2016/06/29 by Victor S. Miller, Miller, Victor S.
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1606.09299
openalex publication_date 2016/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On the math-fun mailing list (7 May 2013), Neil Sloane asked to calculate the number of n × 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 \mathbbF2. 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 an be the number elements of \rm Matn(\mathbbF2) which are squares, and bn be the number of elements of \rm GL(n,\mathbbF2) which are squares. The numerical results strongly suggest that there are constants α,β> 0 such that an ∼ α2n2, bn ∼ β2n2.