Partitions of large multipartites with congruence conditions. I

Abstract:

Let $p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ be the number of partitions of $({n_1}, \ldots ,{n_j})$ where, for $1 \leqslant l \leqslant j$, the lth component of each part belongs to the set ${A_l} = \bigcup \nolimits _{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} }$ and $M,q(l)$ and the ${a_{lh(l)}}$ are positive integers such that $0 < {a_{l1}} < \cdots < {a_{lq(l)}} \leqslant M$. Asymptotic expansions for $p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ are derived, when the ${n_l} \to \infty$ subject to the restriction that ${n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }$ for all l, where $\in$ is any fixed positive number. The case $M = 1$ and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case $j = 1$ for different values of M.