Inverse-conjugate compositions into parts of size at most \(k\)

Yu-Hong Guo1, Augustine O. Munagi2
1School of Mathematics and Statistics, Hexi University, Zhangye, Gansu, 734000, P.R.China
2The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg, South Africa, Augustine.

Abstract

An inverse-conjugate composition of a positive integer \(m\) is an ordered partition of \(m\) whose conjugate coincides with its reversal. In this paper, we consider inverse-conjugate compositions in which the part sizes do not exceed a given integer \(k\). It is proved that the number of such inverse-conjugate compositions of \(2n – 1\) is equal to \(2F_n^{(k-1)}\), where \(F_n^{(k)}\) is a Fibonacci \(k\)-step number. We also give several connections with other types of compositions, and obtain some analogues of classical combinatorial identities.

Keywords: composition, inverse-conjugate, self-inverse, Fibonacci number, combinatorial identity.