Some properties of dihedral compositions

Abstract

Let k, n be integers such that 1 ≤ k ≤ n. Say that two compositions of n into k parts are related if they differ only by an element of D_k (the dihedral group on a set of k elements), that is, if they differ by a cyclic shift (translation) or by reversal of the parts (reflection). This defines an equivalence relation on the set of such compositions. Let _k ⁿ denote the number of distinct corresponding equivalence classes, that is, the number of dihedral compositions of n into k parts. We prove some theorems concerning _k ⁿ .