Arithmetic properties of partitions with even parts distinct

Abstract

In this work, we consider the function ped(n), the number of partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). We prove a number of results for ped(n) including the following: For all n≥0,

$$\mathit{ped}(9n+4)\equiv0\pmod{4}$$

and

$$\mathit{ped}(9n+7)\equiv0\pmod{12}.$$

Indeed, we compute appropriate generating functions from which we deduce these congruences and find, in particular, the surprising result that

$$\sum_{n\geq0}\mathit{ped}(9n+7)q^n=12\frac{ (q^{2};q^{2})_\infty ^{4}(q^{3};q^{3})_\infty ^{6}(q^{4};q^{4})_\infty ^{}}{(q^{};q^{})_\infty ^{11}}.$$

We also show that ped(n) is divisible by 6 at least 1/6 of the time.