New Infinite Families of Congruences Modulo Powers of 2 for 2--Regular Partitions with Designated Summands

TL;DR

This paper establishes new infinite families of congruences modulo powers of 2 for the function counting partitions with designated summands where all parts are odd, using elementary q-series techniques.

Contribution

It proves new Ramanujan-like congruences for the partition function with designated summands, confirming conjectures and expanding understanding of their divisibility properties.

Findings

Proves that PD_2(2^α(4n+3)) ≡ 0 mod 4 for all α ≥ 0

Shows PD_2(2^α(8n+7)) ≡ 0 mod 8 for all α ≥ 0

Uses elementary q-series identities for proofs

Abstract

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called {\it partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd. Recently, Herden, Sepanski, Stanfill, Hammon, Henningsen, Ickes, and Ruiz proved a number of Ramanujan--like congruences for the function $PD_2(n)$ which counts the number of partitions of weight $n$ with designated summands wherein all parts must be odd. In this work, we prove some of the results conjectured by Herden, et. al. by proving the following two infinite families of congruences satisfied by $PD_2(n)$: For all $\alpha\geq 0$ and $n\geq 0,$ \begin{eqnarray*} PD_2(2^\alpha(4n+3)) &\equiv & 0 \pmod{4} \ \ \ \ \ {\textrm and} \\ PD_2(2^\alpha(8n+7)) &\equiv & 0 \pmod{8}. \end{eqnarray*} All of the proof techniques used herein are elementary, relying on classical $q$--series identities and generating function manipulations.