The q, t-symmetry of the generalized q, t-Catalan number C(k1,k2,k3)(q,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{(k_1,k_2,k_3)}(q,t)$$\end{document}

We present two distinct proofs of the q, t-symmetry for the generalized q, t-Catalan number Ck→(q,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\vec {k}}(q,t)$$\end{document}, where k→=(k1,k2,k3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {k}=(k_1,k_2,k_3)$$\end{document}. The first proof is derived through the application of MacMahon’s partition analysis. The second proof is established via a direct bijection.