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 1-3 of 3 results.

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

Original entry on oeis.org

1, 1, 4, 96, 31692, 2147001636, 9223371991763269704, 170141183460469231473432887375376674952, 57896044618658097711785492504343953920509909728243389682424010192567186540224
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jan 16 2001

Keywords

Comments

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)

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

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

Views

Author

Patrick De Causmaecker, Jul 25 2014

Keywords

Comments

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

Examples

			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}}.

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

Definition corrected by Patrick De Causmaecker, Oct 10 2014
a(9), based on A000372, from Patrick De Causmaecker, Jun 01 2023

A326964 Number of connected set-systems covering a subset of {1..n}.

Original entry on oeis.org

1, 2, 7, 112, 32253, 2147316942, 9223372023968335715, 170141183460469231667123699322514272668, 5789604461865809771178549250434395393752402807429031284280914691514037561273
Offset: 0

Views

Author

Gus Wiseman, Aug 10 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets.

Examples

			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}}

Crossrefs

Covering sets of subsets are A000371.
Connected graphs are A001187.
The unlabeled version is A309667.
The BII-numbers of connected set-systems are A326749.
The covering case is A323818.
Cf. A007718, A048143, A058891, A092918, A300913, A304716, A326866, A326948.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[csm[#]]<=1&]],{n,0,4}]

Formula

Binomial transform of A323818.
Showing 1-3 of 3 results.

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