A Filtration of (q,t)-Catalan numbers

TL;DR

This paper introduces new generalizations of (q,t)-Catalan numbers using the nabla operator on k-Schur functions, providing combinatorial and algebraic interpretations in special cases.

Contribution

It defines novel (q,t)-Catalan generalizations and offers combinatorial and algebraic interpretations for specific cases, expanding understanding of these polynomials.

Findings

Combinatorial interpretation via Dyck path configurations

Algebraic interpretation as Hilbert series of sub-modules

Extension of (q,t)-Catalan number framework

Abstract

We define new generalizations of (q,t)-Catalan numbers applying nabla operator on k-Schur functions indexed by column partitions. In some special cases, we give a combinatorial interpretation of these numbers using configurations of Dyck paths. In some other special cases, we also interpret these new polynomial as Hilbert series of explicit sub-modules of the alternating diagonal harmonics built using differential operators.