Let F = GF( q) denote the finite field of order q, and let F n× n denote the algebra of n × n matrices over F. A function f: F n× n → F n× n is called a scalar polynomial function if there exists a polynomial f( x) ϵ F[ x] which represents f when considered as a matrix function under substitution. In this paper a formula is obtained for the number of permutations of F n× n which are scalar polynomial functions.