A212789 Number of endofunctions on [n] with distinct cycle lengths.

1, 1, 3, 20, 186, 2229, 32790, 572018, 11541600, 264370473, 6776462320, 192163455384, 5972728750560, 201906797867085, 7375152706023648, 289473254317393110, 12149690892777901568, 543010240381452000273, 25746662043469525754880, 1290829803802550504743036

Offset: 0

Author

Geoffrey Critzer, May 27 2012

Keywords

• nonn

Examples

			a(3)=20 because there are 27 functions f:{1,2,3}->{1,2,3} but 7 of these have at least two cycles of equal length: (1,2,3);(1,2,1);(1,2,2);(1,1,3);(1,3,3);(2,2,3)(3,2,3) where the functions are represented by their values.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A241980.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `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), j=0..min(1, n/i))))
    end:
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2014

Mathematica

nn = 20; p = Product[1 + t^n/n, {n, 1, nn}]; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Range[0, nn]! CoefficientList[Series[p, {x, 0, nn}], x]

Formula

E.g.f.: A(T(x)) where A(x) is e.g.f. for A007838 and T(x) is e.g.f. for A000169.

Extensions

Terms a(8)-a(19) recomputed by Alois P. Heinz, Aug 10 2014