### 4-core partitions and class numbers

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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 $p{*}_{k,m}(j,n)={p}_{k,m}(j,n)$.

We offer some relations and congruences for two interesting spt-type functions, which together form a relation to Andrews' spt function.