A364233 - OEIS

%I #15 Dec 31 2023 12:42:22

%S 1,0,2,0,0,6,0,0,0,24,0,0,0,0,120,0,0,0,0,2,718,0,0,0,0,0,4,5036,0,0,

%T 0,0,0,1,3,40316,0,0,0,0,0,0,0,18,362862,0,0,0,0,0,0,0,0,14,3628786,0,

%U 0,0,0,0,0,0,0,0,99,39916701,0,0,0,0,0,0,0,0,0,5,78,479001517

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

%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,   0,   4, 5036;

%e   ...

%t T[n_,k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Prime[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,prime(i)), v, f[matrank(MkMat(v))]++); f} \\ _Andrew Howroyd_, Dec 31 2023

%Y Cf. A000142 (row sums), A348891 (minimal nonzero absolute value determinant), A350955 (minimal determinant), A350956 (maximal determinant), A351021 (minimal permanent), A351022 (maximal permanent), A364234 (right diagonal).

%K nonn,tabl

%O 1,3

%A _Stefano Spezia_, Jul 14 2023

%E Terms a(46) and beyond from _Andrew Howroyd_, Dec 31 2023