A000112 -id:A000112 - cp's OEIS frontend

A065066 Triangle T(n,k) read by rows of partially ordered sets ("posets") with n unlabeled nodes and k maximal elements (0 <= k <= n).

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 7, 3, 1, 0, 16, 27, 15, 4, 1, 0, 63, 134, 88, 27, 5, 1, 0, 318, 814, 642, 221, 43, 6, 1, 0, 2045, 6258, 5828, 2319, 477, 64, 7, 1, 0, 16999, 60877, 66612, 30698, 7015, 931, 90, 8, 1, 0, 183231, 755323, 959941, 514525, 133610

Offset: 0

Jens Voß, Nov 07 2001

			1,
0,1,
0,1,1,
0,2,2,1,
0,5,7,3,1,
0,16,27,15,4,1,
0,63,134,88,27,5,1,
0,318,814,642,221,43,6,1,
...

R. J. Mathar, Table of n, a(n) for n = 0..90
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Tables 8, 10, 16, 20, 24, 28, 32, 36, 40...
FindStat - Combinatorial Statistic Finder, The number of minimal elements in a poset., The number of maximal elements of a poset.
Index entries for sequences related to posets

Cf. A000112.

T(n+1, 1) = Sum_{i=0}^n T(n, i) = A000112(n).

Values beyond row 6 from R. J. Mathar, Mar 14 2021

A124482 Number of indecomposable disconnected hook length posets with n elements.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 5, 31

Offset: 1

Franklin T. Adams-Watters, Nov 03 2006

Cheryl A. Gann and Robert A. Proctor, Poset Counts
Cheryl A. Gann and Robert A. Proctor, Definition of Indecomposable Disconnected Hook Length Poset

Cf. A000112, A124480, A124481.

A124775 Number of unlabeled partially ordered sets associated with compositions in standard order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 3, 1, 2, 1, 1

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

The k-th term of the composition is the number of objects with rank k. The rank of an object is one more than the maximum rank of any smaller object in the ordering (1 for a minimal element), or equivalently the size of the largest chain of which the object is the maximal element.

			Composition number 11 is 2,1,1; there are 3 partial orders associated with this (shown below), so a(11) = 3.
..O..*O..*..O
..|..*|..*./|
..O..*O..*O.|
./.\.*|..*|.|
O...O*O.O*O.O
The table starts:
1
1
1 1
1 2 1 1

Cf. A066099, A124776, A124777, A011782 (row lengths), A000112 (row sums).

A301871 Number of N- and bowtie-free posets with n elements.

Original entry on oeis.org

1, 2, 5, 14, 40, 121, 373, 1184, 3823, 12554, 41733, 140301, 475934, 1627440, 5602983, 19406703, 67574371, 236409625, 830582851, 2929246932, 10366380583, 36801225872, 131021870786, 467701875135, 1673584553886, 6002046468815, 21570135722058, 77668429499325, 280167079428684, 1012323004985313

Offset: 1

Stephan Wagner, Mar 28 2018

The number of n-element posets that do not include the two 4-element posets "N" and "bowtie" as induced subposets.

Stephan Wagner, Table of n, a(n) for n = 1..100
T. Hasebe and S. Tsujie, Order quasisymmetric functions distinguish rooted trees, arXiv:1610.03908 [math.CO], 2016-2017.
T. Hasebe and S. Tsujie, Order quasisymmetric functions distinguish rooted trees, Journal of Algebraic Combinatorics 46 (2017), 499-515.
V. Razanajatovo Misanantenaina and S. Wagner, A Tutte-like polynomial for rooted trees and specific posets, arXiv:1803.09623 [math.CO], 2018.

Cf. A000112, A003430, A079144, A079146 for related sequences regarding the enumeration of unlabeled posets.

V=1;Do[V = Normal[Series[(1 - x) Exp[Sum[(2 x^m - x^(2 m)) (V /. x -> x^m)/m, {m, 1, n}]], {x, 0, n}]], {n, 1, 20}]; Table[Coefficient[V,x,n],{n, 1, 20}]

G.f. V(x) = 1 + x + 2x + 5x^2 + ... satisfies V(x) = (1-x)exp[sum_{m >=1} (2x^m-x^(2m))V(x^m)/m] (see Razanajatovo Misanantenaina/Wagner).

A327016 BII-numbers of finite T_0 topologies without their empty set.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 8, 17, 24, 25, 34, 40, 42, 69, 70, 71, 81, 85, 87, 88, 89, 93, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 257, 384, 385, 514, 640, 642, 1029, 1030, 1031, 1281, 1285, 1287, 1408, 1409, 1413, 1538, 1542, 1543, 1664, 1666, 1670, 1920

Offset: 1

Gus Wiseman, Aug 14 2019

nonn

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).

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 finite T_0 topologies without their empty set together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  81: {{1},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  87: {{1},{2},{1,2},{1,3},{1,2,3}}
  88: {{3},{1,3},{1,2,3}}

Gus Wiseman, The first 100 finite T_0 topologies, represented as posets.

T_0 topologies are A001035, with unlabeled version A000112.
BII-numbers of topologies without their empty set are A326876.
BII-numbers of T_0 set-systems are A326947.
Cf. A001930, A048793, A306445, A316978, A319564, A326031, A326872, A326875, A326939, A326941, A326959.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],UnsameQ@@dual[bpe/@bpe[#]]&&SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]

A046907 Number of isomorphism classes of irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 1, 2, 7, 31, 184, 1351, 12524, 146468, 2177570, 41374407, 1008220289, 31559446774, 1269310589336, 65562045668340, 4345161435996517

Offset: 0

John A. Wright

G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table 1).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Cf. A046908.

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
B[x_] = A000112.x^Range[0, Length[A000112] - 1];
A[x_] = 2 - 1/B[x];
CoefficientList[A[x] + O[x]^Length[A000112], x] (* Jean-François Alcover, Jan 01 2020 *)

G.f.: A(x) = 2-1/B(x), where B(x) is g.f. of A000112. - Vladeta Jovovic, Jan 15 2006

More terms from Vladeta Jovovic, Jan 15 2006

A066304 Reduced partially ordered sets (posets) with n unlabeled elements.

Original entry on oeis.org

1, 1, 1, 3, 8, 30, 150

Offset: 0

Christian G. Bower, Dec 12 2001

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 57 (1.4.70).

Index entries for sequences related to posets

Cf. A000112. Euler transform of A066305.

A091070 Number of automorphism groups of partial orders on n points.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 16, 21, 41, 57, 103, 140, 276

Offset: 0

Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

			a(3)=3 because of the 5 partial orders on 3 points, 2 have trivial automorphism group, 2 have an automorphism of order 2 and one has the full symmetric group as its automorphism group; thus 3 different (conjugacy classes of) subgroups occur.

G. Pfeiffer, Subgroups.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A000638 (subgroups of the symmetric group), A000112 (partial orders).

A124016 Number of connected d-complete posets with n elements.

Original entry on oeis.org

1, 1, 2, 5, 11, 28, 69, 181, 474

Offset: 1

Franklin T. Adams-Watters, Nov 03 2006

Cheryl A. Gann and Robert A. Proctor, Poset Counts
Robert A. Proctor, Definition of d-Complete Poset

Cf. A000112, A124480, A124481.

A173311 a(n) is the number of regular D classes in the semigroup of all binary relations on [n].

Original entry on oeis.org

1, 2, 4, 9, 25, 88, 406, 2451, 19450, 202681, 2769965, 49519392, 1154411138, 34978238590, 1373171398361, 69648249299517, 4552778914494604

Offset: 0

Jonathan Vos Post, Feb 16 2010

Previous name was: Partial sums of A000112.

K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.

Cf. A000112, A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A006057, A079263, A079265, A007903.

Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Jan 01 2020 *)

a(n) = Sum_{i=0..n} A000112(i).

New name from Geoffrey Critzer, May 22 2022

cp's OEIS Frontend

A065066 Triangle T(n,k) read by rows of partially ordered sets ("posets") with n unlabeled nodes and k maximal elements (0 <= k <= n).

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A124482 Number of indecomposable disconnected hook length posets with n elements.

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A124775 Number of unlabeled partially ordered sets associated with compositions in standard order.

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A301871 Number of N- and bowtie-free posets with n elements.

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Mathematica

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A327016 BII-numbers of finite T_0 topologies without their empty set.

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Mathematica

A046907 Number of isomorphism classes of irreducible posets with n labeled points.

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Programs

Mathematica

Formula

Extensions

A066304 Reduced partially ordered sets (posets) with n unlabeled elements.

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References

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A091070 Number of automorphism groups of partial orders on n points.

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Author

Keywords

Examples

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A124016 Number of connected d-complete posets with n elements.

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Author

Keywords

Links

Crossrefs

A173311 a(n) is the number of regular D classes in the semigroup of all binary relations on [n].

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