Sequence related to figurate numbers

[Andrew Plewe]

The following statements (suggested by lofar in russian forum
lib.mexmat.ru/forum/) are true:

Let n x n matrices A and B contain the same n^2 variables as elements, then
1) If |det(A)|=|det(B)| then B=PAQ or B=(PAQ)^T where P and Q are some
permutation matrices.
2) If |det(A)|=|det(B)| and trace(A)=trace(B) then B=PAP^{-1} or
B=(PAP^{-1})^T where P is some permutation matrix.

Now it is easy to give an answer to your original question:

> How many different matrices n x n with n^2 different elements occured which have that same characteristic polynomial?

Note that if charpoly(A)=charpoly(B) then det(A)=det(B) and
trace(A)=trace(B), implying that B=PAP^{-1} or B=(PAP^{-1})^T for some
permutation matrix P. Therefore, the answer to your question is 2*n!
(where n! stands for the number of different permutation matrices and
2 accounts for a possible transposition of the matrix).

Max

On 1/8/07, Artur <grafix at csl.pl> wrote:
> Dear Seqfans,
> I'm asking: How many different matrices n x n with n^2 different elements
> occured which have that same characteristic polynomial
> We have to count all permutations n^2 elements in n x n matrix and count
> only these permutations
> which don't changed starting polynomial
> for 2 x 2 case
> we have 4 matrices X^2-(a+d)X+ad-bc
> a b   a c   d c   d b
> c d   b d   b a   c a
>
> BEST WISHES
> ARTUR
>
>