计算矩阵的平方
Counting Matrices that are Squares
摘要：
在数学趣味邮件列表（2013年5月7日）上，Neil Sloane要求计算$n \times n$矩阵的个数，其中$\{0,1\}$中的元素是其他矩阵的平方。本文分析了算法在$\mathbb{F}_{2}$中的情况。我们遵循威尔夫的格言（“什么是答案？）推导出一种“有效”的算法，以比枚举这些矩阵所需的时间少得多的时间来计数这些矩阵。我们使用的算法涉及矩阵的共轭类的分析。出现的受限整数分区由Ramanujan的模拟Theta函数之一的系数计数，这要归功于Sloane的OEIS（在线百科全书）。令$a_n$为${\rm Mat}_n(\mathbb{F}_{2})$中正方形元素的个数，$b_n$为${\rm GL}(n,\mathbb{F}_{2})$中正方形元素的个数。数值结果强烈地表明存在常数$α,β> 0$使得$a_n \sim α2^{n^2}$，$b_n \sim β2^{n^2}$ .
英文摘要：
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 $α,β> 0$ such that $a_n \sim α2^{n^2}$, $b_n \sim β2^{n^2}$.
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