\(d\)-fold partition diamonds (Q6611725)

In the first part of this paper the authors introduce \(d\)-fold partition diamonds and use MacMahon's partition analysis to compute their generating function. They do the same for a ``Schmidt-type'' version, where only certain nodes of the diamond contribute to the weight. For example, if \(s_d(n)\) denotes the number of Schmidt-type \(d\)-fold partition diamonds of weight \(n\), then \N\[ \sum_{n \geq 0} s_d(n)q^n = \prod_{n=1}^{\infty} \frac{A_d(q^n)}{(1-q^n)^{d+1}} = \prod_{n \geq 1}\sum_{j \geq 0} (j+1)^dq^{jn}, \] \Nwhere \(A_d(x)\) is the \(d\)th Eulerian polynomial. The cases \(d=1\) and \(2\) recover known results for ordinary partitions and plane partition diamonds, respectively.\N\NIn the second part of the paper the authors prove a number of families of congruences for \(s_d(n)\). For example, they show that for all \(k,n \geq 0\) one has \N\[ s_{4k+2}(25n+23) \equiv 0 \pmod{5}. \] \NThe proofs are typically done by first establishing the congruence for a base case and then invoking a lemma which implies that \N\[ s_{(p-1)k +r}(n) \equiv s_r(n) \pmod{p} \] \Nif \(p\) is prime. (This lemma -- Lemma 3.5 -- does not appear to the reviewer to be true in the generality the authors claim, though it holds for the special cases used in the paper.)