Arithmetic properties for 7-regular partition triples

Let Tℓ(n) denote the number of ℓ-regular partition triples of n. In this paper, we consider the arithmetic properties of T7(n). An infinite family of congruences modulo powers of 7 and several congruences modulo 7 are established. For instance, we prove that for all n ≥ 0 and α ≥ 1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T_7}\left( {{7^{2\alpha }}n + \frac{{3 \times {7^{2\alpha }} - 3}}{4}} \right) \equiv 0\;(\bmod {7^\alpha })$$\end{document}