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 p(k,m)(j, n) be the number of partitions of n into j parts where each part is = k (mod m), 1 less than or equal to k less than or equal to 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 Ic in the gaps, then p(k,m)*,(j, n) = p(k,m)(j, n).