Towards a Combinatorial proof of Gessel's conjecture on two-sided Gamma positivity: A reduction to simple permutations

Ron M. Adin, Eli Bagno, Estrella Eisenberg, Shulamit Reches, Moriah Sigron

Published 2017-11-17Version 1

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. Unlike the corresponding result for the usual Eulerian polynomial, the proof for the two-sided version was not combinatorial. This paper attempts to set the stage for a combinatorial proof. We represent each permutation as a tree, composed of its simple blocks, and define actions on that tree which induce a combinatorial proof of Gessel's conjecture provided the validity of the conjecture for simple permutations. This reduces the gamma positivity conjecture of general permutations to that of simple ones.