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We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalannumber $C_{\vec{k}}(q,t)$ for $\vec{k}=(k_1,k_2,k_3)$. One is by usingMacMahon'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'spartition analysis.

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