|
|
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.
|
|
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, 40320, 282240, 967680, 2150400, 3440640, 4128768, 3670016, 2097152, 362880, 2903040, 11430720, 29393280, 55112400, 79361856, 89282088, 76527504, 43046721
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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.
|
|
LINKS
|
|
|
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!.
|
|
EXAMPLE
|
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;
|
|
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)):
|
|
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
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|