Recursions for rational q,t-Catalan numbers

TL;DR

This paper introduces a new recursive method for computing rational q,t-Catalan series, connecting link homology with combinatorial structures, and compares it with existing recursions.

Contribution

It presents a simple binary-sequence labeled recursion for rational q,t-Catalan numbers, extending previous work and establishing links to Khovanov-Rozansky homology.

Findings

New recursion for rational q,t-Catalan series

Comparison with Hogancamp-Mellit's recursion

Connection to Khovanov-Rozansky homology of torus links

Abstract

We give a simple recursion labeled by binary sequences which computes rational $q,t$-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of $N,M$-torus links and the rational $q,t$-Catalan power series for general positive $N,M.$