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 1-10 of 15 results. Next

A293606 Number of unlabeled antichains of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 20, 33, 72, 139
Offset: 0

Views

Author

Gus Wiseman, Oct 13 2017

Keywords

Comments

An antichain is a finite set of finite nonempty sets, none of which is a subset of any other. The weight of an antichain is the sum of cardinalities of its elements.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n where every vertex is the unique common element of some subset of the edges. For example, the a(1) = 1 through a(6) = 20 set multipartitions are:
{1} {1}{1} {1}{1}{1} {1}{2}{12} {1}{2}{2}{12} {12}{13}{23}
{1}{2} {1}{2}{2} {1}{1}{1}{1} {1}{2}{3}{23} {1}{2}{12}{12}
{1}{2}{3} {1}{1}{2}{2} {1}{1}{1}{1}{1} {1}{2}{13}{23}
{1}{2}{2}{2} {1}{1}{2}{2}{2} {1}{2}{3}{123}
{1}{2}{3}{3} {1}{2}{2}{2}{2} {1}{1}{2}{2}{12}
{1}{2}{3}{4} {1}{2}{2}{3}{3} {1}{1}{2}{3}{23}
{1}{2}{3}{3}{3} {1}{2}{2}{2}{12}
{1}{2}{3}{4}{4} {1}{2}{3}{3}{23}
{1}{2}{3}{4}{5} {1}{2}{3}{4}{34}
{1}{1}{1}{1}{1}{1}
{1}{1}{1}{2}{2}{2}
{1}{1}{2}{2}{2}{2}
{1}{1}{2}{2}{3}{3}
{1}{2}{2}{2}{2}{2}
{1}{2}{2}{3}{3}{3}
{1}{2}{3}{3}{3}{3}
{1}{2}{3}{3}{4}{4}
{1}{2}{3}{4}{4}{4}
{1}{2}{3}{4}{5}{5}
{1}{2}{3}{4}{5}{6}
(End)

Examples

			Non-isomorphic representatives of the a(5) = 9 antichains are:
((12345)),
((1)(2345)), ((12)(134)), ((12)(345)),
((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),
((1)(2)(3)(45)),
((1)(2)(3)(4)(5)).

Crossrefs

Cf. A006126, A006602, A007411, A007716, A048143, A049311, A283877, A293607.
Cf. A000372, A000612, A003182, A014466, A055621, A293993, A326704, A326972.

Formula

Euler transform of A293607.

A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

Original entry on oeis.org

1, 2, 5, 46, 19181, 2010327182, 9219217424630040409, 170141181796805106025395618012972506978, 57896044618658097536026644159052312978532934306727333157337631572314050272137
Offset: 0

Views

Author

Gus Wiseman, Aug 10 2019

Keywords

Comments

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}}. An antichain is a set-system where no edge is a subset of any other. This sequence counts set-systems whose dual is a (strict) antichain, also called T_1 set-systems.

Examples

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{2},{1,2}}

Crossrefs

Set-systems are A058891.
T_0 set-systems are A326940.
The covering case is A326961.
The version with empty edges allowed is A326967.
Set-systems whose dual is a weak antichain are A326968.
The unlabeled version is A326972.
The BII_numbers of these set-systems are A326979.
Cf. A059052, A326951, A326966, A326970, A326971, A326976, A326977.

Programs

  • Mathematica
    tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],tmQ]],{n,0,3}]

Formula

Binomial transform of A326961.
a(n) = A326967(n)/2.

A326961 Number of set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called covering T_1 set-systems.

Original entry on oeis.org

1, 1, 2, 36, 19020, 2010231696, 9219217412568364176, 170141181796805105960861096082778425120, 57896044618658097536026644159052312977171804852352892309392604715987334365792
Offset: 0

Views

Author

Gus Wiseman, Aug 12 2019

Keywords

Comments

Same as A059523 except with a(1) = 1 instead of 2.
Alternatively, these are set-systems covering n vertices whose dual is a (strict) antichain. 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. An antichain is a set of sets, none of which is a subset of any other.

Examples

			The a(3) = 36 set-systems:
  {{1}{2}{3}}        {{12}{13}{23}{123}}     {{2}{3}{12}{13}{23}}
  {{12}{13}{23}}     {{1}{2}{3}{12}{13}}     {{2}{3}{12}{13}{123}}
  {{1}{2}{3}{12}}    {{1}{2}{3}{12}{23}}     {{2}{12}{13}{23}{123}}
  {{1}{2}{3}{13}}    {{1}{2}{3}{13}{23}}     {{3}{12}{13}{23}{123}}
  {{1}{2}{3}{23}}    {{1}{2}{12}{13}{23}}    {{1}{2}{3}{12}{13}{23}}
  {{1}{2}{13}{23}}   {{1}{2}{3}{12}{123}}    {{1}{2}{3}{12}{13}{123}}
  {{1}{2}{3}{123}}   {{1}{2}{3}{13}{123}}    {{1}{2}{3}{12}{23}{123}}
  {{1}{3}{12}{23}}   {{1}{2}{3}{23}{123}}    {{1}{2}{3}{13}{23}{123}}
  {{2}{3}{12}{13}}   {{1}{3}{12}{13}{23}}    {{1}{2}{12}{13}{23}{123}}
  {{1}{12}{13}{23}}  {{1}{2}{13}{23}{123}}   {{1}{3}{12}{13}{23}{123}}
  {{2}{12}{13}{23}}  {{1}{3}{12}{23}{123}}   {{2}{3}{12}{13}{23}{123}}
  {{3}{12}{13}{23}}  {{1}{12}{13}{23}{123}}  {{1}{2}{3}{12}{13}{23}{123}}

Crossrefs

Covering set-systems are A003465.
Covering T_0 set-systems are A059201.
The version with empty edges allowed is A326960.
The non-covering version is A326965.
Covering set-systems whose dual is a weak antichain are A326970.
The unlabeled version is A326974.
The BII-numbers of T_1 set-systems are A326979.
Cf. A058891, A059052, A059523, A323818, A326972, A326973, A326976, A326977.

Programs

  • Mathematica
    tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&tmQ[#]&]],{n,0,3}]

Formula

Inverse binomial transform of A326965.

A326974 Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.

Original entry on oeis.org

1, 1, 2, 16, 1212
Offset: 0

Views

Author

Gus Wiseman, Aug 11 2019

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 16 set-systems:
  {}  {{1}}  {{1},{2}}        {{1},{2},{3}}
             {{1},{2},{1,2}}  {{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Unlabeled covers are A055621.
The same with T_0 instead of T_1 is A319637.
The labeled version is A326961.
The non-covering version is A326972 (partial sums).
Unlabeled covering set-systems whose dual is a weak antichain are A326973.
Cf. A000612, A059523, A319559, A326946, A326951, A326965, A326970, A326971, A326976, A326977, A326979.

Formula

a(n > 0) = A326972(n) - A326972(n - 1).

A326979 BII-numbers of T_1 set-systems.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 30, 31, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 79, 91, 94, 95, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 158, 159
Offset: 1

Views

Author

Gus Wiseman, Aug 13 2019

Keywords

Comments

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_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a subset of any other.
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.

Examples

			The sequence of all T_1 set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  15: {{1},{2},{1,2},{3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  43: {{1},{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}

Crossrefs

BII-numbers of T_0 set-systems are A326947.
T_1 set-systems are counted by A326965, A326961 (covering), A326972 (unlabeled), and A326974 (unlabeled covering).
BII-numbers of set-systems whose dual is a weak antichain are A326966.
Cf. A059201, A059523, A319639, A326970, A326973, A326976, A326977.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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}];
Select[Range[0,100],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]],SubsetQ]&]

A326976 Number of factorizations of n into factors > 1 such that every prime factor of n is the GCD of some subset of the factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Aug 13 2019

Keywords

Examples

			The a(72) = 5 factorizations:
  (3*4*6)
  (2*3*12)
  (2*2*3*6)
  (2*3*3*4)
  (2*2*2*3*3)

Crossrefs

Factorizations whose dual is a weak antichain are A326975.
T_1 factorizations (whose dual is a strict antichain) are A327012.
T_0 factorizations (whose dual is strict) are A316978.
Cf. A001055, A326947, A326965, A326972, A326974, A326977, A326979.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],n==1||Union[Select[GCD@@@Rest[Subsets[#]],PrimeQ]]==First/@FactorInteger[n]&]],
{n,100}]

A326973 Number of unlabeled set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 3, 19, 1243
Offset: 0

Views

Author

Gus Wiseman, Aug 11 2019

Keywords

Comments

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 19 set-systems:
  {}  {{1}}  {{1,2}}          {{1,2,3}}
             {{1},{2}}        {{1},{2,3}}
             {{1},{2},{1,2}}  {{1},{2},{3}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Unlabeled covering set-systems are A055621.
The labeled version is A326970.
The non-covering case is A326971 (partial sums).
The case that is also T_0 is the T_1 case A326974.
Cf. A000612, A059523, A319637, A326966, A326968, A326972, A326975, A326978.

A326968 Number of set-systems on n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 6, 56, 19446
Offset: 0

Views

Author

Gus Wiseman, Aug 10 2019

Keywords

Comments

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

Examples

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{2},{1,2}}

Crossrefs

The case with strict dual is A326965.
The BII-numbers of these set-systems are A326966.
The version with empty edges allowed is A326969.
The covering case is A326970.
The unlabeled version is A326971.
Cf. A058891, A059523, A326940, A326972, A326973, A326975, A326978, A326979.

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],{1,n}]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Formula

a(n) = A326969(n)/2.
Binomial transform of A326970.

A304996 Number of unlabeled antichains of finite sets spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 6, 24, 166, 3266, 826308
Offset: 0

Views

Author

Gus Wiseman, May 23 2018

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 24 antichains:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{1},{2}}
{{2},{1,2}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{1},{2},{1,2}}
{{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}}

Crossrefs

Partial sums of A304997.
Cf. A006126, A261005, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A014466, A293993, A319639, A319721, A326704, A326972.

Extensions

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A326951 Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

Original entry on oeis.org

2, 4, 8, 40, 2464
Offset: 0

Views

Author

Gus Wiseman, Aug 13 2019

Keywords

Comments

Alternatively, these are unlabeled sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. 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. An antichain is a set of subsets where no edge is a subset of any other.

Examples

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{},{1}}
                  {{1},{2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Crossrefs

Unlabeled sets of subsets are A003180.
Unlabeled T_0 sets of subsets are A326949.
The labeled version is A326967.
The case without empty edges is A326972.
The covering case is A327011 (first differences).
Cf. A003181, A059052, A326960, A326965, A326969, A326974, A326976.

Formula

a(n) = 2 * A326972(n).
a(n) = Sum_{k = 0..n} A327011(k).
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