The Eulerian distribution on self evacuated involutions

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。

The signed Eulerian numbers on involutions

We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for $f_{n,k}$ arising from the properties of its generating function.

The Eulerian distribution on involutions is indeed $γ$-positive

Let $\mathcal I_n$ and $\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\{1, \dots, n\}$, respectively, and let $\text{des}(π)$ denote the number of descents of the permutation $π$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \sum_{π\in \mathcal I_n} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 1$ and $J_{2n}(t) := \sum_{π\in \mathcal J_{2n}} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable …

The Eulerian Distribution on Involutions is Indeed Unimodal

Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$ \sum_{k=0}^{n-1}I_{n,k}t^k=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}a_{n,k}t^{k}(1+t)^{n-2k-1}. $$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right.

A combinatorial proof of a symmetry of $(t,q)$-Eulerian numbers of type $B$ and type $D$

A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type $B$. This involution also proves a symmetry of the generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings and alignments, and hence $q$-Eulerian numbers of type $A$ defined by L. Williams. By considering a restriction of our bijection, we were led to define a new statistic on the permutations of type $D$ and $(t,q)$-Eulerian numbers of type $D$, which is proved to have a nice symmetry as well. We con…

Random decompositions of Eulerian statistics

This paper develops methods to study the distribution of Eulerian statistics defined by second-order recurrence relations. We define a random process to decompose the statistics over compositions of integers. It is shown that the numbers of descents in random involutions and in random derangements are asymptotically normal with rates of convergence $\mathcal{O}(n^{-1/2})$ and $\mathcal{O}(n^{-1/3})$ respectively.