Related papers: Eulerian polynomials and descent statistics

We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…

B\'ona conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Br\"and\'en proved this conjecture by establishing a more general result. In this paper, we give another proof of Br\"and\'en's…

In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y|\alpha,\beta)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of…

A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both…

Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(\pi):=\operatorname{st}(\pi^{-1})$. We give a general approach, based on the theory of symmetric functions, for…

Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers,…

We consider a sequence of four variable polynomials by refining Stieltjes' continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections we derive various combinatorial…

This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of…

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of…

Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…

In a recent paper, Regev and Roichman introduced the <_L order and the L-descent number statistic, des_L, on the group of colored permutations, C_a \wr S_n. Here we define the L-reverse major index statistic, rmaj_L, on the same group and…

We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the…

In this paper we present grammatical interpretations of the alternating Eulerian polynomials of types A and B. As applications, we derive several properties of the type B alternating Eulerian polynomials, including combinatorial expansions,…

Motivated by a conjecture of Savage and Visontai about the equidistribution of the descent statistic on signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$ and the ascent statistic on $(1,4,3,8,\ldots,2n-1,4n)$-inversion sequences,…

We exploit Krattenthaler's bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set $S_n(123)$ of 123-avoiding permutations in $S_n$. In particular, we show that the descents of a…

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…

In this paper, we study Eulerian polynomials for permutations and signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$. Properties of these polynomials, including recurrence relations and unimodality are discussed. In particular, we…

It is known that Bernoulli scheme of independent trials with two outcomes is connected with the binomial coefficients. The aim of this paper is to indicate stochastic processes which are connected with the $q$-polynomial coefficients (in…

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also…

The P-Eulerian polynomial counts the linear extensions of a labeled partially ordered set, P, by their number of descents. It is known that the P-Eulerian polynomials are real-rooted for various classes of posets P. The purpose of this…