On cyclic compositions of positive integers

Abstract

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let \({\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}\) denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We show that the sequence \({\left\langle\begin{array}{c}n \\ k\end{array}\right\rangle}\) is log-concave and prove some results concerning \({\left\langle \begin{array}{c}n \\ k \end{array} \right\rangle}\) modulo two.