Related papers: Two-sided permutation statistics via symmetric fun…
Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to…
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…
A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between…
Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen's sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu's Fs. In previous papers we…
A method of quantum tomography of arbitrary spin particle states is developed on the basis of the root approach. It is shown that the set of mutually complementary distributions of angular momentum projections can be naturally described by…
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 examine the distribution and popularity of different parameters (such as the number of descents, runs, valleys, peaks, right-to-left minima, and more) on the sets of increasing and flattened permutations. For each parameter, we provide…
Savage and Sagan have recently defined a notion of st-Wilf equivalence for any permutation statistic st and any two sets of permutations $\Pi$ and $\Pi'$. In this paper we give a thorough investigation of st-Wilf equivalence for the charge…
A permutation is called {\it {block-wise simple}} if it contains no interval of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of permutations and explore some of its combinatorial properties. We present a generating…
In [S. Kitaev and J. Remmel: Classifying descents according to parity] the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In this paper, we generalize the results of…
We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type…
In this paper, algorithms are developed for computing the Stirling transform and the inverse Stirling transform; specifically, we investigate a class of sequences satisfying a two-term recurrence. We derive a general identity which…
In a series of recent talks Richard Stanley introduced a symmetric function associated to digraphs called the Redei-Berge symmetric function. This symmetric function enumerates descent sets of permutations corresponding to digraphs. We show…
The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(\alpha,\beta)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian…
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases…
Consider a set of order statistics that arise from sorting samples from two different populations, each with their own, possibly different distribution function. The probability that these order statistics fall in disjoint, ordered…
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture…
We consider a few special cases of the more general question: How many permutations $\pi\in\mathcal{S}_n$ have the property that $\pi^2$ has $j$ descents for some $j$? In this paper, we first enumerate Grassmannian permutations $\pi$ by the…
One of the most popular class of tests for independence between two random variables is the general class of rank statistics which are invariant under permutations. This class contains Spearman's coefficient of rank correlation statistic,…
The length $\mathsf{is}(\pi)$ of a longest increasing subsequence in a permutation $\pi$ has been extensively studied. An increasing subsequence is one that has no descents. We study generalizations of this statistic by finding longest…