomial was shown by Chow [3] and Petersen [12]. We refine these results to $\mathfrak{B}_{n}^{+}$ and $\mathfrak{B}_{n}^{-}$ when $n$ is even. We enumerate $B_{n}(t)$ with type-B inversions and find surprisingly that the signed type-B Eulerian polynomial is the same irrespective of whether we use descents or excedances (see Theorem 20).

4.1 Equidistribution of Descent and excedance over $\mathfrak{B}_{n}^{+}$

Let $a_{1},a_{2},\ldots,a_{n}$ be $n$ distinct positive integers with $a_{1}<a_{2}<\cdots<a_{n}$. Let $\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}$ be the Type-B group of $2^{n}n!$ permutations on the letters $a_{1},a_{2},\ldots,a_{n}$. Clearly, when $a_{i}=i$ for all $i$, we get $\mathfrak{B}_{n}=\mathfrak{B}_{\{1,2,\ldots,n\}}$. We write $\pi\in\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}$ in two line notation with $a_{1},a_{2},\ldots,a_{n}$ above and $\pi_{a_{i}}$ below $a_{i}$. Thus, $\pi_{a_{i}}$ is defined for all $i$. For any permutation $\pi\in\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}$ define $\mathsf{inv}_{B}(\pi)=|\{1\leq i<j\leq n:\pi_{a_{i}}>\pi_{a_{j}}\}|+|\{1\leq i<j\leq n:-\pi_{a_{i}}>\pi_{a_{j}}\}|+\}+|\mathsf{Negs}(\pi)|$. Let $\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}^{+}$ denote the subset of even length elements of $\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}$ and $\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}^{-}=\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}-\mathfrak{B}_{\{a_{1},a_{2},\ldots,a_{n}\}}^{+}$. Let $a_{0}=0$ and $\pi(0)=0$ for all $\pi$. Define $\mathsf{des}_{B}(\pi)=|\{i\in\{0,1,2\ldots,n-1\}:\pi_{a_{i}}>\pi_{a_{i+1}}\}|$ and $\mathsf{asc}_{B}(\pi)=|\{i\in\{0,1,2\ldots,n-1\}:\pi_{a_{i}}<\pi_{a_{i+1}}\}|$. Also define $\mathsf{pos}_{a_{i}}(\pi)=k$ if $|\pi_{a_{k}}|=a_{i}$. Define

[TABLE]

We recall the following result from [4, Lemma 26].

Lemma 19

For all positive integers $a_{1}<a_{2}<\ldots<a_{n}$,

[TABLE]

Theorem 20

For all positive integers $a_{1}<a_{2}<\ldots<a_{n}$, the following holds:

[TABLE]

Proof:   We induct on $n$. When $n=1$, clearly $\mathrm{SgnB}_{a_{1}}(s,t,u)=(s-t)u$. Assume that for any $n-1$ integers $a_{1}<a_{2}<\ldots<a_{n-1}$,

[TABLE]

Adding $\pm a_{n}$ at position $n$, we get

[TABLE]

where $s(s-t)^{n-1}u^{n}$ is the contribution of of $\pi\in\mathfrak{B}_{a_{1},a_{2},\ldots,a_{n}}$ with $a_{n}$ appearing in the final position of $\pi$ and $-t(s-t)^{n-1}u^{n}$ is the contribution of $\pi\in\mathfrak{B}_{a_{1},a_{2},\ldots,a_{n}}$ with $\overline{a_{n}}$ appearing in the final position of $\pi$. The proof is complete by summing (26) and (28).

Setting $u=1$ and $a_{i}=i$ in Theorem 20, we get the following:

Corollary 21

[TABLE]

Sivasubramanian in [19] showed that $\mathsf{SgnBExc}_{n}(t)=(1-t)^{n}$ and the same proof shows the following.

[TABLE]

We get the following surprising result about enumerating type-B excedances and type-B descents in $\mathfrak{B}_{n}^{+}$. Note that an analogous result is not true over $\mathfrak{S}_{n}$. As mentioned earlier, Theorem 18 due to Brenti gives a type-B bijective counterpart of Foata’s First Fundamental Transformation. It is easy to check that Brenti’s bijection neither preserves nor reverses parity of $\mathsf{inv}_{B}()$ and so is not directly useful to prove the result below.

Theorem 22

For all positive integers $n$, type-B excedances and type-B descents are equidistributed over $\mathfrak{B}_{n}^{+}$ and $\mathfrak{B}_{n}^{-}$. That is,

[TABLE]

Proof:   Follows from (29) and (30).

In [4], Dey and Sivasubramanian proved several results about gamma positivity of $B_{n}^{+}(s,t)$ and $\mathfrak{B}_{n}^{-}(s,t)$. Due to Theorem 22, these are valid for $\mathsf{BExc}_{n}^{+}(s,t)$ and $\mathsf{BExc}_{n}^{-}(s,t)$. The following three results follow from [4, Theorems 28,29,30].

Lemma 23

The following recurrence relations are true for $\mathsf{BExc}_{n}^{+}(s,t)$ and $\mathsf{BExc}_{n}^{-}(s,t)$ for all positive integers $n\geq 2$.

[TABLE]

Theorem 24

For all positive even integers $n=2m$ with $n\geq 2$, $\mathsf{BExc}_{n}^{+}(s,t)$ and $\mathsf{BExc}_{n}^{-}(s,t)$ are gamma positive polynomials with the same center of symmetry $m$.

Theorem 25

For all positive odd integers $n\geq 3$, $\mathsf{BExc}_{n}^{-}(t)$ and $\mathsf{BExc}_{n}^{+}(t)$ can be written as a sum of $2$ gamma positive polynomials.

5 Type D Coxeter Group

Let $\mathfrak{D}_{n}=\{\sigma\in\mathfrak{B}_{n}:|\mathsf{Negs}(\sigma)|\mbox{ is even}\}$ denote the type-D Coxeter group. For $\pi\in\mathfrak{D}_{n}$, define as before, $\mathsf{Negs}(\pi)=\{\pi_{i}:i>0,\pi_{i}<0\}$ to be the set of elements which occur with a negative sign. Define $\mathsf{inv}_{D}(\pi)=\mathsf{inv}(\pi)+|\{1\leq i<j\leq n:-\pi_{i}>\pi_{j}\}|$. For Type-D Coxeter groups, we have the same definition of excedance as in type-B coxeter groups. Hence, define $\mathsf{exc}_{D}(\pi)=|\{i\in[n]:\pi_{|\pi(i)|}>\pi_{i}\}|+|\{i\in[n]:\pi_{i}=-i\}|$, $\mathsf{nexc}_{D}(\pi)=n-\mathsf{exc}_{D}(\pi)$ and $\mathsf{wkexc}_{D}(\pi)=|\{i\in[n]:\pi_{|\pi(i)|}>\pi_{i}\}|+|\{i\in[n]:\pi_{i}=i\}|$. Let $\mathfrak{D}^{+}_{n}\subseteq\mathfrak{D}_{n}$ denote the subset of even length elements of $\mathfrak{D}_{n}$ and let $\mathfrak{D}_{n}^{-}=\mathfrak{D}_{n}-\mathfrak{D}_{n}^{+}$. Define

[TABLE]

Recall that Brenti showed that type-B excedances and type-B ascents are equidistributed in $\mathfrak{B}_{n}$ (Theorem 18).

Remark 26

It is simple to give a similar bijection $T_{n}:\mathfrak{B}_{n}\mapsto\mathfrak{B}_{n}$ such that $\mathsf{des}_{B}(T_{n}(\pi))=\mathsf{exc}_{B}(\pi)$ and $\mathsf{Negs}((T_{n}(\pi))=\mathsf{Negs}(\pi)$. The bijection is very similar to the proof of [8, Theorem 10.2.3] and [2, Theorem 3.15] and so we omit it.

Remark 27

By Remark 26, as the number of negative entries is preserved by $T_{n}$, enumerating Type-B descents over $\mathfrak{D}_{n}$ is equivalent to enumerating Type-B excedance over $\mathfrak{D}_{n}$ which is equivalent to enumerating Type-D excedance over $\mathfrak{D}_{n}$. That is, for all positive integers $n$, we have

[TABLE]

Recall the operator $\displaystyle D=\left(\frac{d}{ds}+\frac{d}{dt}\right)$. We prove the following recurrence relations.

Lemma 28

For all positive integers $n$, the polynomials $\mathsf{DExc}_{n}(s,t)$ satisfy the following recurrence relations.

[TABLE]

Proof:   We prove (38) first. Using Remark 27, we evaluate $\sum_{\pi\in\mathfrak{D}_{n}}t^{\mathsf{des}_{B}(\pi)}s^{\mathsf{asc}_{B}(\pi)}$ instead. Clearly, $s\sum_{\pi\in\mathfrak{D}_{n-1}}t^{\mathsf{des}_{B}(\pi)}s^{\mathsf{asc}_{B}(\pi)}$ is the contribution from $\pi\in\mathfrak{D}_{n}$ in which the letter ‘$n$’ appears in the last position and $t\sum_{\pi\in\mathfrak{B}_{n-1}-\mathfrak{D}_{n-1}}t^{\mathsf{des}_{B}(\pi)}s^{\mathsf{asc}_{B}(\pi)}$ is the contribution from $\pi\in\mathfrak{D}_{n}$ in which the letter $\overline{n}$ appears in the last position. The term $\displaystyle stD\sum_{\pi\in\mathfrak{D}_{n-1}}t^{\mathsf{des}_{B}(\pi)}s^{\mathsf{asc}_{B}(\pi)}$ accounts for all those permutations $\pi\in\mathfrak{D}_{n}$ in which the letter $n$ appears in all positions except the last position. Likewise, the term $\displaystyle stD\sum_{\pi\in\mathfrak{B}_{n-1}-\mathfrak{D}_{n-1}}t^{\mathsf{des}_{B}(\pi)}s^{\mathsf{asc}_{B}(\pi)}$ accounts for all those permutations $\pi\in\mathfrak{D}_{n}$ in which the letter $\overline{n}$ appears at either the initial position or in the gaps in the permutations of $\mathfrak{B}_{n}-\mathfrak{D}_{n}$. Summing up all these we get

[TABLE]

In an identical manner one can prove (39). We omit the details.

Using Lemma 28, we next show that the polynomials $\mathsf{DExc}_{n}(s,t)$ and $\mathsf{(B\mbox{-}D)Exc}_{n}(s,t)$ are gamma positive.

Theorem 29

For all positive integers $n\geq 2$, $\mathsf{BExc}_{n}^{+}(s,t)=\mathsf{DExc}_{n}(s,t)$ and $\mathsf{BExc}_{n}^{-}(s,t)=\mathsf{(B\mbox{-}D)Exc}_{n}(s,t)$. Hence, for all even natural numbers $n$, $\mathsf{DExc}_{n}(s,t)$ and $\mathsf{(B\mbox{-}D)Exc}_{n}(s,t)$ are gamma positive polynomials with same center of symmetry $n/2$. For all odd positive integers $n\geq 3$, the univariate polynomials $\mathsf{DExc}_{n}(t)$ and $\mathsf{(B\mbox{-}D)Exc}_{n}(t)$ can be written as a sum of $2$ gamma positive polynomials.

Proof:   It is simple to see that $\mathsf{DExc}_{2}(s,t)=s^{2}+2st+t^{2}=\mathsf{BExc}_{2}^{+}(s,t)$ and $\mathsf{(B\mbox{-}D)Exc}_{2}(s,t)=4st=\mathsf{BExc}_{2}^{-}(s,t)$. Further, recurrences (38) and (39) for $\mathsf{DExc}_{n}(s,t)$ and $\mathsf{(B\mbox{-}D)Exc}_{n}(s,t)$ respectively are identical to the recurrences (31) and (32). Thus, the proof of Theorem 24 and Theorem 25 works here.

We next give a recurrence relation satisfied by $\mathsf{DExc}_{n}(s,t)$.

Lemma 30

For all positive integers $n\geq 2$,

[TABLE]

where the following table lists $R_{i}(s,t)$ and its center of symmetry.

\begin{array}[]{l|r}R_{i}(s,t)&\mathrm{\operatorname{c-o-s}}(R_{i}(s,t))\\ \hline\cr R_{1}(s,t)=(s+t)^{4}+8st(s+t)^{2}+16(st)^{2}&2\\ \hline\cr R_{2}(s,t)=16st(s+t)^{2}&2\\ \hline\cr R_{3}(s,t)=4st(s+t)^{3}+32(st)^{2}(s+t)&5/2\\ \hline\cr R_{4}(s,t)=2(st)^{2}(s+t)^{2}+8(st)^{3}&3\\ \hline\cr R_{5}(s,t)=12(st)(s+t)^{2}&2\\ \hline\cr R_{6}(s,t)=8(st)^{2}(s+t)&5/2\\ \hline\cr R_{7}(s,t)=2(st)^{2}&2\end{array}**

Further, let

[TABLE]

Then, $S_{n+4}(s,t)$ is gamma positive with center of symmetry $\frac{1}{2}(n+4)$ and each gamma coefficient of $S_{n+4}(s,t)$ is even.

Proof:   Applying Lemma 28 twice, we get

[TABLE]

Further,

[TABLE]

We get (40) by plugging in (43), (44) and (45) in (42). Further, note that $S_{n+4}(s,t)$ is a sum of $6$ gamma positive polynomials, each with center of symmetry $\frac{1}{2}(n+4)$. Further, from the table, each $R_{i}(s,t)$ clearly has even gamma coefficients and hence, $S_{n+4}(s,t)$ also has even gamma coefficients.

Sivasubramanian in [20] showed the following.

Theorem 31

For all positive integers $n$, \mathsf{SgnDExc}_{n}(t)=\begin{cases}(1-t)^{n}&\text{if neven }.\\ (1-t)^{n-1}&\text{ifn odd }.\end{cases}

A slight modification of this result gives us the following bivariate version of Theorem 31. Since the proof is identical, we omit the details.

Theorem 32

For all positive integers $n$, \mathsf{SgnDExc}_{n}(s,t)=\begin{cases}(s-t)^{n}&\text{if neven }.\\ s(s-t)^{n-1}&\text{ifn odd }.\end{cases}

We use Theorem 32 twice to get the following.

Corollary 33

For positive integers $n$, we have $\mathsf{SgnDExc}_{n+4}(s,t)=(s-t)^{4}\mathsf{SgnDExc}_{n}(s,t).$

We now come to the main Theorem of this section.

Theorem 34

. For all even positive integers $n\geq 4$, $\mathsf{DExc}_{n}^{+}(s,t)$ and $\mathsf{DExc}_{n}^{-}(s,t)$ are gamma positive with center of symmetry $\frac{1}{2}n$.

Proof:   We induct on $n$ with the base cases being $n=4$ and $n=6$. When $n=4$ and $n=6$ one can check the following.

[TABLE]

By (40) and Corollary 33,

[TABLE]

It is simple to see that $R_{1}(s,t)+(s-t)^{4}=2(s+t)^{4}+32(st)^{2}$ and $R_{1}(s,t)-(s-t)^{4}=16st(s+t)^{2}$ are both gamma positive with center of symmetry $2$ and both have even gamma coefficients. Let $n$ be even with $n>7$. By induction, both $\mathsf{DExc}_{n}^{+}(s,t)$ and $\mathsf{DExc}_{n}^{-}(s,t)$ are gamma positive with the same center of symmetry $\frac{1}{2}n$. Further, each of the three terms of (46) have the same center of symmetry $\frac{1}{2}(n+4)$. Thus by Lemma 30, the polynomial $\mathsf{DExc}_{n+4}^{+}(s,t)$ is gamma positive with center of symmetry $\frac{1}{2}(n+4)$. In an identical manner, one can prove that $\mathsf{DExc}_{n+4}^{-}(s,t)$ is gamma positive with center of symmetry $\frac{1}{2}(n+4)$.

Theorem 35

For all odd positive integers $n\geq 5$, $\mathsf{DExc}_{n}^{+}(t)$ and $\mathsf{DExc}_{n}^{-}(t)$ can be written as sum of two gamma positive polynomials.

Proof:   Let $n$ be an odd integer $\geq 5$.

[TABLE]

Hence,

[TABLE]

Now the proof follows in similar lines to the proof of Theorem 2.

6 Excedance in Even and odd Derangements

Let $\mathfrak{SD}_{n}\subseteq\mathfrak{S}_{n}$ be the subset of derangements in $\mathfrak{S}_{n}$. Though descents and excedances in $\mathfrak{S}_{n}$ are equidistributed, they are not equidistributed in $\mathfrak{SD}_{n}\cap\mathcal{A}_{n}$. Recall $\mathsf{ADerExc}_{n}(t)$ as defined in (4). Shin and Zeng [17] and Sharesian and Wachs [15] proved gamma positivity of $\mathsf{ADerExc}_{n}(t)$.

Theorem 36** (Shin and Zeng)**

For all positive integers $n$, $\mathsf{ADerExc}_{n}(t)$ is gamma positive.

Sun and Wang in [21] gave an alternate proof of Theorem 36 based on a semi bijective variant of valley hopping that we call cyclic valley hopping. It can be checked that the cyclic valley hopping proof preserves cycle type and hence sign. Thus, we get the following refinement of Theorem 36. We give a different proof as our proof gives more refined information.

Theorem 37

The polynomials $\mathsf{ADerExc}_{n}^{+}(t)$ and $\mathsf{ADerExc}_{n}^{-}(t)$ are gamma positive for all positive integers $n$.

6.1 Gamma-nonnegativity for even and odd derangements

For $\pi\in\mathfrak{S}_{n}$, let $\mathsf{cyctype}(\pi)=1^{m_{1}}2^{m_{2}}\ldots k^{m_{k}}$ where we have $m_{i}$’s are positive integers and $\pi$ has $m_{i}$ cycles of length $i$. We denote by $\lambda(\pi)\vdash n$, the partition (of the integer $n$) with $m_{i}$ parts having size $i$ and also denote this as $\mathsf{cyctype}(\pi)=\lambda$. It is well known that conjugacy classes in $\mathfrak{S}_{n}$ are indexed by partitions $\lambda\vdash n$ and the conjugacy class $C_{\lambda}=\{\pi\in\mathfrak{S}_{n}:\mathsf{cyctype}(\pi)=\lambda\}$. For $\lambda\vdash n$, define

[TABLE]

We start by summing excedance over cyclic permutations (which are all permutations $\pi$ with $\mathsf{cyctype}(\pi)=n$).

Lemma 38

Let $n\geq 2$ and consider the partition $\mu=n$. Then the following holds.

[TABLE]

Thus $\mathsf{ADerExc}_{\mu}(t)$ is gamma positive with center of symmetry $\frac{1}{2}n$.

Proof:   Let $\pi=a_{1},a_{2},\ldots,a_{n-1}\in\mathfrak{S}_{n-1}$. Define $f:\mathfrak{S}_{n-1}\mapsto C_{\mu}$ by $f(\pi)=(1,n+1-a_{1},n+1-a_{2},\ldots,n+1-a_{n-1})$, where $a_{1},a_{2},\ldots,a_{n-1}$ is in one-line notation and $(1,n+1-a_{1},n+1-a_{2},\ldots,n+1-a_{n-1})$ is in cyclic notation. Clearly, $\pi$ has $k$ descents if and only if $n-a_{1},n-a_{2},\ldots,n-a_{n-1}$ has $k$ ascents. This happens if and only if $f(\pi)=(1,n+1-a_{1},n+1-a_{2},\ldots,n+1-a_{n-1})$ has $k+1$ excedances. $f$ is a bijection with $f^{-1}=g$ where $g:C_{\mu}\mapsto\mathfrak{S}_{n-1}$ is defined by $g((1,a_{1},a_{2},\ldots,a_{n-1}))=n+1-a_{1},n+1-a_{2},\ldots,n+1-a_{n-1}$. $A_{n-1}(t)$ is gamma positive with center of symmetry $\frac{1}{2}(n-2)$ and hence $\mathsf{ADerExc}_{\mu}(t)=tA_{n-1}(t)$ has center of symmetry $\frac{1}{2}n$, completing the proof.

Let $\lambda\vdash n$. We use two notations for $\lambda$. In exponential notation, let $\lambda=1^{m_{1}},2^{m_{2}},\ldots,k^{m_{k}}$. This means $\lambda$ has $m_{i}$ parts equal to $i$ for $1\leq i\leq k$. We also use $\lambda=\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell}$ to denote that the parts of $\lambda$ are $\lambda_{i}$ for $1\leq i\leq\ell$. Define $\binom{n}{\lambda}$ to be the multinomial coefficient $\binom{n}{\lambda_{1},\ldots,\lambda_{\ell}}$. We next consider $C_{\lambda}$, the conjugacy class of $\mathfrak{S}_{n}$ indexed by $\lambda$. It is well known that $C_{\lambda}$ contains all $\pi\in\mathfrak{S}_{n}$ with $\mathsf{cyctype}(\pi)=\lambda$. We start with the case when $\lambda=n_{1},n_{2}$ with both $n_{1},n_{2}\geq 1$. We do not need $n_{1}$ and $n_{2}$ to be distinct.

Lemma 39

Let $\lambda\vdash n$ with $\lambda=n_{1},n_{2}$ where $n_{1},n_{2}\geq 1$. Then,

[TABLE]

Thus, $\sum_{\pi\in C_{\lambda}}t^{\mathsf{exc}(\pi)}$ is gamma positive with center of symmetry $\frac{1}{2}n$ if both $n_{1},n_{2}>1$.

Proof:   Let $T=\{{\pi\in\mathfrak{S}_{n_{1}}:\mathsf{cyctype}(\pi)=n_{1}}\}\times\{{\pi\in\mathfrak{S}_{n_{2}}:\mathsf{cyctype}(\pi)=n_{2}}\}$. Let $\pi\in C_{\lambda}$ with $\pi=(a_{1},a_{2},\ldots,a_{n_{1}}),(b_{1},b_{2},\ldots,b_{n_{2}})$ (written in cycle notation). Define $f:C_{\lambda}\mapsto T$ as $f(\pi)=(a_{1}^{\prime},a_{2}^{\prime},\ldots,a_{n_{1}}^{\prime}),(b_{1}^{\prime},b_{2},\ldots,b_{n_{2}}^{\prime})$ where $(a_{1}^{\prime},a_{2}^{\prime},\ldots,a_{n_{1}}^{\prime})$ is the order preserving restriction of the set $A=\{a_{1},a_{2},\ldots,a_{n_{1}}\}$ to the set $[n_{1}]$. That is we map the least element of $A$ to $1$, the second smallest element to 2 and so on. Similarly $(b_{1}^{\prime},b_{2},\ldots,b_{n_{2}}^{\prime})$ is the order preserving restriction of $B=\{b_{1},b_{2},\ldots,b_{n_{2}}\}$ to $[n_{2}]$. This is an $\displaystyle\frac{\binom{n}{\lambda}}{\prod_{i=1}^{k}(m_{i}!)}$ to $1$ mapping. As excedances are preserved under this map, the proof of (50) is complete. (51) follows from Lemma 38. Further, if both $n_{1},n_{2}>1$, then both $A_{n_{1}-1}(t)$ and $A_{n_{2}-1}(t)$ have positive degree and thus the center of symmetry will clearly be $n/2$.

Remark 40

For a positive integer $n$, let $\lambda\vdash n$ with $\lambda=1^{m_{1}},2^{m_{2}},\ldots,k^{m_{k}}$. Then, $\displaystyle\frac{\binom{n}{\lambda}}{\prod_{i=1}^{k}(m_{i}!)}$ is a positive integer. Indeed, it counts the number of ways to partition the set $[n]$ into parts with sizes $\lambda_{i}$.

Theorem 41

Let $\lambda\vdash n$ with $\lambda=1^{m_{1}}2^{m_{2}}\ldots k^{m_{k}}$. If $m_{1}=0$, then

[TABLE]

Hence, $\mathsf{ADerExc}_{\lambda}(t)$ is gamma positive with center of symmetry $\frac{1}{2}n$. If $m_{1}>0$, then

[TABLE]

Hence, $\mathsf{ADerExc}_{\lambda}(t)$ is a gamma positive polynomial with center of symmetry $\frac{1}{2}(n-m_{1})$.

Proof:   Is an easy application of Lemma 39. By Lemmas 5 and 38, $\prod_{j=2}^{k}(tA_{j-1}(t))^{m_{j}}$ is gamma positive with center of symmetry $\sum_{j=2}^{k}\frac{1}{2}(jm_{j})$. By Remark 40, the multiplication factor is a positive integer and so gamma positivity is preserved. We only need to note that when $m_{1}=0$, $\sum_{j=2}^{k}\frac{1}{2}(jm_{j})=\frac{1}{2}n$. Likewise, when $m_{1}>0$, $\sum_{j=2}^{k}\frac{1}{2}jm_{j}=\frac{1}{2}(n-m_{1})$.

The following is our refinement of Theorem 36.

Theorem 42

For all positive integers $n$, the polynomials $\mathsf{ADerExc}_{n}(t),\mathsf{ADerExc}_{n}^{+}(t)$ and $\mathsf{ADerExc}_{n}^{-}(t)$ are gamma positive with centers of symmetry $\frac{n}{2}$.

Proof:   We denote those partitions $\lambda\vdash n$, with no part being 1 as $1\not\in\lambda$. Clearly $\mathfrak{SD}_{n}=\biguplus_{\lambda\vdash n,1\not\in\lambda}C_{\lambda}$. Thus, by Theorem 41, summing over all conjugacy classes indexed by $\lambda\vdash n$ with $1\not\in\lambda$, we get gamma positivity of $\mathsf{ADerExc}_{n}(t)$.

If $\lambda\vdash n$ with $\lambda=1^{m_{1}}2^{m_{2}}\ldots k^{m_{k}}$, then clearly, if $\pi\in C_{\lambda}$, then $\pi\in\mathcal{A}_{n}$ iff $n-\sum_{j=1}^{k}m_{j}$ is even. For $\lambda\vdash n$ define $\mathsf{sign}(\lambda)=\mathsf{sign}(\pi)$ for any $\pi\in C_{\lambda}$. Thus, summing over conjugacy classes $C_{\lambda}$ with $1\not\in\lambda$ and with $\mathsf{sign}(\lambda)=1$ gives us gamma positivity of $\mathsf{ADerExc}_{n}^{+}(t)$. Likewise, summing over conjugacy classes $C_{\lambda}$ with $1\not\in\lambda$ and with $\mathsf{sign}(\lambda)=-1$ gives us gamma positivity of $\mathsf{ADerExc}_{n}^{-}(t)$.

Define $\mathfrak{SD}_{n,i}\subseteq\mathfrak{S}_{n}$ denote the permutations with $i$ fixed points. Let $\mathfrak{SD}_{n,i}^{+}=\mathfrak{SD}_{n,i}\cap\mathcal{A}_{n}$ and $\mathfrak{SD}_{n,i}^{-}=\mathfrak{SD}_{n,i}\cap(\mathfrak{S}_{n}-\mathcal{A}_{n})$. Define,

[TABLE]

Theorem 43

For all positive integers $n,i$ with $0\leq i\leq n$, the polynomials $\mathsf{ADerExc}_{n,i}^{+}(t)$ and $\mathsf{ADerExc}_{n,i}^{-}(t)$ are gamma positive with center of symmetry $\frac{1}{2}(n-i)$.

Proof:   Sum Theorem 41 over the conjugacy classes $C_{\lambda}$ with $m_{1}=i$ in $\lambda$ and with $\mathsf{sign}(\lambda)=\pm 1$ (with $\mathsf{sign}(\lambda)$ as defined in the proof of Theorem 42).

Recall that $\mathsf{cyc}(w)$ denotes the number of cycles of a permutation $w$. We are now ready to prove the Main result of this Section.

Proof:   (Of Theorem 4) We prove separately for the two statistics $\mathsf{inv}(\pi)$ and $\mathsf{cyc}(\pi)$.

$\underline{\mathsf{stat}(\pi)=\mathsf{inv}(\pi)}:$ Shin and Zeng [17]. showed that

[TABLE]

where $b_{n,i}(q)=\sum_{\pi\in\mathfrak{SD}_{n}(i)}q^{\mathsf{inv}(\pi)}$. Here, $\mathfrak{SD}_{n}(i)$ consists of all elements of $\mathfrak{SD}_{n}$ with exactly $i$ excedances and no double excedances. Consider both terms on either side of the equaliy as polynomials in $\mathbb{R}[t][q]$. The coefficient of $q^{2r}$ and $q^{2r+1}$ for each $r\geq 0$ on either side are the same. Hence, (57) factors nicely for $\mathfrak{SD}_{n}^{+}$ and $\mathfrak{SD}_{n}^{-}$.

$\underline{\mathsf{stat}(\pi)=\mathsf{cyc}(\pi)}:$ Shin and Zeng in [17] also showed

[TABLE]

where $f_{n,i}(q)=\sum_{\pi\in\mathfrak{SD}_{n}(i)}q^{c(\pi)}$. Again comparing the coefficients of $q^{2r}$ and $q^{2r+1}$ in both sides of (58) completes the proof.