Two-sided permutation statistics via symmetric functions

Abstract

Given a permutation statistic $\mathrm{st}$, define its inverse statistic $\mathrm{ist}$ by $\mathrm{ist}(\ddot{I}€):=\mathrm{st}(\ddot{I}{€}^{-1})$. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of ${\mathrm{st}}_1$ and ${\mathrm{st}}_2$ whenever ${\mathrm{st}}_1$ and ${\mathrm{st}}_2$ are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs, and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of ${\mathrm{st}}_1$ and ${\mathrm{ist}}_2$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of ${\mathrm{st}}_1$ and ${\mathrm{st}}_2$. Our work leads to a rederivation of Stanley's generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and ${\hat{I}}^3$-positivity.

44 pages