Generalized Catalan numbers

TL;DR

This paper explores a potential q-deformation of algebraic structures related to Catalan numbers, aiming to categorify identities similar to Rogers-Ramanujan type, with initial evidence supporting its existence.

Contribution

It proposes a conjecture for a q-deformed algebraic structure that generalizes Catalan numbers and provides initial evidence for its validity.

Findings

Evidence for the first two non-trivial cases of the q-deformation

Conjecture of a q-deformed algebraic structure related to Catalan numbers

Potential categorification of Rogers-Ramanujan type identities

Abstract

A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial cases. This $ q $-deformed version will likely categorify series arise from identities of the Roger-Ramanujan type.