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# Problem 626

## Counting Binary Matrices

A binary matrix is a matrix consisting entirely of 0s and 1s. Consider the following transformations that can be performed on a binary matrix:

• Swap any two rows
• Swap any two columns
• Flip all elements in a single row (1s become 0s, 0s become 1s)
• Flip all elements in a single column

Two binary matrices $A$ and $B$ will be considered equivalent if there is a sequence of such transformations that when applied to $A$ yields $B$. For example, the following two matrices are equivalent:

$$A=\begin{pmatrix} 1 & 0 & 1 \ 0 & 0 & 1 \ 0 & 0 & 0 \end{pmatrix}\quad B=\begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}$$

via the sequence of two transformations “Flip all elements in column 3” followed by “Swap rows 1 and 2”.

Define $c(n)$ to be the maximum number of $n\times n$ binary matrices that can be found such that no two are equivalent. For example, $c(3)=3$. You are also given that $c(5)=39$ and $c(8)=656108$.

Find $c(20)$, and give your answer modulo $1\ 001\ 001\ 011$.

## 01矩阵计数

01矩阵指的是只由0或1构成的矩阵。对于任意01矩阵，可以进行以下变形：

• 交换任意两行
• 交换任意两列
• 翻转一行中所有元素（1变成0，0变成1）
• 翻转一列中所有元素

$$A=\begin{pmatrix} 1 & 0 & 1 \ 0 & 0 & 1 \ 0 & 0 & 0 \end{pmatrix}\quad B=\begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}$$