A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

1, 2, 6, 45, 1376, 161587, 64552473, 85987037645, 386933032425826, 6005080379837219319, 328011924848834642962619, 64153024576968812343635391868, 45547297603829979923254392040011994, 118654043008142499115765307533395739785599

Offset: 0

Author

Gus Wiseman, Dec 08 2023

Keywords

• nonn

Examples

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

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.

Cf. A092918, A102896, A133686, A306445, A323818, A355740, A367770, A367869, A367901, A367902, A367905.

Programs

Mathematica

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Length[Union@@#]==Length[#]&]],{n,0,3}]

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

Formula

Binomial transform of A054780.