A combinatorial approach to partitions with parts in the gaps

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 ${\displaystyle p_{k,m}^{\ast }(j,n)}$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let ${\displaystyle p_{k,m}^{\ast }(j,n)}$ 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 ${\displaystyle p_{k,m}^{\ast }(j,n)=p_{k,m}(j,n)}$.