On cyclic compositions of positive integers |
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Authors: | Arnold Knopfmacher Florian Luca Neville Robbins |
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Institution: | 1. The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, P.O. Wits, Johannesburg, 2050, South Africa 2. Instituto de Matem??ticas, Universidad Nacional Autonoma de M??xico, C.P. 58089, Morelia, Michoac??n, Mexico 3. Mathematics Department, San Francisco State University, San Francisco, CA, 94132, USA
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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. |
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