Almost 3-regular overpartitions

Abstract

Let $\overline{ A}_\ell (n)$ be the number of $\ell$-regular overpartitions of n, i.e., overpartitions of n into parts not divisible by $\ell$. Let $\overline{ B}_\ell (n)$ be the number of almost $\ell$-regular overpartitions of n, i.e., overpartitions of n in which none of its overlined parts is divisible by $\ell$. In this paper, we study the connections between $\overline{ A}_3(n)$, respectively $\overline{ B}_3(n)$, and the singular overpartition functions $\overline{ C}_{12,5}(n)$ and $\overline{ C}_{12,1}(n)$ which count the number of overpartitions into parts not divisible by 12 and in which only parts congruent to $\pm 5 \pmod {12}$, respectively $\pm 1 \pmod {12}$, may be overlined. We give a combinatorial proof for the the surprising identity $\overline{ C}_{12,5}(n)=\overline{ C}_{12,1}(n-1)$. We also provide a linear homogeneous recurrence relation for $\overline{ B}_3(n)$ and give an alternate combinatorial interpretation for $\overline{ B}_3(n)$.