A364228 Triangle read by rows: T(n, k) is the number of n X n Hermitian Toeplitz matrices of rank k using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

1, 0, 2, 0, 1, 5, 0, 0, 0, 24, 0, 0, 0, 0, 120, 0, 0, 0, 0, 1, 719, 0, 0, 0, 0, 0, 0, 5040, 0, 0, 0, 0, 0, 2, 1, 40317, 0, 0, 0, 0, 0, 0, 0, 6, 362874, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3628798

Offset: 1

Author

Stefano Spezia, Jul 14 2023

Keywords

• nonn
• more
• tabl

Examples

			The triangle begins:
  1;
  0, 2;
  0, 1, 5;
  0, 0, 0, 24;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   1,  719;
  0, 0, 0,  0,   0,    0, 5040;
  0, 0, 0,  0,   0,    2,    1, 40317;
  0, 0, 0,  0,   0,    0,    0,     6, 362874;
  0, 0, 0,  0,   0,    0,    1,     0,      1, 3628798;
  ...
		

Links

• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A000142 (row sums), A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal permanent), A359617 (maximal permanent), A364229 (right diagonal).

Programs

Mathematica

T[n_,k_]:=Count[Flatten[Table[MatrixRank[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]],{i,(n-1)!},{d,n}]],k]; Table[T[n,k],{n,9},{k,n}]//Flatten