Counting Quasi-Idempotent Irreducible Integral Matrices

TL;DR

This paper investigates the finiteness and enumeration of irreducible matrices satisfying polynomial equations, focusing on the case p(X)=X^2-nX, and explores their connections to combinatorial structures.

Contribution

It proves the finiteness of irreducible matrices satisfying polynomial equations and counts those for p(X)=X^2-nX, linking them to combinatorial sequences.

Findings

Finite set of irreducible matrices for any polynomial p

Explicit count of matrices for p(X)=X^2-nX

Connections to generalized compositions and partitions

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.