A combinatorial approach to partitions with parts in the gaps
Volume 85 / 1998
Acta Arithmetica 85 (1998), 119-133
DOI: 10.4064/aa-85-2-119-133
Abstract
Many links exist between ordinary partitions and partitions with parts in the "gaps". In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let $p^_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p*_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p*_{k,m}(j,n)=p_{k,m}(j,n)$.
