Congruences and density results for partitions into distinct even parts

TL;DR

The paper analyzes ped(n), the number of partitions of n with distinct even parts and unrestricted odd parts, via its generating function $ped(q)=\frac{(-q^2;q^2)_{\infty}}{(q;q^2)_{\infty}}$. Employing Radu’s algorithm, eta-quotient modular forms, and Newman’s and Ono–Taguchi’s results, it proves Nath’s conjectured mod $192$ congruences and establishes infinite families of mod $24$ congruences, as well as lacunarity results for ped($9n+7$) modulo powers of $2$ and $3$. A density result is proven: for each $k\ge1$, ped($9n+7$) is divisible by $2^{k+2}\cdot3$ for almost all $n$, using a modular form ${\mathcal B}_{2,k}(z)$ on $\Gamma_0(2304)$ and Serre’s on coefficient density. Further, a nilpotency theorem for Hecke operators (Ono–Taguchi) yields infinite families of $2$-adic congruences for ped, strengthening the understanding of arithmetic properties of this partition function and its links to modular forms.

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.

Theorems & Definitions (20)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5