The arithmetical combinatorics of k, l-regular partitions

For all positive integers k, l, n, the Little Glaisher theorem states that the number of partitions of n into parts not divisible by k and occurring less than l times is equal to the number of partitions of n into parts not divisible by l and occurring less than k times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case l = 2, we discuss the possibility of constructing a Schur-type companion of the Little Glaisher theorem via the weighted words. (c) 2022 Elsevier B.V. All rights reserved.