The cycle enumerator of unimodal permutations

Descents of $λ$-unimodal cycles in a character formula

We prove an identity conjectured by Adin and Roichman involving the descent set of $λ$-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas have previously been proven algebraically. In this paper, we present a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.

Symmetric unimodal expansions of excedances in colored permutations

We consider several generalizations of the classical $γ$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $γ$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the lite…

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Brändén (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a $(p,q)$-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permuta…

Log-concavity of the Excedance Enumerators in positive elements of Type A and Type B Coxeter Groups

The classical Eulerian Numbers $A_{n,k}$ are known to be log-concave. Let $P_{n,k}$ and $Q_{n,k}$ be the number of even and odd permutations with $k$ excedances. In this paper, we show that $P_{n,k}$ and $Q_{n,k}$ are log-concave. For this, we introduce the notion of strong synchronisation and ratio-alternating which are motivated by the notion of synchronisation and ratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014. We show similar results for Type B Coxeter Groups. We finish with some conjectures to emphasize the following: though strong synchronisation is stronger than …

Signed Countings of Type B and D Permutations and $t,q$-Euler numbers

A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Vergès gives a $q$-analogue with $q$ representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed countings are related to the derivative polynomials of $\tan$ and $\sec$. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol'd showed that the Springe…

On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $γ$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.

The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r

We use the Robinson-Schensted correspondence, followed by symbol-crunching, in order to derive explicit expressions for the quantities mentioned in the title. We follow it by number crunching, in order to compute the first terms of these sequences. As an encore, we cleverly implement Ira Gessel's celebrated determinant formula for the generating functions of these sequences, to crank out many terms. This modest tribute is dedicated to one of the greatest enumerators alive today (and definitely the most modest one!), Ira Martin Gessel, who is turning 64 years-old today

超八面体群对合上的欧拉分布是单峰的

The Eulerian distribution on the involutions of the hyperoctahedral group is unimodal

本文将对称群对合上欧拉分布的单峰性推广到超八面体群（带色置换群）的对合上，并用带符号拟对称函数计算其生成函数。

$k$-交替排列的生成函数的解析推导

An analytic derivation of a generating function for $k$-alternating permutations

通过$q$-指数生成函数的分解和函数方程，解析推导了下降集为$k$倍数排列的反转计数生成函数的闭式表达式。

Arc Permutations

Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, as well as the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircas…

第二类欧拉数的恒等式：通过格点计数

An identity for second Eulerian numbers via lattice-point counting

利用 Ehrhart 理论，给出 Bernoulli 数与第二类欧拉数之间两个恒等式的简单概念性证明。

Inversion polynomials for 321-avoiding permutations: addendum

This addendum contains results about the inversion number and major index polynomials for permutations avoiding 321 which did not fit well into the original paper. In particular, we consider symmetry, unimodality, behavior modulo 2, and signed enumeration.

2-stack sortable permutations with a given number of runs

Using earlier results we prove a formula for the number $W_{(n,k)}$ of 2-stack sortable permutations of length $n$ with $k$ runs, or in other words, $k-1$ descents. This formula will yield the suprising fact that there are as many 2-stack sortable permutations with $k-1$ descents as with $k-1$ ascents. We also prove that $W_{(n,k)}$ is unimodal in $k$, for any fixed $n$.

A note on the number of $k$-roots in $S_n$

The number of $k$-roots of an arbitrary permutation is expressed as an alternating sum of $μ$-unimodal $k$-roots of the identity permutation.

Lipschitz polytopes of posets and permutation statistics

We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define $(P,k)$-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and $h^*$-vectors.

The Weak Bruhat Order and Separable Permutations

In this paper we consider the rank generating function of a separable permutation $π$ in the weak Bruhat order on the two intervals $[\text{id}, π]$ and $[π, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, π]$ and $[π, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets.

Signed Countings of types B and D permutations and $t,q$-Euler Numbers

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Vergès obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès…

Generalized parking functions, descent numbers, and chain polytopes of ribbon posets

We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula I_n(-1) = E_n, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polyto…

避开模式的浅排列

Pattern-avoiding shallow permutations

本文研究了所有$σ\in \mathcal{S}_3$的避开$σ$的浅排列，通过开发更一般的结构结果，成功枚举了避开任何长度为3模式的浅排列。

斯特林-欧拉多项式的一个$q$-模拟

A $q$-analog of the Stirling-Eulerian Polynomials

本文针对排列计数的细化斯特林-欧拉多项式相关研究，利用Gessel的$q$-合成公式证明了Ji公式的$q$-模拟，并为其$γ$-正性展开提供组合方法，推广了相关排列枚举研究成果。

The $q$-tangent and $q$-secant numbers via continued fractions

It is well known that the $(-1)$-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct $q$-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fractions expansions formulae, which permits us to give a unified treatment of Josuat-Vergès' two formulae and also to derive a new $q$-analogue of the aforementioned formulae. Our approach is based on a $(p,q)$-analogue of t…

如何反弹你的因果排列

How to bounce your canon permutation

本文研究了一类新的回文下降多项式，通过Dyck路径和排列生成因果排列，并探讨了其多项式性质及组合解释。

q-Eulerian Polynomials: Excedance Number and Major index

In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been com…

Excedance-type polynomials and gamma-positivity

The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or not, colored or not. Let $p\in [0,1]$ and $q\in [0,1]$ be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-gamma-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly incre…

Eulerian quasisymmetric functions and poset topology

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specialize…

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprisi…

项链、子集和与循环排列

Necklaces, subset sums, and cyclic permutations

本文探讨了子集和与项链之间的关系，引入参数r并研究其周期性条件，同时考虑子集大小k，揭示了项链中1的数量与子集和的关系。

A Refinement of the Eulerian Numbers, and the Joint Distribution of $π(1)$ and Des($π$) in $S_n$

Given a permutation $π$ chosen uniformly from $S_n$, we explore the joint distribution of $π(1)$ and the number of descents in $π$. We obtain a formula for the number of permutations with $\des(π)=d$ and $π(1)=k$, and use it to show that if $\des(π)$ is fixed at $d$, then the expected value of $π(1)$ is $d+1$. We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when $d$ is small the distribution of $π(1)$ among the permutations with $d$ descents is approximately geometric. Applications to Stein's method and the Neggers…

Applications of Symmetric Functions to Cycle and Subsequence Structure after Shuffles

Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…

On some colored Eulerian quasisymmetric functions

Recently, Hyatt introduced some colored Eulerian quasisymmetric function to study the joint distribution of excedance number and major index on colored permutation groups. We show how Hyatt's generating function formula for the fixed point colored Eulerian quasisymmetric functions can be deduced from the Decrease value theorem of Foata and Han. Using this generating function formula, we prove two symmetric function generalizations of the Chung-Graham-Knuth symmetrical Eulerian identity for some flag Eulerian quasisymmetric functions, which are specialized to the flag excedance numbers. Combina…

关于交换置换的无限元组关于轨道数的对数凹性

Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations

本文提出了关于交换置换组轨道数 $A(p,n,k)$ 关于 $k$ 的对数凹性猜想，并基于广义 Turán 数关于 $k$ 的对数凹性，证明了该猜想 $p=\infty$ 的情形。

按指定大小的相邻差异数对 k-ary words 进行计数

Counting k-ary words by number of adjacency differences of a prescribed size

本文研究了 k-ary words 上指定大小相邻差异数的分布问题，计算了相应统计量的常生成函数显式公式，并扩展至有限集合分块和绝对差情形，同时提供了相关递推式的组合证明。

Integer Sequences and Monomial Ideals

Let $\mathfrak{S}_n$ be the set of all permutations of $[n]=\{1,\ldots,n\}$ and let $W$ be the subset consisting of permutations $σ\in \mathfrak{S}_n$ avoiding 132 and 312-patterns. The monomial ideal $I_W = \left\langle \mathbf{x}^σ = \prod_{i=1}^n x_i^{σ(i)} : σ\in W \right\rangle $ in the polynomial ring $R = k[x_1,\ldots,x_n]$ over a field $k$ is called a hypercubic ideal in the article (Certain variants of multipermutohedron ideals, Proc. Indian Acad. Sci.(Math Sci. Vol. 126, No.4, (2016), 479-500). The Alexander dual $I_W^{[\mathbf{n}]}$ of $I_W$ with respect to $\mathbf{n}=(n,\ldots,n)$…

关于下降数与逆下降数联合分布的组合证明

Combinatorial proofs on the joint distribution of descents and inverse descents

本文通过观察排列网格的内部结构，给出了 $A_{n,i,j}$ 递推关系的组合证明，并利用该构造性方法进一步得到对合数 $I_{n,k}$ 与不动点自由对合数 $J_{2n,k}$ 递推关系的组合解释。

二次多项式的项链、排列与周期临界轨道

Necklaces, permutations, and periodic critical orbits for quadratic polynomials

本文通过构造显式双射，证明了二次多项式临界点周期轨道的 Gleason 多项式实根数等于其模 2 不可约因子数，并建立了与循环单峰排列及原始二元项链的联系。

Pattern-Avoiding Polytopes

Two well-known polytopes whose vertices are indexed by permutations in the symmetric group $\mathfrak{S}_n$ are the permutohedron $P_n$ and the Birkhoff polytope $B_n$. We consider polytopes $P_n(Π)$ and $B_n(Π)$, whose vertices correspond to the permutations in $\mathfrak{S}_n$ avoiding a set of patterns $Π$. For various choices of $Π$, we explore the Ehrhart polynomials and $h^*$-vectors of these polytopes as well as other aspects of their combinatorial structure. For $P_n(Π)$, we consider all subsets $Π\subseteq \mathfrak{S}_3$ and are able to provide results in most cases. To illustrate, $…

锯齿形 Eulerian 多项式

Zig-zag Eulerian polynomials

本文利用松弛版 $P$-分拆系统研究自然标号锯齿形偏序集的 Eulerian 多项式，统一既有结果、建立精细化递推，并解释其与交错排列大回报分布的联系。