Extending Congruences for the number of smallest parts in overpartitions with smallest part even
Robson da Silva
TL;DR
The paper broadens the landscape of modulo $4$ congruences for the overpartition statistic $\overline{spt2}(n)$ by presenting new individual congruences at $36n+30$, $48n+34$, and $72n+42$, and by establishing six infinite Ramanujan-like families parameterized by powers of two. Central to the method are 2- and 3-dissections of generating functions, Ramanujan theta functions, and leveraging a key internal congruence $\overline{spt2}(8n+5)\equiv \overline{spt2}(16n+10)\pmod{4}$. The work also proves a further Ramanujan-like congruence in the presence of primes $p>3$ with $p\equiv5$ or $7\pmod{8}$, giving a family $\overline{spt2}(32 p^{2k+1} m + 24 p^{2k+2})\equiv 0\pmod{4}$ under suitable conditions. Collectively, these results deepen our understanding of partition- and overpartition-based statistics modulo small primes and illustrate the power of dissections and modular-analytic techniques in obtaining Ramanujan-type congruences. The findings open avenues for discovering additional congruences and for exploring similar phenomena in related spt-type functions.
Abstract
In a recent paper, Jin, Liu, and Xia \cite{JLX} presented some modulo 4 congruences for $\overline{spt2}(n)$, the number of smallest parts in the overpartitions of $n$ where the smallest part is even and is not overlined. In this paper, we extend the list of such congruences in two directions. First, we prove some new individual congruences for $\overline{spt2}(n)$. Then, we provide a number of infinite families of Ramanujan-like congruences satisfied by $\overline{spt2}(n)$.