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 A348075 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. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS For the statistic "length of the longest cycle", see A211871. LINKS Table of n, a(n) for n=1..66. Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021. D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432. FORMULA T(n,n) = A000142(n-1), n >= 2. T(n,2) = A158243(n), n >= 2. T(n,k) = A145877(n,k) for k >= 2. EXAMPLE 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; ... MAPLE 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)): seq(T(n), n=1..12); # Alois P. Heinz, Sep 27 2021 MATHEMATICA 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]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Oct 03 2021, after Alois P. Heinz *) CROSSREFS Row sums give A000166, n >= 1. Right border gives A000142. Column 1 gives A000004. Column 2 gives A158243. Cf. A145877, A211871. Sequence in context: A082857 A208092 A081155 * A130628 A249431 A331176 Adjacent sequences: A348072 A348073 A348074 * A348076 A348077 A348078 KEYWORD nonn,tabl AUTHOR Steven Finch, Sep 27 2021 STATUS approved

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Last modified April 19 18:05 EDT 2024. Contains 371798 sequences. (Running on oeis4.)