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

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 $n\ge 0$, \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 $k\ge 0$ and all $n\ge 0$, $$PDO(2^{2k+3}n)\equiv PDO(2^{2k+1}n)(\mathrm{mod}{2}^{2k+3}).$$ 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.