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A211871 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.

%I A211871 #52 Nov 26 2024 17:26:41
%S A211871 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 A211871 210,420,504,0,720,0,0,105,1120,2520,2688,3360,0,5040,0,0,0,4760,
%U A211871 15120,27216,20160,25920,0,40320,0,0,945,25200,126000,193536,226800,172800,226800,0,362880
%N A211871 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 A211871 From _Steven Finch_, Sep 27 2021: (Start)
%C A211871 A permutation without fixed points is called a derangement.
%C A211871 For the statistic "length of the smallest cycle", see A348075. (End)
%H A211871 Alois P. Heinz, <a href="/A211871/b211871.txt">Rows n = 0..140, flattened</a>
%H A211871 Steven Finch, <a href="https://arxiv.org/abs/2111.05720">Permute, Graph, Map, Derange</a>, arXiv:2111.05720 [math.CO], 2021.
%H A211871 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 A211871 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 A211871 T(0,0) = 1: (), the empty permutation.
%e A211871 T(2,2) = 1: (2,1).
%e A211871 T(3,3) = 2: (2,3,1), (3,1,2).
%e A211871 T(4,2) = 3: (2,1,4,3), (3,4,1,2), (4,3,2,1).
%e A211871 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 A211871 Triangle T(n,k) begins:
%e A211871   1;
%e A211871   0,  0;
%e A211871   0,  0,   1;
%e A211871   0,  0,   0,    2;
%e A211871   0,  0,   3,    0,    6;
%e A211871   0,  0,   0,   20,    0,   24;
%e A211871   0,  0,  15,   40,   90,    0,  120;
%e A211871   0,  0,   0,  210,  420,  504,    0, 720;
%e A211871   0,  0, 105, 1120, 2520, 2688, 3360,   0, 5040;
%e A211871   ...
%p A211871 A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A211871        add(mul(n-i, i=1..j-1)*A(n-j,k), j=2..k)))
%p A211871     end:
%p A211871 T:= (n, k)-> A(n, k) -`if`(k=0,0,A(n, k-1)):
%p A211871 seq(seq(T(n,k), k=0..n), n=0..12);
%t A211871 A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1,
%t A211871      Sum[Product[n-i, {i, 1, j-1}]*A[n-j, k], {j, 2, k}]]];
%t A211871 T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
%t A211871 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A211871 Columns k=0-3 give: A000007, A000004, A123023 for n>0, A211880.
%Y A211871 Row sums give A000166.
%Y A211871 Diagonal gives: A000142(n-1) for n>1.
%Y A211871 T(n,0) + T(n,2) + T(n,3) gives A055814(n).
%Y A211871 Cf. A348075.
%K A211871 nonn,tabl
%O A211871 0,10
%A A211871 _Alois P. Heinz_, Feb 12 2013