Congruences and density results for partitions into distinct even parts

Abstract

In this paper, we consider the set of partitions $ped(n)$ which counts the number of partitions of $n$ wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences modulo 192 which were conjectured by Nath. Further, we prove a few infinite families of congruences modulo 24 by using a result of Newman. Also, we prove that $ped(9n+7)$ is lacunary modulo $2^{k+2}\cdot 3$ and $3^{k+1}\cdot 4$ for all positive integers $k\geq0$. We further prove an infinite family of congruences for $ped(n)$ modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.