d-Fold Partition Diamonds

TL;DR

This paper introduces new combinatorial objects called d-fold partition diamonds, derives their generating functions, and proves Ramanujan-like congruences for their counting functions, connecting combinatorics with number theory.

Contribution

The work generalizes classical partition concepts to d-fold partition diamonds, derives their generating functions, and establishes new Ramanujan-like congruences.

Findings

Derived generating functions for d-fold and Schmidt type d-fold partition diamonds.

Connected generating functions to Eulerian polynomials.

Proved infinite families of Ramanujan-like congruences for s_d(n).

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 $r_d(n)$ to be their counting function. We also consider the Schmidt type $d$--fold partition diamonds, which have counting function $s_d(n).$ Using partition analysis, we then find the generating function for both, and connect the generating functions $\sum_{n= 0}^\infty s_d(n)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan--like congruences satisfied by $s_d(n)$ for various values of $d$, including the following family: for all $d\geq 1$ and all $n\geq 0,$ $s_d(2n+1) \equiv 0 \pmod{2^d}.$