Related papers: Joint Distributions of Permutation Statistics and …
This paper is concerned with the joint distribution of the number of exterior peaks and the number of proper double descents over permutations on $[n] =\{1,2,\ldots,n\}$. The notion of exterior peaks of a permutation was introduced by…
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…
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…
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…
In this paper, we introduce the notion of a grammatical labeling to describe a recursive process of generating combinatorial objects based on a context-free grammar. For example, by labeling the ascents and descents of a Stirling…
Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is…
We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic…
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right…
Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…
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…
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…
Let $A_{n,i,j}$ be the number of permutations on $[n]$ with $(i-1)$ descents and $(j-1)$ inverse descents.Carlitz, Roselle and Scoville in 1966 first revealed some combinatorial and arithmetic properties of $A_{n,i,j}$,which contain a…
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…
The distributions of the number of occurrences of words (the distributions of words for short) play key roles in information theory, statistics, probability theory, ergodic theory, computer science, and DNA analysis. Bassino et al. 2010 and…
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…
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…
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…
We prove conjectures of the third author [L. Tevlin, Proc. FPSAC'07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by…
We consider the joint distribution of the area and perimeter statistics on the set I_n of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived…
Permutation statistics constitute a classical subject of enumerative combinatorics. In her study of the genus zeta function, Denert discovered a new Mahonian statistic for permutations, which is called the Denert's statistic ({\bf $\den$})…