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

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.