Speaker : Dr. Digjoy Paul

Title of the Talk : Exploring $q,t$-Catalan Polynomials: New Statistics and Dualities

Date, Time and venue : 12 March 2025, 12:00 PM - 1:00 PM at Seminar Hall, Dept of Mathematics.

Abstract : In combinatorics and various other branches of mathematics, $q$-analogs of counting functions have been extensively studied. For instance, the $q$-analog of $n!$ not only corresponds to the generating function for the inversion numbers of permutations in the symmetric group $S_n$ but also serves as the Hilbert series of an $n!$-dimensional coinvariant space.
Catalan numbers enumerate numerous combinatorial objects, including Dyck paths and plane trees. In 2003, Haglund and Haiman introduced a $q,t$-analog of Catalan numbers as a weighted generating function for Dyck paths. These $q,t$-Catalan polynomials also arise as the Hilbert series of certain $S_n$-modules in the diagonal coinvariant ring, whose dimension matches the $n$th Catalan number. A major open problem in this area is finding a combinatorial proof of the symmetry of these polynomials in $q$ and $t$.

In this talk, we begin with an overview of these concepts before presenting recent joint work with Anne Schilling and Joseph Pappe. We introduce two new $q,t$-Catalan polynomials, defined using novel statistics on Dyck paths, and prove their symmetry by developing a new duality on plane trees. We conclude with further applications and open problems in $q,t$-Catalan combinatorics.

About the speaker : Dr. Digjoy Paul is an NBHM postdoctoral fellow at IISc, Bangalore. He obtained his PhD in Mathematics from IMSc, Chennai, in 2020 under Prof. Amritanshu Prasad. He pursued postdoctoral research at TIFR Mumbai, CMI, and later at IISc as a CV Raman Fellow in 2023. His research work focuses on algebraic combinatorics, exploring the interplay between algebraic structures and combinatorial objects.