Related papers: An Eulerian permutation statistic and generalizati…
Let X -> P^1 be a general cyclic cover. We give a simple formula for the number of equivariant meromorphic functions on X subject to ramification conditions at variable points. This generalizes and gives a new proof of a recent result of…
We present some Markovian approaches to prove universality results for some functions on the symmetric group. Some of those statistics are already studied in [Kammoun, 2018, 2020] but not the general case. We prove, in particular, that the…
Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly $m$ descents. It is well known that $A(n,m)$ also counts the number of permutations on $[n]$ with exactly $m$ excedances. In this report,…
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…
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…
A Mahonian d-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern statistics of length at most d. Babson and Steingrimsson classified all Mahonian 3-functions up to trivial bijections and…
Given a finite volume hyperbolic surface, a fundamental polygon and an oriented closed geodesic, we associate a partial covering of the surface. We prove that given a sequence of collections of oriented closed geodesics equidistributing in…
We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure,…
We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove…
Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the Erd\H{o}s--Kac theorem on the distribution of the number of prime factors of a random…
A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our…
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 consider the space of ordered pairs of distinct $\mathbb{C}P^1$-structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space.…
In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein…
Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly $m$ descents, or, due to the Foata transform, the number of permutations on $[n]$ with exactly $m$ excedances. Friends-and-seats graphs,…
In this paper we continue to study {\it quasi associated homogeneous distributions \rm{(}generalized functions\rm{)}} which were introduced in the paper by V.M. Shelkovich, Associated and quasi associated homogeneous distributions…
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 provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences…
We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their…
A reparametrized Dirichlet-multinomial distribution is introduced, and the covariance matrix, as well as, the algorithm for calculating the PDF for n species are provided. The distribution is suited for modelling the joint distribution of…