A367916 - OEIS

%I #11 Dec 29 2023 16:41:17

%S 1,2,6,45,1376,161587,64552473,85987037645,386933032425826,

%T 6005080379837219319,328011924848834642962619,

%U 64153024576968812343635391868,45547297603829979923254392040011994,118654043008142499115765307533395739785599

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

%H Andrew Howroyd, <a href="/A367916/b367916.txt">Table of n, a(n) for n = 0..50</a>

%F Binomial transform of A054780.

%e The a(0) = 1 through a(2) = 6 set-systems:

%e   {}  {}     {}

%e       {{1}}  {{1}}

%e              {{2}}

%e              {{1},{2}}

%e              {{1},{1,2}}

%e              {{2},{1,2}}

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

%o (PARI) \\ Here b(n) is A054780(n).

%o b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(2^k-1, n))

%o a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ _Andrew Howroyd_, Dec 29 2023

%Y The covering case is A054780.

%Y For graphs we have A367862, covering A367863, unlabeled A006649.

%Y These set-systems have ranks A367917.

%Y A000372 counts antichains, covering A006126, nonempty A014466.

%Y A003465 counts set-systems covering {1..n}, unlabeled A055621.

%Y A058891 counts set-systems, unlabeled A000612.

%Y A059201 counts covering T_0 set-systems.

%Y A136556 counts set-systems on {1..n} with n edges.

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

%K nonn

%O 0,2

%A _Gus Wiseman_, Dec 08 2023