Related papers: A new encoding of permutations by Laguerre histori…

Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of…

This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics. Properties of the enumerative polynomials for permutations, signed permutations and…

We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove…

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…

We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are Meixner-Pollaczek polynomials with certain arguments and constant factors. Moreover, through a general connection between moments…

The $(q,r)$-Eulerian polynomials are the $(\maj-$$\exc,\fix,\exc)$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical…

The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by…

Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations, where a Cayley permutation is a word of positive integers such that if…

The use of permutation polynomials has appeared, along to their compositional inverses, as a good choice in the implementation of cryptographic systems. Hence, there has been a demand for constructions of these polynomials which…

We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted…

Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingr\'imsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a permutation…

We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an…

We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial…

In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent…

A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters of any prefix is non-negative, together with a permutation \sigma of {1,2,...,n} such…

Motivated by recent work on (re)mixed Eulerian numbers, we provide a combinatorial interpretation of a subfamily of the remixed Eulerian numbers introduced by Nadeau and Tewari. More specifically, we show that these numbers can be realized…

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family…

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of pattern-avoiding permutations into classical statistics on Dyck…

We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences, to obtain a bijection with a new class of labeled trees, which we call…