A218868 Triangular array read by rows: T(n,k) is the number of n-permutations that have exactly k distinct cycle lengths.

1, 2, 3, 3, 10, 14, 25, 95, 176, 424, 120, 721, 3269, 1050, 6406, 21202, 12712, 42561, 178443, 141876, 436402, 1622798, 1418400, 151200, 3628801, 17064179, 17061660, 2162160, 48073796, 177093256, 212254548, 41580000, 479001601, 2293658861, 2735287698, 719072640

Offset

1,2

Comments

T(A000217(n),n) gives A246292. - Alois P. Heinz, Aug 21 2014

Links

Alois P. Heinz, Rows n = 1..170, flattened

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009

Formula

E.g.f.: Product_{i>=1} (1 + y*exp(x^i/i) - y).

Example

:      1;
:      2;
:      3,       3;
:     10,      14;
:     25,      95;
:    176,     424,     120;
:    721,    3269,    1050;
:   6406,   21202,   12712;
:  42561,  178443,  141876;
: 436402, 1622798, 1418400, 151200;

Maple

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2014

Mathematica

nn=10; a=Product[1-y+y Exp[x^i/i], {i, 1, nn}]; f[list_]:=Select[list, #>0&]; Map[f, Drop[Range[0, nn]!CoefficientList[Series[a , {x, 0, nn}], {x, y}], 1]]//Grid

Crossrefs

Columns k=1-3 give: A005225, A005772, A133119.

Row sums are: A000142.

Row lengths are: A003056.

Cf. A208437, A242027 (the same for endofunctions), A246292, A317327.

Sequence in context: A123027 A100652 A094416 * A329874 A152300 A117030

Adjacent sequences: A218865 A218866 A218867 * A218869 A218870 A218871

Keyword

nonn,tabf

Author

Geoffrey Critzer, Nov 07 2012

Status

approved