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A364790
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Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 0, 1, 2, ..., n-1.
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1
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1, 0, 2, 0, 0, 6, 0, 0, 1, 23, 0, 0, 0, 0, 120, 0, 0, 0, 0, 2, 718, 0, 0, 0, 0, 4, 31, 5005, 0, 0, 0, 0, 0, 2, 44, 40274, 0, 0, 0, 0, 0, 0, 4, 284, 362592, 0, 0, 0, 0, 0, 0, 0, 111, 769, 3627920, 0, 0, 0, 0, 0, 0, 2, 14, 244, 7056, 39909484, 0, 0, 0, 0, 0, 0, 0, 4, 64, 742, 9667, 478991123
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OFFSET
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1,3
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LINKS
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EXAMPLE
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The triangle begins:
1;
0, 2;
0, 0, 6;
0, 0, 1, 23;
0, 0, 0, 0, 120;
0, 0, 0, 0, 2, 718;
0, 0, 0, 0, 4, 31, 5005;
0, 0, 0, 0, 0, 2, 44, 40274;
0, 0, 0, 0, 0, 0, 4, 284, 362592;
...
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MATHEMATICA
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T[n_, k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]], {i, n!}], k]; Join[{1}, Table[T[n, k], {n, 2, 9}, {k, n}]]//Flatten
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PROG
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(PARI)
MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
row(n)={if(n==1, [1], my(f=vector(n)); forperm(vector(n, i, i-1), v, f[matrank(MkMat(v))]++); f)} \\ Andrew Howroyd, Jan 07 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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