A318131 -id:A318131 - cp's OEIS frontend

A318128 Number of set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

1, 0, 2, 84, 31478, 2147000136, 9223371998203475474, 170141183460469231537996491257596836636, 57896044618658097711785492504343953922551603929769020459976077632195066756398

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			The a(2) = 2 set-systems are {{1},{2}}, and {{1},{2},{1,2}}.

Cf. A000371, A001146, A003465, A119563, A131288.

Cf. A318128, A318129, A318130, A318131, A318132.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]],{n,4}]

Inverse binomial transform of A318129.

A318129 Number of sets of nonempty subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

1, 1, 3, 91, 31827, 2147158387, 9223372011085950171, 170141183460469231602560095290109272523, 57896044618658097711785492504343953923912733397452774312538303978325772978595

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			The a(2) = 3 sets of sets are {}, {{1},{2}}, {{1},{2},{1,2}}.

Cf. A000371, A001146, A003465,A040996, A119563, A131288.

Cf. A318128, A318130, A318131, A318132.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Binomial transform of A318128.

a(n) = A318130(n) - 2^(2^n - 1). [corrected]

A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 26, 1884, 18660728, 12813206113141264, 33758171486592987125648226573752576, 1435913805026242504952006868879460423733630400489039411798068453617852416

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			Non-isomorphic representatives of the a(3) = 26 set-systems:
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..12

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318131.

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

a(n) = A055621(n) - 2*A055621(n-1) = A000612(n) - 3*A000612(n-1) + 2*A000612(n-2) for n >= 2. - Andrew Howroyd, Jan 29 2024

a(5) onwards from Andrew Howroyd, Jan 29 2024

A318130 Number of sets of subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

2, 3, 11, 219, 64595, 4294642035, 18446744047940725979, 340282366920938463334247399005993378251, 115792089237316195423570985008687907850547725730273056332267095982282337798563

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			The a(2) = 11 sets of sets:
  {}
  {{}}
  {{},{1}}
  {{},{2}}
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Cf. A000371, A001146, A003465, A119563, A131288, A318128, A318129.

Cf. A318128, A318129, A318131, A318132.

Table[Length[Select[Subsets[Subsets[Range[n]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Binomial transform of A131288.

Inverse binomial transform of A119563(n) = 2^(2^n) + 2^n - 1.

cp's OEIS Frontend

A318128 Number of set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

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A318129 Number of sets of nonempty subsets of {1,...,n} with intersection {}.

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A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

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A318130 Number of sets of subsets of {1,...,n} with intersection {}.

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