Generating Function for $K$-Restricted Jagged Partitions

We present a natural extension of Andrews' multiple sums counting partitions of the form $(\lambda_1,\cdots,\lambda_m)$ with $\lambda_i\geq \lambda_{i+k-1}+2$. 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 $(n_1,n_2,\cdots , n_m)$ with $n_m\geq 1$ subject to the weakly decreasing conditions $n_i\geq n_{i+1}-1$ and $n_i\geq n_{i+2}$. The $K$-restriction refers to the following additional conditions: $n_i \geq n_{i+K-1} +1$ or $n_i = n_{i+1}-1 = n_{i+K-2}+1= n_{i+K-1}$. The corresponding generalization of the Rogers-Ramunjan identities is displayed, together with a novel combinatorial interpretation.