A222029 Triangle of number of functions in a size n set for which the sequence of composition powers ends in a length k cycle.

1, 1, 3, 1, 16, 9, 2, 125, 93, 32, 6, 1296, 1155, 480, 150, 24, 20, 16807, 17025, 7880, 3240, 864, 840, 262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420, 4782969, 5752131, 3009888, 1692180, 653184, 773920, 46080, 5040, 0, 32256, 0, 26880, 0, 0, 2688

Offset: 0

Author

Chad Brewbaker, May 14 2013

Keywords

• nonn
• look
• tabf

Comments

If you take the powers of a finite function you generate a lollipop graph. This table organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is A225725.

Warning: For T(n,k) after the sixth row there are zero entries and k can be greater than n: T(7,k) = |{1=>262144, 2=>292383, 3=>145320, 4=>71610, 5=>24192, 6=>26250, 7=>720, 8=>0, 9=>0, 10=>504, 11=>0, 12=>420}|.

Examples

			T(1,1) = |{[0]}|, T(2,1) = |{[0,0],[0,1],[1,1]}|, T(2,2) = |{[0,1]}|.
Triangle starts:
       1;
       1;
       3,      1;
      16,      9,      2;
     125,     93,     32,     6;
    1296,   1155,    480,   150,    24,    20;
   16807,  17025,   7880,  3240,   864,   840;
  262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..30, flattened
• Chad Brewbaker, Ruby program for A222029

Crossrefs

Columns k=1-10 give A000272, A163951, A163952, A291110, A291111, A291112, A291113, A291114, A291115, A291116.

Rows sums give A000312.

Row lengths are A000793.

Number of nonzero elements of rows give A009490.

Last elements of rows give A162682.

Main diagonal gives A290961.

Cf. A057731 (the same for permutations), A290932.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
      b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(
         b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 14 2017

Mathematica

b[n_, m_]:=b[n, m]=If[n==0, x^m, Sum[(j - 1)!*b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, n}]]; T[n_]:=If[n==0, {1}, Drop[CoefficientList[Sum[b[j, 1]n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], x], 1]]; Table[T[n], {n, 0, 10}]//Flatten (* Indranil Ghosh, Aug 17 2017 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial, Symbol, lcm, factorial as f, Poly, flatten
x=Symbol('x')
@cacheit
def b(n, m): return x**m if n==0 else sum([f(j - 1)*b(n - j, lcm(m, j))*binomial(n - 1, j - 1) for j in range(1, n + 1)])
def T(n): return Poly(sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)]),x).all_coeffs()[::-1][1:]
print([T(n) for n in range(11)]) # Indranil Ghosh, Aug 17 2017

Formula

Sum_{k=1..A000793(n)} k * T(n,k) = A290932. - Alois P. Heinz, Aug 14 2017

Extensions

T(0,1)=1 prepended by Alois P. Heinz, Aug 14 2017