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A348075
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Triangular array read by rows: T(n,k) is the number of derangements whose shortest cycle has exactly k nodes; n >= 1, 1 <= k <= n.
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1
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0, 0, 1, 0, 0, 2, 0, 3, 0, 6, 0, 20, 0, 0, 24, 0, 105, 40, 0, 0, 120, 0, 714, 420, 0, 0, 0, 720, 0, 5845, 2688, 1260, 0, 0, 0, 5040, 0, 52632, 22400, 18144, 0, 0, 0, 0, 40320, 0, 525105, 223200, 151200, 72576, 0, 0, 0, 0, 362880, 0, 5777090, 2522520, 1425600, 1330560, 0, 0, 0, 0, 0, 3628800
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OFFSET
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1,6
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COMMENTS
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For the statistic "length of the longest cycle", see A211871.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
0;
0, 1;
0, 0, 2;
0, 3, 0, 6;
0, 20, 0, 0, 24;
0, 105, 40, 0, 0, 120;
0, 714, 420, 0, 0, 0, 720;
0, 5845, 2688, 1260, 0, 0, 0, 5040;
0, 52632, 22400, 18144, 0, 0, 0, 0, 40320;
...
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
b(n-j, min(m, j))*binomial(n-1, j-1), j=2..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
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MATHEMATICA
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b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[(j - 1)!*
b[n - j, Min[m, j]]*Binomial[n - 1, j - 1], {j, 2, n}]];
T[n_] := If[n == 1, {0}, CoefficientList[b[n, n], x] // Rest];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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