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

We present two distinct proofs of the q, t-symmetry for the generalized q, t-Catalan number \(C_{\vec {k}}(q,t)\) $C_{\overrightarrow{k}}(q,t)$ , where \(\vec {k}=(k_1,k_2,k_3)\) $\overrightarrow{k}=(k_1,k_2,k_3)$ . The first proof is derived through the application of MacMahon’s partition analysis. The second proof is established via a direct bijection.