Distributions of statistics on separable permutations

TL;DR

This paper develops functional equations to analyze the distributions of classical permutation statistics on separable permutations, enabling explicit calculations and revealing distributional equivalences and conjectures.

Contribution

It introduces a systematic method to derive distributions of permutation statistics on separable permutations, generalizing previous results and providing explicit joint distributions.

Findings

Derived functional equations for six classical statistics.

Obtained a third degree equation for joint ascent-descent distribution.

Identified two distribution equivalence classes for maxima and minima statistics.

Abstract

We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left minima) on separable and irreducible separable permutations. The equations are used to find a third degree equation for joint distribution of ascents and descents on separable permutations that generalizes the respective known result for the descent distribution. Moreover, our general functional equations allow us to derive explicitly (joint) distribution of any subset of maxima and minima statistics on irreducible, reducible and all separable permutations. In particular, there are two equivalence classes of distributions of a pair of maxima or minima statistics. Finally, we present three unimodality conjectures about distributions of statistics on separable permutations.