cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A326946 Number of unlabeled T_0 set-systems on n vertices.

Original entry on oeis.org

1, 2, 5, 34, 1919, 18660178
Offset: 0

Views

Author

Gus Wiseman, Aug 08 2019

Keywords

Comments

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).

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{1},{2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

Crossrefs

The non-T_0 version is A000612.
The antichain case is A245567.
The covering case is A319637.
The labeled version is A326940.
The version with empty edges allowed is A326949.
Cf. A003180, A055621, A059052, A059201, A316978, A319559, A319564, A326942.

Programs

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

Formula

Partial sums of A319637.
a(n) = A326949(n)/2.

Extensions

a(5) from Max Alekseyev, Oct 11 2023

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

Views

Author

Gus Wiseman, Aug 07 2019

Keywords

Comments

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).

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

a(n) = 2 * A059201(n).
Inverse binomial transform of A326941.

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

Views

Author

Gus Wiseman, Aug 08 2019

Keywords

Comments

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).

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

Inverse binomial transform of A326945.
a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k). - Andrew Howroyd, Aug 14 2019

Extensions

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

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

Views

Author

Gus Wiseman, Aug 08 2019

Keywords

Comments

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).

Examples

			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}}

Crossrefs

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.

Formula

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

Extensions

a(5) from Max Alekseyev, Oct 11 2023

A327011 Number of unlabeled sets of subsets covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 sets of subsets.

Original entry on oeis.org

2, 2, 4, 32, 2424
Offset: 0

Views

Author

Gus Wiseman, Aug 13 2019

Keywords

Comments

Alternatively, these are unlabeled sets of subsets covering n vertices whose dual is a (strict) antichain. 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}}. An antichain is a set of subsets where no edge is a subset of any other.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Crossrefs

Unlabeled covering sets of subsets are A003181.
The same with T_0 instead of T_1 is A326942.
The non-covering version is A326951 (partial sums).
The labeled version is A326960.
The case without empty edges is A326974.
Cf. A001146, A055621, A059523, A319637, A326961, A326972, A326973, A326976, A326977, A326979.

Formula

a(n) = A326974(n) / 2.
a(n > 0) = A326951(n) - A326951(n - 1).
Showing 1-5 of 5 results.

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