A006126 -id:A006126 - cp's OEIS frontend

A299353 Number of labeled connected uniform hypergraphs spanning n vertices.

1, 1, 1, 5, 50, 1713, 1101990, 68715891672, 1180735735356264714926, 170141183460507906731293351306487161569, 7237005577335553223087828975127304177495735363998991435497132228228565768846

Offset: 0

Gus Wiseman, Jun 18 2018

nonn

A hypergraph is uniform if all edges have the same size.

Let T be the regular triangle A299354, where column k is the logarithmic transform of the inverse binomial transform of c(d) = 2^binomial(d,k). Then a(n) is the sum of row n.

			The a(3) = 5 hypergraphs:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}

Wikipedia, Hypergraph

Cf. A000005, A001315, A006126, A006129, A038041, A048143, A298422, A299354, A299471, A306020, A306021.

nn=10;Table[Sum[SeriesCoefficient[Log[Sum[x^m/m!*(-1)^(m-d)*Binomial[m,d]*2^Binomial[d,k],{m,0,n},{d,0,m}]],{x,0,n}]*n!,{k,n}],{n,nn}]

A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 18, 142, 3100, 823042

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 18 antichains:
{{1,2,3}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

A322451 Number of unlabeled 3-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 29, 2102, 7011184, 1788775603336, 53304526022885280592, 366299663378889804782337225824, 1171638318502622784366970315264281830913536, 3517726593606524901243694560022510194223171115509135178240

Offset: 0

Gus Wiseman, Dec 09 2018

nonn

3-uniform means that every edge consists of 3 vertices. - Brendan McKay, Sep 03 2023

			Non-isomorphic representatives of the a(5) = 29 hypergraphs:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{123}{145}{245}{345}}
  {{124}{135}{245}{345}}
  {{125}{135}{245}{345}}
  {{134}{235}{245}{345}}
  {{145}{235}{245}{345}}
  {{123}{124}{135}{245}{345}}
  {{123}{145}{235}{245}{345}}
  {{124}{134}{235}{245}{345}}
  {{134}{145}{235}{245}{345}}
  {{135}{145}{235}{245}{345}}
  {{145}{234}{235}{245}{345}}
  {{123}{124}{134}{235}{245}{345}}
  {{123}{134}{145}{235}{245}{345}}
  {{123}{145}{234}{235}{245}{345}}
  {{124}{135}{145}{235}{245}{345}}
  {{125}{135}{145}{235}{245}{345}}
  {{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{235}{245}{345}}
  {{124}{135}{145}{234}{235}{245}{345}}
  {{125}{135}{145}{234}{235}{245}{345}}
  {{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{135}{145}{234}{235}{245}{345}}
  {{125}{134}{135}{145}{234}{235}{245}{345}}
  {{124}{125}{134}{135}{145}{234}{235}{245}{345}}
  {{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}

Andrew Howroyd, Table of n, a(n) for n = 0..25

First differences of A000665.

Cf. A006126, A006129, A038041, A299471, A301922, A302374, A302394, A306017, A306021.

a(12) from Andrew Howroyd, Dec 15 2018

Name corrected by Brendan McKay, Sep 03 2023

A326359 Number of maximal antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 6, 28, 375, 31745, 123805913

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(0) = 1 through a(4) = 28 antichains:
  {}   {1}    {12}      {123}           {1234}
              {1}{2}    {1}{23}         {1}{234}
                        {2}{13}         {2}{134}
                        {3}{12}         {3}{124}
                        {1}{2}{3}       {4}{123}
                        {12}{13}{23}    {1}{2}{34}
                                        {1}{3}{24}
                                        {1}{4}{23}
                                        {2}{3}{14}
                                        {2}{4}{13}
                                        {3}{4}{12}
                                        {1}{2}{3}{4}
                                        {12}{134}{234}
                                        {13}{124}{234}
                                        {14}{123}{234}
                                        {23}{124}{134}
                                        {24}{123}{134}
                                        {34}{123}{124}
                                        {1}{23}{24}{34}
                                        {2}{13}{14}{34}
                                        {3}{12}{14}{24}
                                        {4}{12}{13}{23}
                                        {12}{13}{14}{234}
                                        {12}{23}{24}{134}
                                        {13}{23}{34}{124}
                                        {14}{24}{34}{123}
                                        {123}{124}{134}{234}
                                        {12}{13}{14}{23}{24}{34}

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.

Antichains of nonempty sets are A014466.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of sets are A326358.
Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A307249, A326360, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{1,n}],SubsetQ]]],{n,0,5}]

For n > 0, a(n) = A326358(n) - 1.

a(6) from Andrew Howroyd, Aug 14 2019

a(7) from Dmitry I. Ignatov, Oct 12 2021

A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

Original entry on oeis.org

1, 1, 1, 2, 12, 133, 11386, 12143511

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 12 antichains:
  {{1,2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[stableSets[Subsets[Range[n]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],Union@@#==Range[n]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = Select[FindClique[g, Infinity, All], BitOr @@ # == n - 1 &];
Length[sets] (* Elijah Beregovsky, May 05 2020 *)

a(6)-a(7) from Elijah Beregovsky, May 05 2020

A326880 BII-numbers of set-systems that are closed under nonempty intersection.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 46, 47, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88

Offset: 1

Gus Wiseman, Jul 29 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

The enumeration of these set-systems by number of covered vertices is A326881.

			Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:
  20: {{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}
  49: {{1},{1,3},{2,3}}
  50: {{2},{1,3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  84: {{1,2},{1,3},{1,2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Select[Tuples[bpe/@bpe[#],2],Intersection@@#!={}&]]&]

from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E,f = [bin_i(k) for k in bin_i(n)],0
        for i in combinations(E,2):
            x = list(set(i[0])&set(i[1]))
            if x not in E and len(x) > 0:
                f += 1
                break
        if f < 1:
            yield n
A326880_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 07 2025

A293607 Number of unlabeled clutters of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 7, 8, 23, 42

Offset: 0

Gus Wiseman, Oct 13 2017

A clutter is a connected antichain of finite sets. The weight of a clutter is the sum of cardinalities of its edges.

			Non-isomorphic representatives of the a(7) = 8 clutters are:
((1234567)),
((12)(13456)), ((123)(1245)), ((123)(1456)),
((12)(13)(145)), ((12)(13)(234)), ((12)(13)(245)), ((13)(24)(125)).

Cf. A006126, A007716, A048143, A049311, A283877, A293606.

A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 16, 163, 11742, 12160640

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 16 maximal intersecting antichains:
  {{1,2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{2,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets]-Log[2,n]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.

a(7) from Elijah Beregovsky, May 06 2020

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

Original entry on oeis.org

0, 0, 1, 23, 1105, 154941, 66072394, 88945612865, 396990456067403

Offset: 0

Gus Wiseman, Dec 12 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(3) = 23 set-systems:
  {{1,2}}
  {{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For at least one choice we have A367902.
For no choices we have A367903, no singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367909.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n,0,3}]

A367903(n) + A367904(n) + a(n) = A058891(n).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A120338 Number of disconnected antichain covers of a labeled n-set.

Original entry on oeis.org

0, 1, 4, 30, 546, 41334, 54502904, 19317020441804

Offset: 1

Greg Huber, Jun 22 2006

nonn

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices. - Gus Wiseman, Sep 26 2019

			a(3)=4: the four disconnected covers are {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}} and {{1},{2},{3}}.

Cf. A006126, A007153, A048143, A327355.
Column k = 0 of A327351, if we assume a(0) = 1.
Column k = 0 of A327357, if we assume a(0) = 1.
The non-covering version is A327354.
The unlabeled version is A327426.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n]],SubsetQ],Union@@#==Range[n]&&Length[csm[#]]!=1&]],{n,4}] (* Gus Wiseman, Sep 26 2019 *)

cp's OEIS Frontend

A299353 Number of labeled connected uniform hypergraphs spanning n vertices.

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A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

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A322451 Number of unlabeled 3-uniform hypergraphs spanning n vertices.

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A326359 Number of maximal antichains of nonempty subsets of {1..n}.

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A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

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A326880 BII-numbers of set-systems that are closed under nonempty intersection.

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A293607 Number of unlabeled clutters of weight n.

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A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

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A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.