An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands

TL;DR

This paper proves an infinite family of divisibility congruences for the number of partitions into odd parts with designated summands, extending previous results and employing modular forms, generating functions, and induction techniques.

Contribution

It introduces a new infinite family of internal congruences for PDO(n) modulo powers of 2, expanding the understanding of divisibility properties of these specialized partitions.

Findings

Established a family of congruences for PDO(2^{2k+3}n) modulo 2^{2k+3}.

Used modular relations and generating function dissections to prove the results.

Demonstrated novel dissection techniques differing from past literature.

Abstract

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{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, and they denoted the number of such partitions of size $n$ by the function $PDO(n)$. Since then, numerous authors have proven a variety of divisibility properties satisfied by $PDO(n)$. Recently, the second author proved the following internal congruences satisfied by $PDO(n)$: For all $n\geq 0$, \begin{align*} PDO(4n) &\equiv PDO(n) \pmod{4},\\ PDO(16n) &\equiv PDO(4n) \pmod{8}. \end{align*} In this work, we significantly extend these internal congruence results by proving the following new infinite family of congruences: For all $k\geq 0$ and all $n\geq 0$, $$PDO(2^{2k+3}n) \equiv PDO(2^{2k+1}n) \pmod{2^{2k+3}}.$$ We utilize several classical tools to prove this family, including generating function dissections via the unitizing operator of degree two, various modular relations and recurrences involving a Hauptmodul on the classical modular curve $X_0(6)$, and an induction argument which provides the final step in proving the necessary divisibilities. It is notable that the construction of each $2$-dissection slice of our generating function bears an entirely different nature to those studied in the past literature.