Congruences for two-color partitions with odd smallest part

Abstract

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)(\mathrm{mod}4)$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from ${\lim }_{k\to \infty }{C}_k(q)$.

20 pages, submitted. This a replacement for the earlier version ''On two color partitions with odd smallest part"