%I #49 Nov 11 2021 10:44:27
%S 1,0,0,0,0,1,0,0,0,2,0,0,3,0,6,0,0,0,20,0,24,0,0,15,40,90,0,120,0,0,0,
%T 210,420,504,0,720,0,0,105,1120,2520,2688,3360,0,5040,0,0,0,4760,
%U 15120,27216,20160,25920,0,40320,0,0,945,25200,126000,193536,226800,172800,226800,0,362880
%N Number T(n,k) of permutations of n elements with no fixed points and largest cycle of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C From _Steven Finch_, Sep 27 2021: (Start)
%C A permutation without fixed points is called a derangement.
%C For the statistic "length of the smallest cycle", see A348075. (End)
%H Alois P. Heinz, <a href="/A211871/b211871.txt">Rows n = 0..140, 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. of column k>1: (exp(x^k/k)-1) * exp(Sum_{j=2..k-1} x^j/j); e.g.f. of column k<=1: 1-k.
%e T(0,0) = 1: (), the empty permutation.
%e T(2,2) = 1: (2,1).
%e T(3,3) = 2: (2,3,1), (3,1,2).
%e T(4,2) = 3: (2,1,4,3), (3,4,1,2), (4,3,2,1).
%e T(4,4) = 6: (2,4,1,3), (2,3,4,1), (3,1,4,2), (3,4,2,1), (4,1,2,3), (4,3,1,2).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 0;
%e 0, 0, 1;
%e 0, 0, 0, 2;
%e 0, 0, 3, 0, 6;
%e 0, 0, 0, 20, 0, 24;
%e 0, 0, 15, 40, 90, 0, 120;
%e 0, 0, 0, 210, 420, 504, 0, 720;
%e 0, 0, 105, 1120, 2520, 2688, 3360, 0, 5040;
%e ...
%p A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p add(mul(n-i, i=1..j-1)*A(n-j,k), j=2..k)))
%p end:
%p T:= (n, k)-> A(n, k) -`if`(k=0,0,A(n, k-1)):
%p seq(seq(T(n,k), k=0..n), n=0..12);
%t A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1,
%t Sum[Product[n-i, {i, 1, j-1}]*A[n-j, k], {j, 2, k}]]];
%t T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y Columns k=0-3 give: A000007, A000004, A123023(n+2) for n>0, A211880.
%Y Row sums give A000166.
%Y Diagonal gives: A000142(n-1) for n>1.
%Y T(n,0) + T(n,2) + T(n,3) gives A055814(n).
%Y Cf. A348075.
%K nonn,tabl
%O 0,10
%A _Alois P. Heinz_, Feb 12 2013