Related papers: Major index distribution over permutation classes

A signed permutation \pi = \pi_1\pi_2 \ldots \pi_n in the hyperoctahedral group B_n is a word such that each \pi_i \in {-n, \ldots, -1, 1, \ldots, n} and {|\pi_1|, |\pi_2|, \ldots, |\pi_n|} = {1,2,\ldots,n}. An index i is a peak of \pi if…

Centrosymmetric involutions in the symmetric group S_{2n} are permutations \pi such that \pi=\pi^{-1} and \pi(i)+\pi(2n+1-i)=2n+1 for all i, and they are in bijection with involutions of the hyperoctahedral group. We describe the…

We compute the PI-exponent of the matrix ring with coefficients in an associative algebra. As a consequence, we prove the following. Let $\mathcal{R}$ be a PI-algebra with a positive PI-exponent. If $M_n(\mathcal{R})$ and $M_m(\mathcal{R})$…

The study of Mahonian statistics dated back to 1915 when MacMahon showed that the major index and the inverse number have the same distribution on a set of permutations with length n. Since then, many Mahonian statistics have been…

A probabilistic generative network model with $n$ nodes and $m$ overlapping layers is obtained as a superposition of $m$ mutually independent Bernoulli random graphs of varying size and strength. When $n$ and $m$ are large and of the same…

MacMahon's classic theorem states that the 'length' and 'major index' statistics are equidistributed on the symmetric group S_n. By defining natural analogues or generalizations of those statistics, similar equidistribution results have…

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…

We investigate dynamical properties of the set of permutations of $\mathbb{Z}^d$ with restricted movement, i.e., permutations $\pi $ of $\mathbb{Z}^d$ such that $\pi (\mathbf{n})-\mathbf{n}$ lies, for every $\mathbf{n}\in \mathbb{Z}^d$, in…

A pure excedance in a permutation $\pi=\pi_1\pi_2\ldots \pi_n$ is a position $i<\pi_i$ such that there is no $j<i$ with $i\leq \pi_j<\pi_i$. We present a one-to-one correspondence on the symmetric group that transports pure excedances to…

In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper…

We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The…

For a given permutation $\tau$, let $P_N^{\tau}$ be the uniform probability distribution on the set of $N$-element permutations $\sigma$ that avoid the pattern $\tau$. For $\tau=\mu_k:=123\cdots k$, we consider $P_N^{\mu_k}(\sigma_I=J)$…

Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of…

We study statistical properties of the random variables $X_{\sigma}(\pi)$, the number of occurrences of the pattern $\sigma$ in the permutation $\pi$. We present two contrasting approaches to this problem: traditional probability theory and…

In this article, we establish the mod-$\phi$ convergence of the major index of a uniform random standard tableau whose shape converges in the Thoma simplex. This implies various probabilistic estimates, in particular speed of convergence…

We give a detailed analysis of the proportion of elements in the symmetric group on $n$ points whose order divides $m$, for $n$ sufficiently large and $m \ge n$ with $m = O(n)$.

Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…

The fundamental bijection is a bijection $\theta:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations…

In this paper, we study the distribution of difference of multiplicative and additive characters modulo $p$ at consecutive polynomial values. More precisely, for an interval $I$ over finite field and $0<m<1$, we investigate the following…

We introduce the notion of a weighted inversion statistic on the symmetric group, and examine its distribution on each conjugacy class. Our work generalizes the study of several common permutation statistics, including the number of…