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A364230 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.

%I A364230 #17 Dec 30 2023 23:04:23
%S A364230 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 A364230 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 A364230 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 A364230 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 A364230 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A364230 The triangle begins:
%e A364230   1;
%e A364230   0, 2;
%e A364230   0, 0, 6;
%e A364230   0, 0, 0, 24;
%e A364230   0, 0, 0,  0, 120;
%e A364230   0, 0, 0,  0,   2, 718;
%e A364230   0, 0, 0,  0,   4,  31, 5005;
%e A364230   ...
%t A364230 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 A364230 (PARI)
%o A364230 MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
%o A364230 row(n)={my(f=vector(n)); forperm(vector(n,i,i), v, f[matrank(MkMat(v))]++); f} \\ _Andrew Howroyd_, Dec 30 2023
%Y A364230 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 A364230 nonn,tabl
%O A364230 1,3
%A A364230 _Stefano Spezia_, Jul 14 2023
%E A364230 Terms a(46) and beyond from _Andrew Howroyd_, Dec 30 2023