Inverse-conjugate compositions into parts of size at most k

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 inverseconjugate 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(k-1)n , where F(k)n is a Fibonacci k-step number. We also give several connections with other types of compositions, and obtain some analogues of classical combinatorial identities.