Inverse-Conjugate Compositions Modulo $m$

Abstract

We consider inverse-conjugate compositions of a positive integer $n$ in which the parts belong to the residue class of 1 modulo an integer $m>0$. It is proved that such compositions exist only for values of $n$ that belong to the residue class of 1 modulo 2m. An enumerations results is provided using the properties of inverse-conjugate compositions. This work extends recent results for inverse-conjugate compositions with odd parts.