The $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{(k_1,k_2,k_3)}(q,t)$ and $C_{(k,k,k,k)}(q,t)$

TL;DR

This paper proves the $q,t$-symmetry of generalized $q,t$-Catalan numbers for specific parameter sets using two different methods, including MacMahon's partition analysis and bijections, confirming symmetry properties.

Contribution

It provides two novel proofs of the $q,t$-symmetry for generalized $q,t$-Catalan numbers, expanding understanding of their combinatorial properties.

Findings

Established $q,t$-symmetry for $C_{oldsymbol{k}}(q,t)$ with $oldsymbol{k}=(k_1,k_2,k_3)$

Proved $C_{(k,k,k,k)}(q,t) = C_{(k,k,k,k)}(t,q)$ using MacMahon's partition analysis

Demonstrated bijective proof for symmetry in specific cases

Abstract

We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{\vec{k}}(q,t)$ for $\vec{k}=(k_1,k_2,k_3)$. One is by using MacMahon's partition analysis as we proposed; the other is a direct bijection. We also prove $C_{(k,k,k,k)}(q,t) = C_{(k,k,k,k)}(t,q)$ by using MacMahon's partition analysis.