Let T_ℓ(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,$${T_7}\left( {{7^{2\alpha }}n + \frac{{3 \times {7^{2\alpha }} - 3}}{4}} \right) \equiv 0\;(\bmod {7^\alpha })$$

中文翻译：

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7个规则分区三元组的算术性质

让Ť _ℓ（Ñ）表示的ℓ正规分区三元组的数量Ñ。在本文中，我们考虑了T ₇（n）的算术性质。建立了7的幂等幂的无限家族和7的几个幂等模。例如，我们证明了对所有Ñ ≥0和α≥1，$$ {T_7} \左（{{7 ^ {2 \阿尔法}}的n + \压裂{{3 \倍{7 ^ {2 \阿尔法}}-3}} {4}} \ right）\ equiv 0 \;（\ bmod {7 ^ \ alpha}）$$