d-fold partition diamonds

Abstract

In this work we introduce new combinatorial objects called d–fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d\left(n\right)$ to be their counting function. We also consider the Schmidt type d–fold partition diamonds, which have counting function $s_d\left(n\right)$. Using partition analysis, we then find the generating function for both, and connect the generating functions ${\sum }_{n=0}^{\infty }{s}_d\left(n\right)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan–like congruences satisfied by $s_d\left(n\right)$ for various values of d, including the following family: for all $d\ge 1$ and all $n\ge 0$, $s_d(2n+1)\equiv 0\left(\mathrm{mod}{2}^d\right)$.