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
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
Programs
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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