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A367916
Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.
15
1, 2, 6, 45, 1376, 161587, 64552473, 85987037645, 386933032425826, 6005080379837219319, 328011924848834642962619, 64153024576968812343635391868, 45547297603829979923254392040011994, 118654043008142499115765307533395739785599
OFFSET
0,2
LINKS
FORMULA
Binomial transform of A054780.
EXAMPLE
The a(0) = 1 through a(2) = 6 set-systems:
{} {} {}
{{1}} {{1}}
{{2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
MATHEMATICA
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Length[Union@@#]==Length[#]&]], {n, 0, 3}]
PROG
(PARI) \\ Here b(n) is A054780(n).
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n, k) * binomial(2^k-1, n))
a(n) = sum(k=0, n, binomial(n, k) * b(k)) \\ Andrew Howroyd, Dec 29 2023
CROSSREFS
The covering case is A054780.
For graphs we have A367862, covering A367863, unlabeled A006649.
These set-systems have ranks A367917.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A136556 counts set-systems on {1..n} with n edges.
Sequence in context: A359659 A374874 A370818 * A136557 A092662 A371341
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2023
STATUS
approved