Ramanujan-type Congruences for Overpartitions Modulo 16

TL;DR

This paper establishes new Ramanujan-type congruences for overpartition counts modulo 16, including a 16-dissection of the generating function and several infinite families of congruences, advancing understanding of overpartition arithmetic properties.

Contribution

It provides the first 16-dissection of the overpartition generating function modulo 16 and derives multiple infinite families of congruences using 2-adic expansions.

Findings

Proves $ar{p}(16n+14) ot ext{ } ext{divisible by } 16$ for all $n$.

Establishes $ar{p}( ext{ell}^2 n + r ext{ell}) ot ext{ } ext{divisible by } 16$ for certain primes $ ext{ell}$.

Derives explicit congruences such as $ar{p}(4n) ot ext{ } ext{divisible by } 16$ with specific parity and residue conditions.

Abstract

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overline{p}(5n+2)\equiv 0 \pmod{4}$, $\overline{p}(4n+3)\equiv 0 \pmod{8}$ and $\overline{p}(8n+7)\equiv 0 \pmod{64}$. By employing dissection techniques, Yao and Xia obtained congruences for $\overline{p}(n)$ modulo $8, 16$ and $32$, such as $\overline{p}(48n+26) \equiv 0 \pmod{8}$, $\overline{p}(24n+17)\equiv 0 \pmod{16}$ and $\overline{p}(72n+69)\equiv 0 \pmod{32}$. In this paper, we give a 16-dissection of the generating function for $\overline{p}(n)$ modulo 16 and we show that $\overline{p}(16n+14)\equiv0\pmod{16}$ for $n\ge 0$. Moreover, by using the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg, we obtain that $\overline{p}(\ell^2n+r\ell)\equiv0\pmod{16}$, where $n\ge 0$, $\ell \equiv -1\pmod{8}$ is an odd prime and $r$ is a positive integer with $\ell \nmid r$. In particular, for $\ell=7$, we get $\overline{p}(49n+7)\equiv0\pmod{16}$ and $\overline{p}(49n+14)\equiv0\pmod{16}$ for $n\geq 0$. We also find four congruence relations: $\overline{p}(4n)\equiv(-1)^n\overline{p}(n) \pmod{16}$ for $n\ge 0$, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{32}$ for $n$ being not a square of an odd positive integer, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{64}$ for $n\not\equiv 1,2,5\pmod{8}$ and $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{128}$ for $n\equiv 0\pmod{4}$.