Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities

Abstract: Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up, we investigate a handful of double sum identities appeared in recent works of Cao-Wang, Wang-Wang, Wei-Yu-Ruan, Andrews-Uncu, Chern, and Wang, finding partition theoretical interpretations to all of these identities, and in most cases supplying Franklin-type involutive proofs. This approach dates back more than a century to P. A. MacMahon's interpretations of the celebrated Rogers-Ramanujan identities, and has been further developed by Kur\c{s}ung\"{o}z in the last decade.