1,4

Row sums are the factorials (A000142).

Sum(T(n,k), k=2..n) = A000166(n) (the derangement numbers).

T(n,1) = A002467(n).

T(n,n) = (n-1)! (A000142).

Sum(k*T(n,k),k=1..n) = A028417(n).

For the statistic "length of the longest cycle", see A126074.

Alois P. Heinz, Rows n = 1..141, flattened

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.

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

T(4,2)=3 because we have 3412=(13)(24), 2143=(12)(34) and 4321=(14)(23).

Triangle starts:

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, 0, 0, 720;

25487, 5845, 2688, 1260, 0, 0, 0, 5040;

...

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

Rest[Transpose[Table[Range[0, 16]! CoefficientList[

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 *)

Cf. A000142, A000166, A002467, A028417, A126074.

T(2n,n) gives A110468(n-1) (for n>0). - Alois P. Heinz, Apr 21 2017

Sequence in context: A134895 A318468 A346492 * A373984 A283572 A057075

Adjacent sequences: A145874 A145875 A145876 * A145878 A145879 A145880

nonn,tabl

Emeric Deutsch, Oct 27 2008

approved