Inverse-conjugate compositions into parts of size at most $k$

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.