A000112 -id:A000112 - cp's OEIS frontend

A263861 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and whose order polytope has k facets.

1, 1, 1, 1, 1, 3, 1, 1, 7, 6, 2, 1, 13, 26, 17, 4, 2, 1, 22, 85, 112, 60, 27, 7, 3, 1

Offset: 0

Christian Stump, Oct 28 2015

Row sums give A000112.

The order polytope of a poset P is given by all points in the unit cube [0,1]^P such that xp

			Triangle begins:
1,
1,
1,1,
1,3,1,
1,7,6,2,
1,13,26,17,4,2,
1,22,85,112,60,27,7,3,1,
...

FindStat - Combinatorial Statistic Finder, The number of facets in the order polytope of this poset.
Richard Stanley, Two poset polytopes, Discrete & Computational Geometry 1 (1986), DOI: 10.1007/BF02187680.

Cf. A000112.

A263862 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and whose chain polytope has k facets.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 7, 6, 2, 1, 13, 25, 18, 4, 2, 1, 22, 80, 111, 60, 32, 7, 4, 1

Offset: 0

Christian Stump, Oct 28 2015

Row sums give A000112.

Is this the same as A263858? - R. J. Mathar, Nov 03 2015

			Triangle begins:
1,
1,
1,1,
1,3,1,
1,7,6,2,
1,13,25,18,4,2,
1,22,80,111,60,32,7,4,1,
...

FindStat - Combinatorial Statistic Finder, The number of facets in the chain polytope of the poset.

Cf. A000112.

A263863 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and k chains.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 3, 3, 2, 2, 0, 3, 0, 0, 0, 1, 1, 1, 3, 5, 8, 5, 7, 4, 8, 2, 5, 1, 5, 0, 3, 0, 0, 0, 4, 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,
1,1,2,0,1,
1,1,3,3,2,2,0,3,0,0,0,1,
1,1,3,5,8,5,7,4,8,2,5,1,5,0,3,0,0,0,4,0,0,0,0,0,0,0,1,
...

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

Cf. A000112.

A340318 Minimum size of a partial order that contains all partial orders of size n.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 16

Offset: 0

Caleb Stanford, Jan 04 2021

a(n) is the minimum number of elements in a poset P such that every poset of size n is isomorphic to a subset of P, where the subset inherits the order from P.
Elementary bounds are a(n) >= 2n-1 because it must contain a chain and an antichain, and a(n) <= 2^n-1 because every partial order embeds into the powerset partial order on n elements. It is shown in the MathOverflow link that a(n) has no polynomial upper bound. This is in particular derived from binomial(a(n),n) >= A000112(n).
a(4) = 8 verified using a computer-assisted proof with a SAT solver.
a(5) = 11 proven on MathOverflow.
a(6) = 16 and 16 <= a(7) <= 25 proven on MathOverflow. - Jukka Kohonen, Jan 15 2021

			a(2) = 3 because there are 2 nonisomorphic posets on two elements, and both embed into the poset of three elements {a, b, c} with ordering a < b (and other pairs are incomparable).
a(3) = 5 because all posets on three elements can be embedded into {a, b, c, d, e} with ordering a < d, b < c < d, and b < e.

Joel David Hamkins and Fedor Petrov, What is the minimal size of a partial order that is universal for all partial orders of size n?, MathOverflow.
Jukka Kohonen, What is the minimum size of a partial order containing all partial orders of size 5? (proofs of a(5)=11, a(6)=16 and 16 <= a(7) <= 25), MathOverflow.
Caleb Stanford, Alloy program to verify a(n) for small n, GitHub.

Cf. A000112, A001035, A004401, A097911.

# Find an u-poset that contains all n-posets as induced posets.
def find_universal_poset(n,u):
    PP = list(Posets(n))
    for U in Posets(u):
        ok = True
        for P in PP:
            if not U.has_isomorphic_subposet(P):
                ok = False
                break
        if ok:
            return U
    return None

a(6) from Jukka Kohonen, Jan 15 2021

A354693 Number of unlabeled prime posets with n elements.

Original entry on oeis.org

1, 0, 0, 1, 4, 28, 234, 2585, 36326, 646405, 14528011, 412212506

Offset: 1

Salah Uddin Mohammad, Jun 03 2022

A poset P is called prime if it is not decomposable. A poset Q is called decomposable if Q can be obtained as the composition (lexicographic product) of the outer poset Q' and the inner posets Qi, 1 <= i <= r, where |Q'| = r > 1 and at least one of the posets Qi is nonsingleton.

S. M. Khamis, On numerical counting of prime, UPO, and the general type of posets according to heights, Congressus Numerantium, 146 (2000), 157-171.
S. M. Khamis, Recognition of prime posets and one of its applications, J. Egypt. Math. Soc., 14 (1) (2006), 5-13.

Cf. A000112, A003430, A202182.

A363911 n! times the number of posets with n unlabeled elements.

Original entry on oeis.org

1, 1, 4, 30, 384, 7560, 228960, 10306800, 685399680, 66490865280, 9316160179200, 1866087527673600, 529244914160793600, 210621677079215001600, 116661392964364363315200, 89281569344544938769408000, 93799600948326479830880256000

Offset: 0

Geoffrey Critzer, Jun 27 2023

nonn

Let H be Green's H relation on the semigroup of binary relations on [n]. Then a(n) is the number of elements that are H-related to a poset.

There are A000112(n) D-classes containing the nonsingular relations. There are A001035(n) L-classes in these D-classes. Each such L-class contains exactly one idempotent relation (which is necessarily a poset).

Wikipedia, Green's relations

Cf. A003425, A000142, A001035, A000112.

nn = 10; A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt",
    "Table"], {, }][[All, 2]];Range[0, 16]! Table[A000112[[i]], {i, 1, 17}]

a(n) = A000142(n)*A000112(n).

A376894 Stationary differences in A342447: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) which does not depend on k if k>= 2n-2 (for n>0).

Original entry on oeis.org

1, 3, 14, 61, 273, 1228, 5631, 26141, 123261, 589251, 2855815, 14021038, 69707192

Offset: 1

Rico Zöllner and Konrad Handrich, Oct 22 2024

Number of unlabeled posets A342447(j,k) with j points, without isolated points, with k arcs in the Hasse diagramm missing n points to achieve saturation of the poset i.e. j=2k-n+1.
A342447 is the number of unlabeled posets of j points with k arcs in the Hasse diagram.
A342447(j,k)-A342447(j-1,k) = 0 if j > 2k.
For k >= 2n-2, A342447(2k-n+1,k)-A342447(2k-n,k) does not depend on k.
Therefore we define: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k).
A342447(2k-n,k) = A022016(k) - a(1)-...-a(n) for k >= 2n-2, n>0
Proof will soon be submitted to JOIS.

			See the table of A342447
 1 ;
 1 ;
 1 1 ;
 1 1 3 ;
 1 1 4  8  2 ;
 1 1 4 11 29  12   5 ;
 1 1 4 12 43 105  92   45   12    3 ;
 1 1 4 12 46 156 460  582  487  204   71   14   7 ;
 1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ;
 ...
The differences between row j and j-1 of column k (convergence indicated by | |):
 0 ;
 0 ;
 0 |1| ;
 0  0 |3| ;
 0  0 |1| 8    2 ;
 0  0  0 |3|  27    12     5 ;
 0  0  0 |1| |14|   93    87      45    12   ... ;
 0  0  0  0   |3|   51   368     537   475   ... ;
 0  0  0  0   |1|  |14|  210    1515  3335   ... ;
 0  0  0  0    0    |3|  |61|    857  6691   ... ;
 0  0  0  0    0    |1|  |14|    258  3683   ... ;
 0  0  0  0    0     0    |3|    |61| 1127   ... ;
 0  0  0  0    0     0    |1|    |14| |273|  ... ;
a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) for n>=1
e.g. for n = 2 -> k = 2n-2 = 2
a(2) = A342447(3,2) - A342447(2,2) = 3 - 0 = 3
for n = 3 -> k >= 2n-2 = 6
a(3) = A342447(10,6) - A342447(9,6) = 745 - 731 = 14

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

Cf. A000112, A022016.

Differences of A342447.

a(8)-a(13) from Konrad Handrich, Jan 07 2025

A379608 Number of unlabeled Riordan posets with n elements.

Original entry on oeis.org

1, 2, 5, 11, 33, 74, 144, 232, 639

Offset: 1

Salah Uddin Mohammad, Dec 27 2024

Posets associated to binary Riordan matrices are called Riordan posets.

			For example, all the posets up to 3 elements are Riordan posets.

G. S. Cheon et al., Riordan posets and associated incidence matrices, Linear Algebra and its Applications, 632 (2022), 308-331.

Cf. A000112, A000608, A349401

a(9) from Salah Uddin Mohammad, Jun 28 2025

A381121 Number of partially ordered sets ("posets") covering n unlabeled elements.

Original entry on oeis.org

1, 0, 1, 3, 11, 47, 255, 1727, 14954, 166232, 2384053, 44182143, 1058142319, 32718935706, 1304369332319, 66936884741385, 4414855587293931

Offset: 0

Peter Dolland, Feb 14 2025

			For n = 3 there are 5 posets, but 2 of them have at least 1 unrelated element, so there are 5 - 2 = 3  = a(3) posets without an unrelated element.

Partial differences of A000112.

a(0) = 1; a(n) = A000112(n) - A000112(n - 1) for n > 0.

A382829 Number of distinct rank vectors of distributive lattices of height n.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 197, 864, 4325, 24922

Offset: 0

Ludovic Schwob, Apr 06 2025

Distributive lattices are ranked posets, and we define the rank vector of a ranked poset P as the vector whose k-th coordinate (starting at k = 0) is the number of elements of rank k in P.

By Birkhoff's representation theorem, elements of a finite distributive lattice L are in bijection with lower sets of the poset of join-irreducible elements of L, an element of rank k corresponding to a lower of set size k.

			The rank vectors corresponding to a(4) = 15 are:
  (1, 1, 1, 1, 1),   (1, 1, 1, 2, 1),   (1, 1, 2, 1, 1),
  (1, 1, 2, 2, 1),   (1, 1, 3, 3, 1),   (1, 2, 1, 1, 1),
  (1, 2, 1, 2, 1),   (1, 2, 2, 1, 1),   (1, 2, 2, 2, 1),
  (1, 2, 3, 2, 1),   (1, 2, 3, 3, 1),   (1, 3, 3, 1, 1),
  (1, 3, 3, 2, 1),   (1, 3, 4, 3, 1),   (1, 4, 6, 4, 1).
Two non-isomorphic distributive lattices have for rank vector (1, 2, 2, 2, 1).

Cf. A000112, A006982.

Previous Showing 51-60 of 60 results.

cp's OEIS Frontend

A263861 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and whose order polytope has k facets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A263862 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and whose chain polytope has k facets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A263863 Triangle read by rows: T(n,k) (n>=0, k>=n+1) is the number of posets with n elements and k chains.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A340318 Minimum size of a partial order that contains all partial orders of size n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Extensions

A354693 Number of unlabeled prime posets with n elements.

Views

Author

Keywords

Comments

Links

Crossrefs

A363911 n! times the number of posets with n unlabeled elements.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A376894 Stationary differences in A342447: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) which does not depend on k if k>= 2n-2 (for n>0).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Extensions

A379608 Number of unlabeled Riordan posets with n elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A381121 Number of partially ordered sets ("posets") covering n unlabeled elements.

Views

Author

Keywords

Examples

Crossrefs

Formula

A382829 Number of distinct rank vectors of distributive lattices of height n.

Views