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 partitions with designated summands. These are built by taking unrestricted integer partitions and designating exactly one part of each size. In the 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 ${\rm PDO}(n)$. Since then, numerous authors have proven a variety of divisibility properties satisfied by ${\rm PDO}(n)$. Recently, the second author proved the following internal congruences satisfied by ${\rm PDO}(n)$: For all $n\geq 0$, $${\rm PDO}(4n) \equiv {\rm PDO}(n) \pmod {4},$$ $${\rm PDO}(16n) \equiv {\rm PDO}(4n) \pmod {8}.$$ In the present work, we significantly extend these results by proving the following new infinite family of congruences: For all $k\geq 0$ and all $n\geq 0$, $${\rm PDO}(2^{2k+3}n) \equiv {\rm PDO}(2^{2k+1}n) \pmod {2^{2k+3}}.$$ To do so, we utilize several classical tools, including generating function dissections via the unitizing operator of degree 2, 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.
