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.

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, 0, 0, 0, 2, 44, 40274, 0, 0, 0, 0, 0, 0, 4, 284, 362592, 0, 0, 0, 0, 0, 0, 0, 111, 769, 3627920, 0, 0, 0, 0, 0, 0, 2, 14, 244, 7056, 39909484, 0, 0, 0, 0, 0, 0, 0, 4, 64, 742, 9667, 478991123

Offset: 1

Author

Stefano Spezia, Aug 08 2023

Keywords

• nonn
• tabl

Examples

			The triangle begins:
  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, 0,  0,   0,   2,   44, 40274;
  0, 0, 0,  0,   0,   0,    4,   284, 362592;
  ...
		

Links

• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A000142 (row sums), A358323 (minimal determinant), A358324 (maximal determinant), A358326 (minimal permanent), A358327 (maximal permanent), A364791 (right diagonal).

Programs

Mathematica

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

PARI

MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
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

Extensions

Terms a(46) and beyond from Andrew Howroyd, Jan 07 2024