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

Author

Geoffrey Critzer, Nov 07 2012

Keywords

• nonn
• tabf

Comments

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

Examples

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

Links

• Alois P. Heinz, Rows n = 1..170, flattened
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009

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.

Programs

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

Formula

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