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%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
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