On two-sided gamma-positivity for simple permutations

TL;DR

This paper explores a conjecture about gamma-positivity in simple permutations, extending a known result about the two-sided Eulerian polynomial and using permutation decomposition trees for proof support.

Contribution

It proposes a new conjecture on gamma-positivity for simple permutations and links it to existing results through decomposition tree analysis.

Findings

Supporting evidence for the gamma-positivity conjecture in simple permutations

Connection established between simple permutation structure and Gessel-Lin result

Extension of gamma-positivity concepts to broader permutation classes

Abstract

Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin. We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.