Variations on Descents and Inversions in Permutations

TL;DR

This paper introduces new permutation statistics based on alternating descents and inversions, explores their distributional properties, and connects them to classical combinatorial objects like Euler numbers and Dyck paths.

Contribution

It defines novel alternating descent and inversion statistics, proves their equidistribution, and links these to Eulerian polynomials and q-analogs of Euler numbers.

Findings

Alternating descent set is equidistributed with a 3-descent set on permutations starting with 1.

Joint distributions of the new descent-inversion pairs are shown to be the same.

A new q-analog of the Euler number E_n is derived from alternating inversions.

Abstract

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices i such that either i is odd and sigma_i > sigma_{i+1}, or i is even and sigma_i < sigma_{i+1}. We show that this statistic is equidistributed with the 3-descent set statistic on permutations sigma = sigma_1sigma_2...sigma_{n+1} with sigma_1 = 1, defined to be the set of indices i such that the triple sigma_i sigma_{i+1} sigma_{i+2} forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials sum_{sigma in S_n} t^{des(sigma)+1} using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the ab-index of the Boolean algebra B_n, and make observations about it. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new q-analog of the Euler number E_n and show how it emerges in a q-analog of an identity expressing E_n as a weighted sum of Dyck paths.