On the $LU$ Factorization of M-Matrices: Cardinality of the Set $\mathcal{P}_n^g ( A )$

An $n \times n$M-matrix A is said to admit an $LU$ factorization into $n \times n$M-matrices if A can be expressed as $A = LU$ where L is an $n \times n$ lower triangular M-matrix and where U is an upper triangular M-matrix. Then, for any given $n \times n$M-matrix A, let $\mathcal{P}_n^g ( A )$ denote the set of all $n \times n$ permutation matrices P such that $PAP^T $ admits an LU factorization into M-matrices with nonsingular L. Our aim here is to determine upper and lower bounds for $| \mathcal{P}_n^g ( A ) |$, the cardinality of the set $\mathcal{P}_n^g ( A )$. This is done in Theorem 4, while in Theorem 2, $| \mathcal{P}_n^g ( A ) |$ is precisely determined for a special class of $n \times n$M-matrices.