A364230 - OEIS

%I #17 Dec 30 2023 23:04:23

%S 1,0,2,0,0,6,0,0,0,24,0,0,0,0,120,0,0,0,0,2,718,0,0,0,0,4,31,5005,0,0,

%T 0,0,0,2,44,40274,0,0,0,0,0,0,4,272,362604,0,0,0,0,0,0,0,111,774,

%U 3627915,0,0,0,0,0,0,2,14,244,6974,39909566,0,0,0,0,0,0,0,4,64,743,9533,478991256

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>

%e The triangle begins:

%e   1;

%e   0, 2;

%e   0, 0, 6;

%e   0, 0, 0, 24;

%e   0, 0, 0,  0, 120;

%e   0, 0, 0,  0,   2, 718;

%e   0, 0, 0,  0,   4,  31, 5005;

%e   ...

%t T[n_,k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Range[n]], i]]],{i,n!}],k]; Table[T[n,k],{n,8},{k,n}]//Flatten

%o (PARI)

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

%o row(n)={my(f=vector(n)); forperm(vector(n,i,i), v, f[matrank(MkMat(v))]++); f} \\ _Andrew Howroyd_, Dec 30 2023

%Y Cf. A000142 (row sums), A350953 (minimal determinant), A350954 (maximal determinant), A351019 (minimal permanent), A351020 (maximal permanent), A356865 (minimal nonzero absolute value determinant), A364231 (right diagonal).

%K nonn,tabl

%O 1,3

%A _Stefano Spezia_, Jul 14 2023

%E Terms a(46) and beyond from _Andrew Howroyd_, Dec 30 2023