Generating Function for K-Restricted Jagged Partitions

We present a natural extension of Andrews’ multiple sums counting partitions of the form (λ1, · · · , λm) with λi ≥ λi+k−1 + 2. The multiple sum that we construct is the generating function for the so-called K-restricted jagged partitions. A jagged partition is a sequence of non-negative integers (n1, n2, · · · , nm) with nm ≥ 1 subject to the weakly decreasing conditions ni ≥ ni+1− 1 and ni ≥ ni+2. The K-restriction refers to the following additional conditions: ni ≥ ni+K−1 + 1 or ni = ni+1 − 1 = ni+K−2 + 1 = ni+K−1. The corresponding generalization of the Rogers-Ramunjan identities is displayed, together with a novel combinatorial interpretation.