Arithmetic Properties Satisfied by a Recent Integer Partition Function of Dombos

Abstract

In recent work of Dombos, the set of integer partitions of $n$ wherein the parts are either divisible by 4 or congruent to $\pm 1(\mathrm{mod}6)$ arose in a natural way. In this work, we will denote the function which counts the number of such partitions of $n$ by $dp(n)$. Using elementary generating function manipulations and classical $q$--series results, we prove several congruences satisfied by $dp(n)$. As an example, we prove that, for all $\hat{I}\pm \ge 1$ and all $n\ge 0$, \begin{equation*} dp \left( 3^{2α+ 1}n + \frac{7 \cdot 9^α+ 1}{4} \right) \equiv 0 \pmod{3}. \end{equation*}