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%I #29 Nov 11 2021 10:23:23
%S 1,1,1,4,0,2,15,3,0,6,76,20,0,0,24,455,105,40,0,0,120,3186,714,420,0,
%T 0,0,720,25487,5845,2688,1260,0,0,0,5040,229384,52632,22400,18144,0,0,
%U 0,0,40320,2293839,525105,223200,151200,72576,0,0,0,0,362880,25232230
%N Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).
%C Row sums are the factorials (A000142).
%C Sum(T(n,k), k=2..n) = A000166(n) (the derangement numbers).
%C T(n,1) = A002467(n).
%C T(n,n) = (n-1)! (A000142).
%C Sum(k*T(n,k),k=1..n) = A028417(n).
%C For the statistic "length of the longest cycle", see A126074.
%H Alois P. Heinz, <a href="/A145877/b145877.txt">Rows n = 1..141, flattened</a>
%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 E.g.f. for column k is (1-exp(-x^k/k))*exp( -sum(j=1..k-1, x^j/j ) ) / (1-x). - _Vladeta Jovovic_
%e T(4,2)=3 because we have 3412=(13)(24), 2143=(12)(34) and 4321=(14)(23).
%e Triangle starts:
%e 1;
%e 1, 1;
%e 4, 0, 2;
%e 15, 3, 0, 6;
%e 76, 20, 0, 0, 24;
%e 455, 105, 40, 0, 0, 120;
%e 3186, 714, 420, 0, 0, 0, 720;
%e 25487, 5845, 2688, 1260, 0, 0, 0, 5040;
%e ...
%p F:=proc(k) options operator, arrow: (1-exp(-x^k/k))*exp(-(sum(x^j/j, j = 1 .. k-1)))/(1-x) end proc: for k to 16 do g[k]:= series(F(k),x=0,16) end do: T:= proc(n,k) options operator, arrow: factorial(n)*coeff(g[k],x,n) end proc: for n to 11 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form
%t Rest[Transpose[Table[Range[0, 16]! CoefficientList[
%t Series[(Exp[x^n/n] -1) (Exp[-Sum[x^k/k, {k, 1, n}]]/(1 - x)), {x, 0, 16}],x], {n, 1, 8}]]] // Grid (* _Geoffrey Critzer_, Mar 04 2011 *)
%Y Cf. A000142, A000166, A002467, A028417, A126074.
%Y T(2n,n) gives A110468(n-1) (for n>0). - _Alois P. Heinz_, Apr 21 2017
%K nonn,tabl
%O 1,4
%A _Emeric Deutsch_, Oct 27 2008
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