A Three-way Correspondence Between Partitions

Abstract

In a recent paper a 1-1 correspondence was established between partitions of a positive integer n of the form (p1p2p3...pn) where pj − pj+k−1 ⩾ 2 and at most i − 1 of the pj equal 1, and partitions of n whose hook differences dj all lie in the range −(i − 2) ⩽ dj ⩽ 2k − i − 1. The correspondence uses on a sequence of operations for constructing partitions as an intermediate state, and the size of the partition produced is the greater index of the corresponding sequence. In this paper the correspondence is extended to a three-way correspondence by showing how the greater index of a sequence may be decomposed and reassembled to form a Durfee dissection partition. These correspondences, together with other rearrangements, are then used to interpret some of Slater's identities as generating functions for sets of partitions. © 1982, Academic Press Inc. (London) Limited. All rights reserved.