A000112 -id:A000112 - cp's OEIS frontend

A263857 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and k antichains.

1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 3, 3, 2, 3, 2, 0, 1, 0, 0, 0, 1, 1, 4, 6, 6, 9, 8, 7, 4, 5, 2, 2, 2, 3, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Christian Stump, Oct 28 2015

Row sums give A000112.

			Triangle begins:
1,
1,
1,1,
1,2,1,0,1,
1,3,3,2,3,2,0,1,0,0,0,1,
1,4,6,6,9,8,7,4,5,2,2,2,3,0,2,0,0,0,1,0,0,0,0,0,0,0,1,
...
The two element poset with 2 incomparable elements x, y has 4 antichains: {}, {x}, {y}, and {x,y}. The two element poset with 2 comparable elements x, y has 3 antichains: {}, {x}, and {y}. So T(2,1) = 1 and T(2,2) = 1.

FindStat - Combinatorial Statistic Finder, The number of antichains of a poset.

Cf. A000112.

A263858 Triangle read by rows: T(n,k) (n>=0, k>=1) is the number of posets with n elements and k maximal chains.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 7, 6, 2, 1, 13, 25, 18, 4, 2

Offset: 0

Christian Stump, Oct 28 2015

Row sums give A000112.

			Triangle begins:
1,
1,
1,1,
1,3,1,
1,7,6,2,
1,13,25,18,4,2,
...

FindStat - Combinatorial Statistic Finder, The number of maximal chains in a poset.

Cf. A000112.

A263860 Triangle read by rows: T(n,k) (n>=0, k>=1) is the number of posets with n elements and k linear extensions.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 1, 3, 2, 2, 1, 3, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 3, 5, 3, 7, 2, 5, 2, 4, 1, 4, 0, 2, 2, 2, 0, 2, 0, 4, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Christian Stump, Oct 28 2015

Row sums give A000112.

			Triangle begins:
1,
1,
1,1,
1,2,1,0,0,1,
1,3,2,2,1,3,0,2,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,
...

FindStat - Combinatorial Statistic Finder, The number of linear extensions of a poset.

Cf. A000112.

A349401 Number of unlabeled disconnected posets with n elements.

Original entry on oeis.org

0, 0, 1, 2, 6, 19, 80, 395, 2487, 19890, 206565, 2804453, 49872647, 1158843214, 35049606566, 1374685228988, 69690886873398, 4554367168841547

Offset: 0

Salah Uddin Mohammad, Nov 15 2021

Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 points.
Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, An Exact Enumeration of the Unlabeled Disconnected Posets, J. Int. Seq., Vol. 25 (2022), Article 22.5.4.

Cf. A000112, A000608, A342500.

a(n) = A000112(n) - A000608(n).

A376633 T(n,k) is the number of nonisomorphic n-element self-dual posets (or partially ordered sets) with k arcs in the Hasse diagram, irregular triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 5, 2, 1, 1, 1, 2, 4, 9, 11, 12, 5, 4, 1, 1, 1, 2, 4, 10, 16, 26, 22, 21, 10, 5, 0, 1, 1, 1, 2, 4, 11, 20, 44, 65, 98, 86, 79, 41, 25, 8, 4, 2, 2, 1, 1, 2, 4, 11, 21, 51, 92, 175, 220, 276, 237, 208, 103, 67, 25, 18, 5, 3, 0, 1, 1, 1, 2, 4, 11, 22, 55, 114, 264, 462, 798, 1015, 1294, 1180, 1035, 676, 477, 243, 149, 57, 36, 13, 8, 2, 4, 1, 1, 1, 2, 4, 11, 22, 56, 121, 303, 614, 1264, 2042, 2348, 3995, 4755, 4272, 3910, 2680, 1977, 1078, 697, 300, 189, 60, 50, 15, 12, 0, 3, 0, 1

Offset: 1

Rico Zöllner and Konrad Handrich, Sep 30 2024

Posets whose Hasse diagram looks the same if it is turned upside down.

The dual poset P* of the poset P is defined by: s ≤ t in P* if and only if t ≤ s in P. If P and P* are isomorphic, then P is called self-dual.

			The table starts:
1 ;
1 1 ;
1 1 1 ;
1 1 2 2 2 ;
1 1 2 3 5 2 1 ;
1 1 2 4 9 11 12 5 4 1 ;
1 1 2 4 10 16 26 22 21 10 5 0 1 ;
1 1 2 4 11 20 44 65 98 86 79 41 25 8 4 2 2 ;
1 1 2 4 11 21 51 92 175 220 276 237 208 103 67 25 18 5 3 0 1 ;
1 1 2 4 11 22 55 114 264 462 798 1015 1294 1180 1035 676 477 243 149 57 36 13 8 2 4 1;
...

R. P. Stanley, Enumerative Combinatorics I, 2nd. ed., pp. 277.

Cf. A000112, A022016, A022017, A342447.

Row sums: A133983.

A173399 Partial sums of A000798.

Original entry on oeis.org

1, 2, 6, 35, 390, 7332, 216859, 9752100, 652531454, 63912820877, 9040966693920, 1825887005430112, 521181458071204133, 208402575114740157174, 115825454552169007964634, 88852094573138413500449755, 93499965506283177978710943850, 134231450058844163850300579669696, 261626767193693218968916427480786899

Offset: 0

Jonathan Vos Post, Feb 17 2010

nonn

Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.

The subsequence of primes in this partial sum begins: 2, 216859.

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

Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Dec 30 2019 *)

a(n) = Sum_{i=0..n} A000798(i).

Missing term 7332 added and extended by Jean-François Alcover, Dec 30 2019

A174118 Partial sums of A000608.

Original entry on oeis.org

1, 2, 3, 6, 16, 60, 298, 1948, 16460, 179801, 2540520, 46485494, 1101504593, 33766488831, 1336910042036, 68237302714204, 4481677081035893

Offset: 1

Jonathan Vos Post, Mar 08 2010

Partial sums of number of connected partially ordered sets with n unlabeled elements. The subsequence of primes in this partial sum begins: 2, 3, 179801. Partial sum of Inverse Euler transform of A000112.

Cf. A000608, A000112.

A185349 Number of isomorphism classes of partially ordered sets of length n that occur as intervals in weak Bruhat order of some Coxeter group.

Original entry on oeis.org

1, 1, 2, 6, 22, 93

Offset: 0

Matthew J. Samuel, Feb 15 2011

Björner, Anders; Brenti, Francesco. Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.

Matthew J. Samuel, Word posets, complexity, and Coxeter groups, arXiv:1101.4655 [math.CO]
Matthew J. Samuel, Enumeration of small weak order intervals

a(n) >= A000112(n).

A230090 A certain restricted class of posets on n points.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 5, 11, 35635

Offset: 1

N. J. A. Sloane, Oct 09 2013

See Guay-Paquet for precise definition.

M. Guay-Paquet, A modular relation for the chromatic symmetric functions of (3+1)-free posets, arXiv preprint arXiv:1306.2400 [math.CO], 2013.

Cf. A000112, A079146.

A236756 Number of (unlabeled) partially ordered sets on n elements which are Cohen-Macaulay over the integers.

Original entry on oeis.org

1, 1, 2, 4, 9, 24, 73, 261, 1103

Offset: 0

Sam DeHority, Jan 30 2014

			For n=1, there is one poset up to isomorphism which corresponds to the simplicial complex with one facet of a single element. The homology groups over Z vanish for this, and it is therefore Cohen-Macaulay, thus a(1) = 1.
For n=2, there are two posets up to isomorphism. The first of which, the sum of two singletons, also has order complex with vanishing homology groups over Z. The second has an order complex with a single facet, and any interval has a contractible order complex, which therefore has vanishing homology groups. They are both Cohen-Macaulay thus a(2) = 2.

R. P. Stanley, Enumerative Combinatorics, Volume 1, Second edition, Cambridge, page 273.

Cf. A000112.

for n in range(66):
    j=0;
    for p in Posets(n):
        if p.order_complex().is_cohen_macaulay():
            j = j+1;
    print(j)

cp's OEIS Frontend

A263857 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and k antichains.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A263858 Triangle read by rows: T(n,k) (n>=0, k>=1) is the number of posets with n elements and k maximal chains.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A263860 Triangle read by rows: T(n,k) (n>=0, k>=1) is the number of posets with n elements and k linear extensions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A349401 Number of unlabeled disconnected posets with n elements.

Views

Author

Keywords

Links

Crossrefs

Formula

A376633 T(n,k) is the number of nonisomorphic n-element self-dual posets (or partially ordered sets) with k arcs in the Hasse diagram, irregular triangle read by rows.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A173399 Partial sums of A000798.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A174118 Partial sums of A000608.

Views

Author

Keywords

Comments

Crossrefs

A185349 Number of isomorphism classes of partially ordered sets of length n that occur as intervals in weak Bruhat order of some Coxeter group.

Views

Author

Keywords

References

Links

Crossrefs

A230090 A certain restricted class of posets on n points.

Views

Author

Keywords

Comments

Links

Crossrefs

A236756 Number of (unlabeled) partially ordered sets on n elements which are Cohen-Macaulay over the integers.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Sage