A136722 Number of preferential arrangements (or hierarchical orderings) on the connected graphs on n unlabeled nodes.

1, 1, 2, 8, 48, 336, 3584, 54592, 1422976, 66836480, 5998884352, 1030861378560, 335994532814848, 206175878632321024, 237596569295651315712, 514414692643000188272640, 2096154545790162572944244736, 16113456361117058761983824232448, 234269143891823701379016369973493760

Offset: 0

Author

Thomas Wieder, Jan 19 2008

Keywords

• easy
• nonn

Examples

			There are A001349(3)=2 connected graphs for n=3 unlabeled elements:
The chain
o-o-o
and the triangle
. o
/..\
o - o.
There are a(3)=8 hierarchical orders on these two graphs.
The chain gives us 6 orderings:
o-o-o
o
|
o-o
. o
/..\
o . o
o . o
.\./
. o
o-o
|
o
o
|
o
|
o
The triangle gives us two orderings:
. o
/..\
o - o
o - o
\../
. o
		

Crossrefs

Cf. A001349, A011782, A136723, A034691, A075729.

Programs

Python

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A136722(n):
    if n == 0: return 1
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n<Chai Wah Wu, Jul 03 2024

Formula

a(n)=A001349(n)*A011782(n).

Extensions

More terms from Alois P. Heinz, Apr 21 2012