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%I #31 Nov 11 2021 10:25:21
%S 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,
%T 0,5845,2688,1260,0,0,0,5040,0,52632,22400,18144,0,0,0,0,40320,0,
%U 525105,223200,151200,72576,0,0,0,0,362880,0,5777090,2522520,1425600,1330560,0,0,0,0,0,3628800
%N 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.
%C For the statistic "length of the longest cycle", see A211871.
%H Steven Finch, <a href="https://arxiv.org/abs/2111.05720">Permute, Graph, Map, Derange</a>, arXiv:2111.05720 [math.CO], 2021.
%H D. Panario and B. Richmond, <a href="https://doi.org/10.1007/s00453-001-0047-1">Exact largest and smallest size of components</a>, Algorithmica, 31 (2001), 413-432.
%F T(n,n) = A000142(n-1), n >= 2.
%F T(n,2) = A158243(n), n >= 2.
%F T(n,k) = A145877(n,k) for k >= 2.
%e Triangle begins:
%e 0;
%e 0, 1;
%e 0, 0, 2;
%e 0, 3, 0, 6;
%e 0, 20, 0, 0, 24;
%e 0, 105, 40, 0, 0, 120;
%e 0, 714, 420, 0, 0, 0, 720;
%e 0, 5845, 2688, 1260, 0, 0, 0, 5040;
%e 0, 52632, 22400, 18144, 0, 0, 0, 0, 40320;
%e ...
%p b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
%p b(n-j, min(m, j))*binomial(n-1, j-1), j=2..n))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p seq(T(n), n=1..12); # _Alois P. Heinz_, Sep 27 2021
%t b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[(j - 1)!*
%t b[n - j, Min[m, j]]*Binomial[n - 1, j - 1], {j, 2, n}]];
%t T[n_] := If[n == 1, {0}, CoefficientList[b[n, n], x] // Rest];
%t Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Oct 03 2021, after _Alois P. Heinz_ *)
%Y Row sums give A000166, n >= 1.
%Y Right border gives A000142.
%Y Column 1 gives A000004.
%Y Column 2 gives A158243.
%Y Cf. A145877, A211871.
%K nonn,tabl
%O 1,6
%A _Steven Finch_, Sep 27 2021
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