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A364228 Triangle read by rows: T(n, k) is the number of n X n Hermitian Toeplitz matrices of rank k using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

%I A364228 #10 Aug 02 2023 14:42:03
%S A364228 1,0,2,0,1,5,0,0,0,24,0,0,0,0,120,0,0,0,0,1,719,0,0,0,0,0,0,5040,0,0,
%T A364228 0,0,0,2,1,40317,0,0,0,0,0,0,0,6,362874,0,0,0,0,0,0,1,0,1,3628798
%N A364228 Triangle read by rows: T(n, k) is the number of n X n Hermitian Toeplitz matrices of rank k using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.
%H A364228 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A364228 The triangle begins:
%e A364228   1;
%e A364228   0, 2;
%e A364228   0, 1, 5;
%e A364228   0, 0, 0, 24;
%e A364228   0, 0, 0,  0, 120;
%e A364228   0, 0, 0,  0,   1,  719;
%e A364228   0, 0, 0,  0,   0,    0, 5040;
%e A364228   0, 0, 0,  0,   0,    2,    1, 40317;
%e A364228   0, 0, 0,  0,   0,    0,    0,     6, 362874;
%e A364228   0, 0, 0,  0,   0,    0,    1,     0,      1, 3628798;
%e A364228   ...
%t A364228 T[n_,k_]:=Count[Flatten[Table[MatrixRank[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]],{i,(n-1)!},{d,n}]],k]; Table[T[n,k],{n,9},{k,n}]//Flatten
%Y A364228 Cf. A000142 (row sums), A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal permanent), A359617 (maximal permanent), A364229 (right diagonal).
%K A364228 nonn,more,tabl
%O A364228 1,3
%A A364228 _Stefano Spezia_, Jul 14 2023