Ira Gessel introduced a multivariate formal power series tracking the distribution of ascents and descents in labeled binary trees. In addition to showing that it was a symmetric function, he conjectured it was Schur positive. In this talk, I will sketch a proof of this conjecture using a weighted generalization of a bijection due to Préville-Ratelle and Viennot called the Push-Gliding algorithm. We will then explore connections to symmetric group actions on certain hyperplane arrangements. If time allows, I will discuss a polynomial considered by Postnikov in his work on alternating trees and show how gamma-positivity of the coefficients of this polynomial naturally follows as a corollary of our work.
This work is joint with Ira Gessel and Vasu Tewari.