Limits of Modified Higher (q,t)-Catalan Numbers

TL;DR

This paper investigates the limits of modified higher q,t-Catalan numbers, revealing their connection to partition generating functions and proposing conjectures on their algebraic and combinatorial properties.

Contribution

It computes the limits of various modified higher q,t-Catalan numbers and links these limits to partition generating functions, advancing understanding of their algebraic-combinatorial equivalence.

Findings

Limits of modified higher q,t-Catalan numbers equal partition generating functions.

Certain coefficients count specific integer partitions.

Conjectures on the homological significance of the nabla operator.

Abstract

The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of several versions of the modified higher $q,t$-Catalan numbers and show that these limits equal the generating function for integer partitions. We also identify certain coefficients of the higher $q,t$-Catalan numbers as enumerating suitable integer partitions, and we make some conjectures on the homological significance of the Bergeron-Garsia nabla operator.