The Inverse-conjugate Compositions With Odd Parts

Abstract

In this paper, we first present a combinatorial proof of the recurrence relation about the number of the inverse-conjugate compositions of $2n+1$, $n>1$. And then we get some counting results about the inverse-conjugate compositions for special compositions. In particular, we show that the number of the inverse-conjugate compositions of $4k+1$, $k>0$ with odd parts is $2^k$, and provide an elegant combinatorial proof. Lastly, we give a relation between the number of the inverse-conjugate odd compositions of $4k+1$ and the number of the self-inverse odd compositions of $4k+1$.