Abstract:
We enumerate distinct integer partitions according to the number of parts and the number of consecutive runs with odd length. This outputs a double-sum expression for a bi-variate generating function denoted as D(x,y;q). We discuss in this talk how D(x,y;q) fits well with existing literature on integer partitions as well as some recent Rogers-Ramanujan type identities. First off, we give partition theoretical interpretations and combinatorial proofs to a pair of (1,2)-indexed double-sum Rogers-Ramanujan type identities, previously derived by Cao and Wang using an integration method. Secondly, we make a connection with the 2-measure of a strict partition, a notion recently introduced by Andrews, Chern and Li. Finally, we explain how similar idea leads to partition theoretical interpretations to a pair of (1,3)-indexed double-sum Rogers-Ramanujan type identities due to Andrews-Uncu, and Chern-Wang, and we note that this interpretation extends to a (1,2b+1)-indexed generalization made by Andrews and Uncu as well. The talk is based on joint work with Haijun Li.