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A364790 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 0, 1, 2, ..., n-1.

%I A364790 #13 Jan 08 2024 01:09:18
%S A364790 1,0,2,0,0,6,0,0,1,23,0,0,0,0,120,0,0,0,0,2,718,0,0,0,0,4,31,5005,0,0,
%T A364790 0,0,0,2,44,40274,0,0,0,0,0,0,4,284,362592,0,0,0,0,0,0,0,111,769,
%U A364790 3627920,0,0,0,0,0,0,2,14,244,7056,39909484,0,0,0,0,0,0,0,4,64,742,9667,478991123
%N A364790 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 0, 1, 2, ..., n-1.
%H A364790 Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A364790 The triangle begins:
%e A364790   1;
%e A364790   0, 2;
%e A364790   0, 0, 6;
%e A364790   0, 0, 1, 23;
%e A364790   0, 0, 0,  0, 120;
%e A364790   0, 0, 0,  0,   2, 718;
%e A364790   0, 0, 0,  0,   4,  31, 5005;
%e A364790   0, 0, 0,  0,   0,   2,   44, 40274;
%e A364790   0, 0, 0,  0,   0,   0,    4,   284, 362592;
%e A364790   ...
%t A364790 T[n_, k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Join[{0},Range[n-1]]], i]]], {i, n!}], k]; Join[{1},Table[T[n, k], {n,2,9}, {k, n}]]//Flatten
%o A364790 (PARI)
%o A364790 MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
%o A364790 row(n)={if(n==1, [1], my(f=vector(n)); forperm(vector(n, i, i-1), v, f[matrank(MkMat(v))]++); f)} \\ _Andrew Howroyd_, Jan 07 2024
%Y A364790 Cf. A000142 (row sums), A358323 (minimal determinant), A358324 (maximal determinant), A358326 (minimal permanent), A358327 (maximal permanent), A364791 (right diagonal).
%K A364790 nonn,tabl
%O A364790 1,3
%A A364790 _Stefano Spezia_, Aug 08 2023
%E A364790 Terms a(46) and beyond from _Andrew Howroyd_, Jan 07 2024