Research Projects

This project focuses on the properties of combinatorial sequences, specifically gamma positivity and log-concavity. The Eulerian polynomial $A_n(t)$ has been proven gamma positive by Foata and Schützenberger, and recent research has shown that the polynomials $A_n^+(t)$ are gamma positive if $n ≡ 0, 1 (mod 4)$. However, no combinatorial interpretation for the gamma coefficients has been found. The same question is raised about descent-based type B and type D Eulerian polynomials over positive elements in Coxeter groups. Guo and Zeng conjectured that these polynomials are gamma positive for all positive integers $n ≥ 9$, but no combinatorial interpretation of the gamma coefficients is known yet. Gessel conjectured that the joint distribution of descents and inverse descents over the symmetric group is two-sided gamma positive, which was recently proved by Lin. The project also seeks to find an alternating analogue for $A_n^+(t)$, which would provide a geometric connection and potentially lead to a geometric interpretation for the gamma coefficients. Log-concave polynomials are known, and the notion of strong synchronisation has been introduced to prove that excedance enumerating polynomials over alternating groups are indeed log-concave. Proving some of these conjectures is a substantial part of the project. Additionally, the project is interested in the signed enumeration of various statistics over Coxeter groups and standard young tableaux.