A326941 -id:A326941 - cp's OEIS frontend

A059201 Number of T_0-covers of a labeled n-set.

1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 16 2001

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.
From Gus Wiseman, Aug 13 2019: (Start)
A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges). For example, the a(2) = 4 covers are:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}
(End)

G. C. Greubel, Table of n, a(n) for n = 0..11
Vladeta Jovovic, T_0-covers of a labeled 3-set

Row sums of A059202.
Cf. A059203, A059084, A059085, A059086, A059087, A059088, A059089.
Covering set-systems are A003465.
The unlabeled version is A319637.
The version with empty edges allowed is A326939.
The non-covering version is A326940.
BII-numbers of T_0 set-systems are A326947.
The same with connected instead of covering is A326948.
The T_1 version is A326961.
Cf. A245567, A316978, A319559, A319564, A323818, A326941, A326946, A326970.

Table[Sum[StirlingS1[n + 1, k]*2^(2^(k - 1) - 1), {k, 0, n + 1}], {n,0,5}] (* G. C. Greubel, Dec 28 2016 *)
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 13 2019 *)

a(n) = Sum_{i=0..n+1} stirling1(n+1, i)*2^(2^(i-1)-1).
a(n) = Sum_{m=0..2^n-1} A059202(n,m).
Inverse binomial transform of A326940 and exponential transform of A326948. - Gus Wiseman, Aug 13 2019

A326947 BII-numbers of T_0 set-systems.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 70, 71, 73, 74, 75, 77, 78

Offset: 1

Gus Wiseman, Aug 08 2019

The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

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. Elements of a set-system are sometimes called edges.

			The sequence of all T_0 set-systems together with their BII numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}

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

T_0 set-systems are counted by A326940, with unlabeled version A326946.

Cf. A059201, A316978, A319559, A319564, A326939, A326941, A326949.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
TZQ[sys_]:=UnsameQ@@dual[sys];
Select[Range[0,100],TZQ[bpe/@bpe[#]]&]

from itertools import count, chain, islice
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):
        a,b,s = [bin_i(k) for k in bin_i(n)],[],set()
        for i in {i for i in chain.from_iterable(a)}:
            b.append([])
            for j in range(len(a)):
                if i in a[j]:
                    b[-1].append(j)
            s.add(tuple(b[-1]))
        if len(s) == len(b):
            yield n
A326947_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jul 25 2024

A326940 Number of T_0 set-systems on n vertices.

Original entry on oeis.org

1, 2, 7, 112, 32105, 2147161102, 9223372004645756887, 170141183460469231537996491362807709908, 57896044618658097711785492504343953921871039195927143534469727707459805807105

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(2) = 7 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The non-T_0 version is A058891 shifted to the left.
The covering case is A059201.
The version with empty edges is A326941.
The unlabeled version is A326946.
Cf. A003180, A316978, A319559, A319564, A319637, A326939, A326947, A326949.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&]],{n,0,3}]

Binomial transform of A059201.

A326939 Number of T_0 sets of subsets of {1..n} that cover all n vertices.

Original entry on oeis.org

2, 2, 8, 192, 63384, 4294003272, 18446743983526539408, 340282366920938462946865774750753349904, 115792089237316195423570985008687907841019819456486779364848020385134373080448

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A000371.
The case without empty edges is A059201.
The non-covering version is A326941.
The unlabeled version is A326942.
The case closed under intersection is A326943.
Cf. A003180, A003181, A003465, A316978, A319564, A319637, A326940, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&UnsameQ@@dual[#]&]],{n,0,3}]

a(n) = 2 * A059201(n).

Inverse binomial transform of A326941.

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

Original entry on oeis.org

2, 1, 1, 5, 76, 5993, 7689745, 2414465044600, 56130437141763247212112, 286386577668298408602599478477358234902247

Offset: 0

Patrick De Causmaecker, Jul 25 2014

This is the number of antichain covers such that the induced partition contains only singletons. The induced partition of {{1,2},{2,3},{1,3},{3,4}} is {{1},{2},{3},{4}}, while the induced partition of {{1,2,3},{2,3,4}} is {{1},{2,3},{4}}.
This sequence is related to A006126. See 1st formula.
The sequence is also related to Dedekind numbers through Stirling numbers of the second kind. See 2nd formula.
Sets of subsets of the described type are said to be T_0. - Gus Wiseman, Aug 14 2019

			For n = 0, a(0) = 2 by the antisets {}, {{}}.
For n = 1, a(1) = 1 by the antiset {{1}}.
For n = 2, a(2) = 1 by the antiset {{1},{2}}.
For n = 3, a(3) = 5 by the antisets {{1},{2},{3}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.

Cf. A000372 (Dedekind numbers), A006126 (Number of antichain covers of a labeled n-set).
Sequences counting and ranking T_0 structures:
  A000112 (unlabeled topologies),
  A001035 (topologies),
  A059201 (covering set-systems),
  A245567 (antichain covers),
  A309615 (covering set-systems closed under intersection),
  A316978 (factorizations),
  A319559 (unlabeled set-systems by weight),
  A319564 (integer partitions),
  A319637 (unlabeled covering set-systems),
  A326939 (covering sets of subsets),
  A326940 (set-systems),
  A326941 (sets of subsets),
  A326943 (covering sets of subsets closed under intersection),
  A326944 (covering sets of subsets with {} and closed under intersection),
  A326945 (sets of subsets closed under intersection),
  A326946 (unlabeled set-systems),
  A326947 (BII-numbers of set-systems),
  A326948 (connected set-systems),
  A326949 (unlabeled sets of subsets),
  A326950 (antichains),
  A326959 (set-systems closed under intersection),
  A327013 (unlabeled covering set-systems closed under intersection),
  A327016 (BII-numbers of topologies).

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 14 2019 *)

A000372(n) = Sum_{k=0..n} S(n+1,k+1)*a(k).
a(n) = A006126(n) - Sum_{k=1..n-1} S(n,k)*a(k).
Were n > 0 and S(n,k) is the number of ways to partition a set of n elements into k nonempty subsets.
Inverse binomial transform of A326950, if we assume a(0) = 1. - Gus Wiseman, Aug 14 2019

Definition corrected by Patrick De Causmaecker, Oct 10 2014

a(9), based on A000372, from Patrick De Causmaecker, Jun 01 2023

A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

Original entry on oeis.org

2, 2, 6, 70, 4078, 2704780, 151890105214, 28175292217767880450

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(3) = 6 sets of subsets:
  {}    {{1}}     {{1},{1,2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A326906.
The case without empty edges is A309615.
The non-covering version is A326945.
The version not closed under intersection is A326939.
Cf. A003180, A003181, A003465, A059052, A059201, A245567, A316978, A319564, A319637, A326940, A326941, A326942, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

Original entry on oeis.org

1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(2) = 4 sets of subsets:
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The version not closed under intersection is A059201.
The non-T_0 version is A326881.
The version where {} is not necessarily an edge is A326943.
Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

Original entry on oeis.org

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A102897.
The version not closed under intersection is A326941.
The covering case is A326943.
The case without empty edges is A326959.
Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A326943.

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

2, 4, 12, 112, 38892

Offset: 0

Gus Wiseman, Aug 10 2019

The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1,2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Sets of subsets whose dual is strict are A326941.
The BII-numbers of set-systems whose dual is a weak antichain are A326966.
Sets of subsets whose dual is a (strict) antichain are A326967.
The case without empty edges is A326968.
Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

a(n) = 2 * A326968(n).

a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).

A326949 Number of unlabeled T_0 sets of subsets of {1..n}.

Original entry on oeis.org

2, 4, 10, 68, 3838, 37320356

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{},{1}}
                  {{1},{2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A003180.
The labeled version is A326941.
The covering case is A326942 (first differences).
The case without empty edges is A326946.
Cf. A000371, A000612, A003181, A059052, A245567, A316978, A319559, A319564, A319637, A326939, A326940.

a(n) = 2 * A326946(n).

a(5) from Max Alekseyev, Oct 11 2023

cp's OEIS Frontend

A059201 Number of T_0-covers of a labeled n-set.

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A326947 BII-numbers of T_0 set-systems.

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A326940 Number of T_0 set-systems on n vertices.

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A326939 Number of T_0 sets of subsets of {1..n} that cover all n vertices.

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A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

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A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

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A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

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A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

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