A matrix for which horizontal and vertical dimensions are the same (i.e., an 
matrix).
A matrix 
may be tested to determine if it is square in Wolfram
Language using SquareMatrixQ[m].
Consider the numbers of 
matrices on 
distinct symbols. The number of distinct matrices modulo rotations and reflections
for 
, 2, ... are given by 1, 3, 45360,
... (OEIS A086829).
Consider an 
matrix consisting of the integers 1 to 
arranged in any order. Then the maximal determinants possible
for 
, 2, ... are 1, 10, 412, 40800, 6839492,
... (OEIS A085000).
Consider an 
matrix with single copies of the digits 1, 2, ..., 
and the rest of the elements zero. Then the triangle of 
matrices with digits 
, 1, ..., 
that are rotationally and reflectively distinct is 1, 1;
1, 1, 2, 3, 3; 1, 3, 12, 66, 378, 1890, 7560, 22680, 45360, 45360; ... (OEIS A087074).
