Congruence properties for Schmidt type d-fold partition diamonds

Abstract

Recently, Dockery, Jameson, Sellers and Wilson introduced new combinatorial objects called d-fold partition diamonds, which generalize both the classical partition function and the plane partition diamonds of Andrews, Paule and Riese. They also investigated a partition function $s_d\left(n\right)$ which counts the number of Schmidt type d-fold partition diamonds of n. They presented the generating functions of $s_d\left(n\right)$ and proved several congruences for $s_d\left(n\right)$. At the end of their paper, they posed a conjecture on congruences modulo 7 for $s_{6k+1}\left(n\right)$ and $s_{6k+2}\left(n\right)$. In this paper, we prove the conjectural congruences for $s_{6k+1}\left(n\right)$ by using two methods: an elementary proof based on a result of Garvan and an algorithmic proof based on the Mathematica package RaduRK by Smoot. We also give an algorithmic proof of the conjectural congruences for $s_{6k+2}\left(n\right)$ by utilizing Smoot's Mathematica package RaduRK. In addition, we prove new infinite families of congruences modulo 7 for $s_{6k+1}\left(n\right)$ and prove that $\frac{s_{6k+1}(7n+3)}{7}$ takes integer values with probability 1 for $n\ge 0$. Moreover, we show that there exist infinitely many integers $r_i$ such that $s_{12k+1}\left(r_i\right)\equiv i\left(\mathrm{mod}13\right)$ with $0\le i\le 12$.