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Experimental Mathematics Seminar

Counting Matrices that are Squares

Victor S. Miller, Center for Communications Research, Princeton.

Location:  zoom
Date & time: Thursday, 28 January 2021 at 5:00PM - 6:00PM

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 GF(2) We follow the dictum of Wilf (in the paper 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).