Combinatorial identities for restricted set partitions

Abstract

Motivated by a problem of Knuth we consider set partitions in which every pair of elements a,b in a block satisfies |a−b|≠d>0. We explore enumeration results and combinatorial identities which arise from imposition of related distance restrictions on partitions. Our tools include elementary bijection techniques and the concept of generalized successions, that is, pairs of consecutive elements in a sequence. The underlying theme is a correspondence between successions and the cardinality of a distinguished block in a partition. As an application we obtain a new proof of a combinatorial identity found by Chu and Wei (2008).