Related papers: Equidistributions of MAJ and STAT over pattern avo…

The distribution of higher order level spacings, i.e. the distribution of $\{s_{i}^{(n)}=E_{i+n}-E_{i}\}$ with $n\geq 1$ is derived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poisson…

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial…

A notion of generalized $n$-semimodularity is introduced, which extends that of (sub/super)mod\-ularity in four ways at once. The main result of this paper, stating that every generalized $(n\colon\!2)$-semimodular function on the $n$th…

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…

A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally $O(n^{-1/2})$ and can be very considerable when the distribution is heavily biased or…

The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set…

The Poisson distribution is the default choice of likelihood for probabilistic models of count data. However, due to the equidispersion contraint of the Poisson, such models may have predictive uncertainty that is artificially inflated.…

A general challenge in statistics is prediction in the presence of multiple candidate models or learning algorithms. Model aggregation tries to combine all predictive distributions from individual models, which is more stable and flexible…

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more…

In 2019, B\'ona and Smith introduced the notion of strong pattern avoidance, saying that a permutation $\pi$ strongly avoids a pattern $\sigma$ if $\pi$ and $\pi^2$ both avoid $\sigma$. Recently, Archer and Geary generalized the idea of…

On the set of permutations of a finite set, we construct a bijection which maps the 3-vector of statistics $(maj-exc,des,exc)$ to a 3-vector $(maj\_2,\widetilde{des\_2},inv\_2)$ associated with the $q$-Eulerian polynomials introduced by…

A desarrangement is a permutation whose first ascent is even. Desarrangements were introduced in the 1980s by Jacques D\'{e}sarm\'{e}nien, who proved that they are in bijection with derangements. We revisit the study of desarrangements,…

For a permutation $\pi$ the major index of $\pi$ is the sum of all indices $i$ such that $\pi_i > \pi_{i+1}$. It is well known that the major index is equidistributed with the number of inversions over all permutations of length $n$. In…

In this paper, a new mixed Poisson distribution is introduced. This new distribution is obtained by utilizing mixing process, with Poisson distribution as mixed distribution and Transmuted Exponential distribution as mixing distribution.…

There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is an open problem for about $40$ years now. So…

In their paper \cite{DokosDwyer:Permutat12}, Dokos et al. conjecture that the major index statistic is equidistributed among 1423-avoiding, 2413-avoiding, and 2314-avoiding permutations. In this paper we confirm this conjecture by…

Exponential families comprise a broad class of statistical models and parametric families like normal distributions, binomial distributions, gamma distributions or exponential distributions. Thereby the formal representation of its…

A descent $k$ of a permutation $\pi=\pi_{1}\pi_{2}\dots\pi_{n}$ is called a big descent if $\pi_{k}>\pi_{k+1}+1$; denote the number of big descents of $\pi$ by $\operatorname{bdes}(\pi)$. We study the distribution of the…

A set partition $\sigma$ of $[n]=\{1,\cdots ,n\}$ contains another set partition $\omega$ if a standardized restriction of $\sigma$ to a subset $S\subseteq[n]$ is equivalent to $\omega$. Otherwise, $\sigma$ avoids $\omega$. Sagan and Goyt…

In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in…