A327075 Number of non-connected unlabeled simple graphs covering n vertices.

1, 0, 0, 0, 1, 2, 10, 35, 185, 1242, 13929, 292131, 12344252, 1032326141, 166163019475, 50671385831320, 29105332577409883, 31455744378606296280, 64032559078724993894492, 245999991257359808853560276, 1787823917424909126688749033668, 24639597815428343970034635549911427

Offset: 0

Author

Gus Wiseman, Aug 26 2019

Keywords

• nonn

Comments

We consider the empty graph to be neither connected (one component) nor disconnected (more than one component).

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(6) = 10 graphs (empty columns not shown):
  {}  {12,34}  {12,35,45}     {12,34,56}
               {12,34,35,45}  {12,35,46,56}
                              {12,36,46,56}
                              {13,23,46,56}
                              {12,34,35,46,56}
                              {12,36,45,46,56}
                              {13,23,45,46,56}
                              {12,13,23,45,46,56}
                              {12,35,36,45,46,56}
                              {12,34,35,36,45,46,56}
		

Links

• Gus Wiseman, The a(4) = 1 through a(6) = 10 non-connected covering graphs.

Crossrefs

Column k = 0 of A327201.

The labeled version is A327070.

Disconnected graphs are A000719.

Cf. A000088, A001187, A001349, A002494, A006129, A054592.

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 A327075(n):
    if n <= 1: return 1-n
    @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 b(n)-b(n-1)-sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 03 2024

Formula

a(n) = A002494(n) - A001349(n), if we assume A001349(0) = A001349(1) = 0.

Extensions

a(20)-a(21) from Chai Wah Wu, Jul 03 2024