Partitions of large multipartites with congruence conditions. I

Abstract

Let p ( n 1 , … , n j : A 1 , … , A j ) p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j}) be the number of partitions of ( n 1 , … , n j ) ({n_1}, \ldots ,{n_j}) where, for 1 ⩽ l ⩽ j 1 \leqslant l \leqslant j , the lth component of each part belongs to the set A l = ⋃ h ( l ) = 1 q ( l ) { a l h ( l ) + M v : v = 0 , 1 , 2 , … } {A_l} = \bigcup \nolimits _{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} } and M , q ( l ) M,q(l) and the a l h ( l ) {a_{lh(l)}} are positive integers such that 0 > a l 1 > ⋯ > a l q ( l ) ⩽ M 0 > {a_{l1}} > \cdots > {a_{lq(l)}} \leqslant M . Asymptotic expansions for p ( n 1 , … , n j : A 1 , … , A j ) p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j}) are derived, when the n l → ∞ {n_l} \to \infty subject to the restriction that n 1 ⋯ n j ⩽ n l j + 1 − ∈ {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 M = 1 and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case j = 1 j = 1 for different values of M.