Counting Quasi-Idempotent Irreducible Integral Matrices

Abstract

Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences and their asymptotics. Such matrices turn out to be related to generalized compositions and generalized partitions.

11 pages