A219694 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k nonrecurrent elements; n>=1, 0<=k<=n-1.

1, 2, 2, 6, 12, 9, 24, 72, 96, 64, 120, 480, 900, 1000, 625, 720, 3600, 8640, 12960, 12960, 7776, 5040, 30240, 88200, 164640, 216090, 201684, 117649, 40320, 282240, 967680, 2150400, 3440640, 4128768, 3670016, 2097152, 362880, 2903040, 11430720, 29393280, 55112400, 79361856, 89282088, 76527504, 43046721

Offset: 1

Author

Geoffrey Critzer, Nov 25 2012

Keywords

• nonn
• tabl

Comments

x in {1,2,...,n} is a recurrent element if there is some k such that f^k(x) = x where f^k(x) denotes iterated functional composition. In other words, a recurrent element is in a cycle of the functional digraph. An element that is not recurrent is a nonrecurrent element.

Examples

			T(2,1) = 2 because we have 1->1 2->1; and 1->2 2->2.
:    1;
:    2,     2;
:    6,    12,     9;
:   24,    72,    96,     64;
:  120,   480,   900,   1000,    625;
:  720,  3600,  8640,  12960,  12960,   7776;
: 5040, 30240, 88200, 164640, 216090, 201684, 117649;
		

Links

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

Crossrefs

Cf. A216971.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      (j-1)!*b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(add(
    b(j)*(x*n)^(n-j)*binomial(n-1, j-1), j=0..n)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 22 2016

Mathematica

nn=8;f[list_]:=Select[list,#>0&];t=Sum[n^(n-1)x^n y^n/n!,{n,1,nn}];Drop[Map[f,Range[0,nn]!CoefficientList[Series[1/(1-x Exp[t]),{x,0,nn}],{x,y}]],1]//Grid

Formula

E.g.f.: 1/(1-x*exp(A(x,y))), where A(x,y) = Sum_{n>=1} n^(n-1)*(y*x)^n/n!.