Refinements of Beck-type partition identities

Franklin's identity generalizes Euler's and states that the number of partitions n with j different parts divisible by r equals repeated at least times. In this article, we give a refinement when j=1. We prove j=1, r=2 for fixed perimeter, i.e., largest hook. also derive Beck-type perimeter: excess in all into odd perimeter M over distinct whose set even is singleton. provide analytic combinatorial proofs our results.