Related papers: A Simple Proof that Major Index and Inversions are…

A permutation is called mod-k-alternating if its entries are restricted to having the same remainder as the index, modulo some integer $k \geq 1.$ In this paper, we find the sign-balance for mod-k-alternating permutations with respect to…

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular,…

The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study…

We prove the universality of the large deviations for conjugacy invariant permutations with few cycles. As an application, we establish the universality of large deviation at speeds $n$ and $\sqrt{n}$ for the length of monotone subsequences…

We show that the distribution of the number of peaks at height $i$ modulo $k$ in $k$-Dyck paths of a given length is independent of $i\in[0,k-1]$ and is the reversal of the distribution of the total number of peaks. Moreover, these…

Using classical transformations on the symmetric group and two transformations constructed in Fix-Mahonian Calculus I, we show that several multivariable statistics are equidistributed either with the triplet (fix,des,maj), or the pair…

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…

We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formulas for…

We combinatorially prove that the number $R(n,k)$ of permutations of length $n$ having $k$ runs is a log-concave sequence in $k$, for all $n$. We also give a new combinatorial proof for the log-concavity of the Eulerian numbers.

We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.

Let $R(n,k)$ denote the number of permutations of ${1,2,...,n}$ with $k$ alternating runs. In this note we present an explicit formula for the numbers $R(n,k)$.

We find the range of the major index on the various conjugacy classes of involutions in the symmetric group $S_n$. In addition to indicating the minimum and the maximum values, we show that except for the case of involutions without fixed…

In this paper, we introduce the concept of the over-Mahonian number, which counts the overlined permutations of length $n$ with $k$ inversions, allowing the first elements associated with the inversions to be independently overlined or not.…

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…

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…

We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length $n$, counted by their number of cycles. We then use this result to prove that if $k$ is the number of…

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a $q$-analogue of…

In this paper, we give a new bijective proof of a multiset analogue of even-odd permutations identity. This multiset version is equivalent to the original coin arrangements lemma which is a key combinatorial lemma in the Sherman's Proof of…

Let $P(n)$ be the probability that two independent, uniformly random permutations of $[n]$ have the same order, and let $K(n)$ be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, we prove…

Let $s$ denote West's stack-sorting map. A permutation is called $t-\textit{sorted}$ if it is of the form $s^t(\mu)$ for some permutation $\mu$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can…