Related papers: Positivity and divisibility of alternating descent…
We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices…
Descent polynomials and peak polynomials, which enumerate permutations with given descent and peak sets respectively, have recently received considerable attention. We give several formulas for $q$-analogs of these polynomials which refine…
In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these…
Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a $q$-analogue of…
In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. For instance, we show that the…
Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any…
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…
A ballot permutation is a permutation $\pi$ such that in any prefix of $\pi$ the descent number is not more than the ascent number. By using a reversal concatenation map, we give a formula for the joint distribution (pk, des) of the peak…
In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the…
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…
An open conjecture in pattern avoidance theory is that the distribution of the major index among 321-avoiding permutations is distributed unimodally. We construct a formula for this distribution, and in the case of 2 descents prove…
We introduce a statistic $\pmaj$ on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This…
It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The…
In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack…
We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left minima) on separable and irreducible separable…
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…
Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that…
We continue to explore cyclotomic factors in the descent set polynomial $Q_{n}(t)$, which was introduced by Chebikin, Ehrenborg, Pylyavskyy and Readdy. We obtain large classes of factors of the form $\Phi_{2s}$ or $\Phi_{4s}$ where $s$ is…
Inversion sequences, also known as subexcedant sequences, form a fundamental class of objects in enumerative combinatorics. In this paper, we study the joint distribution of five statistics on inversion sequences. While several statistics…
We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when…