Related papers: A new decomposition of ascent sequences and Euler-…
A Fishburn permutation is a permutation which avoids the bivincular pattern $(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of…
Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and…
We show that two classes of combinatorial objects--inversion tables with no subsequence of decreasing consecutive numbers and matchings with no 2-nestings--are enumerated by the Fishburn numbers. In particular, we give a simple bijection…
It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by…
One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010),…
In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…
A sequence x=x_1 x_2...x_n $ is said to be an ascent sequence of length $n$ if it satisfies x_1=0 and $0\leq x_i\leq asc(x_1x_2...x_{i-1})+1$ for all $2\leq i\leq n$, where $asc(x_1x_2... x_{i-1})$ is the number of ascents in the sequence…
We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative…
The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in…
An unlabeled poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Let $p_n$ denote the number of (2+2)-free posets of size $n$. In a recent paper,…
We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2+2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not…
A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equi-distributed with the standard one. This concept is then applied to construct explicit…
A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle…
Carlitz and Scoville introduced the polynomials $A_n(x,y|{\alpha},{\beta})$, which we refer to as the $(\alpha, \beta)$-Eulerian polynomials. These polynomials count permutations based on Eulerian-Stirling statistics, including descents,…
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…
Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only…
In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent…
We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can…
This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics…
The aim of this paper is to derive new representations for the Hankel functions, the Bessel functions and their derivatives, exploiting the reformulation of the method of steepest descents by M. V. Berry and C. J. Howls (Berry and Howls,…