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

1,3

Links

Table of n, a(n) for n=1..55.

Wikipedia, Toeplitz Matrix

Example

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;
  ...

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

Crossrefs

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

Sequence in context: A244128 A016584 A293961 * A112899 A212808 A337085

Adjacent sequences: A364225 A364226 A364227 * A364229 A364230 A364231

Keyword

nonn,more,tabl

Author

Stefano Spezia, Jul 14 2023

Status

approved