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.

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A059083 Number of T_0-antichains on a labeled n-set.

Original entry on oeis.org

2, 3, 3, 8, 96, 6373, 7725703, 2414518872815, 56130437161078967568912
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

Keywords

Comments

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point.

Examples

			a(0) = 1 + 1, a(1) = 1 + 2, a(2) = 2 + 1, a(3) = 6 + 2, a(4) = 12 + 52 + 25 + 6 + 1, a(5) = 520 + 1770 + 2086 + 1370 + 490 + 115 + 20 + 2.

References

  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A059079-A059082, A059048-A059052.
Cf. A000372.

Formula

a(n) = Sum_{m=0..binomial(n, floor(n/2))} A(m, n), where A(m, n) is number of m-element T_0-antichains on a labeled n-set. Cf. A059080.
a(n) = column sums of A059080.

Extensions

More terms from Vladeta Jovovic, Nov 28 2003

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

A326950 Number of T_0 antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 2, 4, 12, 107, 6439, 7726965, 2414519001532, 56130437161079183223017, 286386577668298409107773412840148848120595
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

			The a(0) = 1 through a(3) = 12 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{2}}      {{2}}
             {{1},{2}}  {{3}}
                        {{1},{2}}
                        {{1},{3}}
                        {{2},{3}}
                        {{1,2},{1,3}}
                        {{1,2},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}

Crossrefs

Antichains of nonempty sets are A014466.
T_0 set-systems are A326940.
The covering case is A245567.
Cf. A006126, A059201, A059052, A245567, A319559, A319564, A326030, A326946, A326947.

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[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}]

Formula

Binomial transform of A245567, if we assume A245567(0) = 1.

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 14 2019
a(9), based on A245567, from Patrick De Causmaecker, Jun 01 2023

A059049 Number of 6-element ordered T_0-antichains on an unlabeled n-set; T_1-hypergraphs on 6 labeled nodes with n (not necessarily empty) distinct hyperedges (n=0,1,...,64).

Original entry on oeis.org

0, 0, 0, 0, 30, 8220, 738842, 25256626, 464670831, 5570534392, 48655319306, 332222541564, 1859009659336, 8811850222304, 36244568422086, 131710639199900, 428697293437675, 1263065928235140, 3396450715952370
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

Keywords

Comments

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

References

  • V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A059048, A059050-A059052.

Formula

a(n)=C(64, n) - 30*C(48, n) + 120*C(40, n) + 60*C(36, n) + 60*C(34, n) - 12*C(33, n) - 345*C(32, n) - 720*C(30, n) + 810*C(28, n) + 120*C(27, n) + 480*C(26, n) + 360*C(25, n) - 480*C(24, n) - 720*C(23, n) - 240*C(22, n) - 540*C(21, n) + 1380*C(20, n) + 750*C(19, n) + 60*C(18, n) - 210*C(17, n) - 1535*C(16, n) - 1820*C(15, n) + 2250*C(14, n) + 1800*C(13, n) - 2820*C(12, n) + 300*C(11, n) + 2040*C(10, n) + 340*C(9, n) - 1815*C(8, n) + 510*C(7, n) - 1350*C(6, n) + 1350*C(5, n) + 274*C(4, n) - 548*C(3, n) + 120*C(2, n).

A059050 Number of 7-element ordered T_0-antichains on an unlabeled n-set; T_1-hypergraphs on 7 labeled nodes with n (not necessarily empty) distinct hyperedges (n=0,1,...,128).

Original entry on oeis.org

0, 0, 0, 0, 0, 20580, 9106440, 1058272828, 56671435220, 1819138009252, 40526220292026, 685404291322800, 9333711208757096, 106588763704012184, 1051025434717631806, 9144977489478933912, 71381946761468630363
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

Keywords

Comments

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

References

  • V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A059048, A059049, A059051, A059052.

Formula

a(n)=C(128, n) - 42*C(96, n) + 210*C(80, n) + 140*C(72, n) + 210*C(68, n) - 84*C(66, n) + 14 *C(65, n) - 819*C(64, n) - 2520*C(60, n) + 2730*C(56, n) + 840*C(54, n) + 840*C(52, n) - 420*C(51, n) + 2940*C(50, n) + 630*C(48, n) - 5040*C(46, n) + 840*C(45, n) - 1260*C(44, n) + 1680*C(43, n) - 9660*C(42, n) + 1260*C(41, n) + 3360*C(40, n) - 7560*C(39, n) + 11130*C(38, n) + 5880*C(37, n) + 9240*C(36, n) + 2982*C(35, n) - 6300*C(34, n) - 8652 *C(33, n) - 9905*C(32, n) - 8400*C(31, n) - 8540*C(30, n) + 13860*C(29, n) + 14490 *C( 28, n) - 5040*C(27, n) + 10500*C(26, n) + 10080*C(25, n) - 8120*C(24, n) - 15050*C(23, n) - 5040*C(22, n) - 11340*C(21, n) + 20580*C(20, n) + 15750*C(19, n) - 1540*C(18, n) - 5810*C(17, n) - 16485*C(16, n) - 21420*C(15, n) + 26250*C(14, n) + 21000*C(13, n) - 29820*C(12, n) + 3500*C(11, n) + 17640*C(10, n) + 2940*C(9, n) - 16016*C(8, n) + 4410*C(7, n) - 9744*C(6, n) + 9744*C(5, n) + 1764*C(4, n) - 3528*C(3, n) + 720*C(2, n).

A059079 Number of n-element T_0-antichains on a labeled set.

Original entry on oeis.org

2, 5, 19, 16654, 2369110564675, 5960531437586238714806902334250676, 479047836152505670895481840783987408043359908583921478726185296900312296071642855730299
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Dec 23 2000

Keywords

Comments

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point.

Examples

			a(0) = (1/0!)*[1!*e] = 2; a(1) = (1/1!)*[2!*e] = 5; a(2) = (1/2!)*([4!*e] - 2*[3!*e] + [2!*e]) = 19; a(3) = (1/3!)*([8!*e] - 6*[6!*e] + 6*[5!*e] + 3*[4!*e] - 6*[3!*e] + 2*[2!*e]) = 16654, where [n!*e]=floor(n!*exp(1)).

References

  • V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A059080-A059083, A059048-A059052, A000522.

A059080 Triangle A(n,m) of numbers of n-element T_0-antichains on a labeled m-set, m=0,...,2^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 0, 1, 6, 12, 0, 0, 0, 2, 52, 520, 2640, 6720, 6720, 0, 0, 0, 0, 25, 1770, 53940, 1012620, 13487040, 136745280, 1094688000, 7025356800, 36084787200, 145297152000, 435891456000, 871782912000, 871782912000
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Dec 29 2000

Keywords

Comments

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. Row sums give A059079. Column sums give A059083.

Examples

			[1, 1], [1, 2, 2], [0, 0, 1, 6, 12], [0, 0, 0, 2, 52, 520, 2640, 6720, 6720], ...; there are 2 3-element T_0-antichains on a 3-set: {{1}, {2}, {3}}, {{1, 2}, {1, 3}, {2, 3}}.

References

  • V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A059079, A059081-A059083, A059048-A059052.

A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 2, 12, 232, 19230, 16113300, 1063117943398, 225402329237199496416
Offset: 0

Views

Author

Gus Wiseman, Aug 11 2019

Keywords

Comments

First differs from A182507 at a(5) = 19230, A182507(5) = 12848.
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).

Examples

			The a(0) = 1 through a(3) = 12 set-systems:
  {}  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
             {{2},{1,2}}  {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Crossrefs

The version with empty edges allowed is A326943.
Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.

Programs

  • Mathematica
    dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Formula

a(n) = A326943(n) - A326944(n).
a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.
a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.

A326967 Number of 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, 10, 92, 38362, 4020654364, 18438434849260080818, 340282363593610212050791236025945013956, 115792089237316195072053288318104625957065868613454666314675263144628100544274
Offset: 0

Views

Author

Gus Wiseman, Aug 10 2019

Keywords

Comments

Alternatively, these are 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. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other.

Examples

			The a(0) = 2 through a(2) = 10 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Crossrefs

Sets of subsets are A001146.
The unlabeled version is A326951.
The covering version is A326960.
The case without empty edges is A326965.
Sets of subsets whose dual is a weak antichain are A326969.
Cf. A059052, A059523, A326941, A326966, A326972, A326976, A326977, A326979.

Programs

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

Formula

a(n) = 2 * A326965(n).
Binomial transform of A326960.

A059051 Number of ordered T_0-antichains on an unlabeled n-set; labeled T_1-hypergraphs with n (not necessarily empty) distinct hyperedges.

Original entry on oeis.org

2, 3, 2, 4, 99, 190866
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Dec 19 2000

Keywords

Comments

An antichain on a set is a T_0-antichain if for every two distinct points of the set there exists a member of the antichain containing one but not the other point. T_1-hypergraph is a hypergraph which for every ordered pair (u,v) of distinct nodes has a hyperedge containing u but not v.

Examples

			a(0) = 1 + 1, a(1) = 1 + 2, a(2) = 1 + 1, a(3) = 2 + 2, a(4) = 1 + 13 + 25 + 30 + 30, a(5) = 26 + 354 + 2086 + 8220 + 20580 + 38640 + 60480 + 60480.

References

  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A059048-A059050, A059052.

Formula

a(n) = Sum_{m=0..C(n, floor(n/2))} A(m, n), where A(m, n) is number of m-element ordered T_0-antichains on an unlabeled n-set. Cf. A059048.
a(n) = column sums of A059048.
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