Counting Matrices that are Squares

Abstract

On the math-fun mailing list (7 May 2013), Neil Sloane asked to calculate the number of $n \times n$ matrices with entries in $\{0,1\}$ which are squares of other such matrices. In this paper we analyze the case that the arithmetic is in $\mathbb{F}_{2}$. We follow the dictum of Wilf ("What is an answer?") to derive a "effective" algorithm to count such matrices in much less time than it takes to enumerate them. The algorithm which we use involves the analysis of conjugacy classes of matrices. The restricted integer partitions which arise are counted by the coefficients of one of Ramanujan's mock Theta functions, which we found thanks to Sloane's OEIS (Online Encyclopedia of Integer Sequences). Let $a_n$ be the number elements of ${\rm Mat}_n(\mathbb{F}_{2})$ which are squares, and $b_n$ be the number of elements of ${\rm GL}(n,\mathbb{F}_{2})$ which are squares. The numerical results strongly suggest that there are constants $\alpha,\beta > 0$ such that $a_n \sim \alpha 2^{n^2}$, $b_n \sim \beta 2^{n^2}$.