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In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objectswhich they called \emph{partitions with designated summands}. These are builtby taking unrestricted integer partitions and designating exactly one of eachoccurrence of a part. In that same work, Andrews, Lewis, and Lovejoy alsostudied such partitions wherein all parts must be odd, and they denoted thenumber 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 congruencessatisfied 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, wesignificantly extend these internal congruence results by proving the followingnew 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 severalclassical tools to prove this family, including generating function dissectionsvia the unitizing operator of degree two, various modular relations andrecurrences involving a Hauptmodul on the classical modular curve $X_0(6)$, andan induction argument which provides the final step in proving the necessarydivisibilities. It is notable that the construction of each $2$-dissectionslice of our generating function bears an entirely different nature to thosestudied in the past literature.

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