A000608 -id:A000608 - cp's OEIS frontend

A174118 Partial sums of A000608.

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.

A000112 Number of partially ordered sets ("posets") with n unlabeled elements.

Original entry on oeis.org

1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231, 2567284, 46749427, 1104891746, 33823827452, 1338193159771, 68275077901156, 4483130665195087

Offset: 0

N. J. A. Sloane

Also number of fixed effects ANOVA models with n factors, which may be both crossed and nested.

			R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 (or 2nd. ed., Fig. 3.1, p. 243) shows the unlabeled posets with <= 4 points.
From _Gus Wiseman_, Aug 14 2019: (Start)
Also the number of unlabeled T_0 topologies with n points. For example, non-isomorphic representatives of the a(4) = 16 topologies are:
  {}{1}{12}{123}{1234}
  {}{1}{2}{12}{123}{1234}
  {}{1}{12}{13}{123}{1234}
  {}{1}{12}{123}{124}{1234}
  {}{1}{2}{12}{13}{123}{1234}
  {}{1}{2}{12}{123}{124}{1234}
  {}{1}{12}{13}{123}{124}{1234}
  {}{1}{2}{12}{13}{123}{124}{1234}
  {}{1}{2}{12}{13}{123}{134}{1234}
  {}{1}{2}{3}{12}{13}{23}{123}{1234}
  {}{1}{2}{12}{13}{24}{123}{124}{1234}
  {}{1}{12}{13}{14}{123}{124}{134}{1234}
  {}{1}{2}{3}{12}{13}{23}{123}{124}{1234}
  {}{1}{2}{12}{13}{14}{123}{124}{134}{1234}
  {}{1}{2}{3}{12}{13}{14}{23}{123}{124}{134}{1234}
  {}{1}{2}{3}{4}{12}{13}{14}{23}{24}{34}{123}{124}{134}{234}{1234}
(End)

G. Birkhoff, Lattice Theory, 1961, p. 4.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.
J. L. Davison, Asymptotic enumeration of partial orders. Proceedings of the seventeenth Southeastern international conference on combinatorics, graph theory, and computing (Boca Raton, Fla., 1986). Congr. Numer. 53 (1986), 277--286. MR0885256 (88c:06001)
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, pages 96ff; Vol. I, 2nd. ed., Chap. 3, pp. 241ff; Vol. 2, Problem 5.39, p. 88.
For further references concerning the enumeration of topologies and posets see under A001035.

David Wasserman, Table of n, a(n) for n = 0..16
R. Bayon, N. Lygeros, and J.-S. Sereni, New progress in enumeration of mixed models, Applied Mathematics E-Notes, 5 (2005), 60-65.
R. Bayon, N. Lygeros, and J.-S. Sereni, Nouveaux progrès dans l'énumération des modèles mixtes, in Knowledge discovery and discrete mathematics: JIM'2003, INRIA, Université de Metz, France, 2003, pp. 243-246.
Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points [On Brendan McKay's home page]
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
Kim Ki-Hang Butler, The number of partially ordered sets, Journal of Combinatorial Theory, Series B 13.3 (1972): 276-289.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
Kim Ki-Hang Butler and Gaoacs Markowsky. The number of partially ordered sets. II., J. Korean Math. Soc 11 (1974): 7-17.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C. Chaunier, Letter to N. J. A. Sloane, Jun 22 1993, with several attachments
C. Chaunier and N. Lygeros, The Number of Orders with Thirteen Elements, Order 9:3 (1992) 203-204. [See Chaunier letter]
C. Chaunier and N. Lygeros, Le nombre de posets à isomorphie près ayant 12 éléments Theoretical Computer Science, 123 p. 89-94, 1994.
C. Chaunier and N. Lygeros, Progrès dans l'énumération des posets, C. R. Acad. Sci. Paris 314 série I (1992) 691-694. [See Chaunier letter]
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date [Annotated scanned copy]
Gábor Czédli, Minimum-sized generating sets of the direct powers of free distributive lattices, arXiv:2309.13783 [math.CO], 2023. See p. 14. See also CUBO, A Mathematical Journal, Vol. 26, no. 2, pp. 217-237, August 2024.
M. Erné and K. Stege, The number of partially ordered (labeled) sets, Preprint, 1989. (Annotated scanned copy)
Uli Fahrenberg, Christian Johansen, Georg Struth, and Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
FindStat - Combinatorial Statistic Finder, Posets
R. Fraisse and N. Lygeros, Petits posets: dénombrement, représentabilité par cercles et compenseurs C. R. Acad. Sci. Paris, 313, I, 417-420, 1991.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
G. Grekos, Letter to N. J. A. Sloane, Oct 31 1994, with attachments
M. Guay-Paquet, A modular relation for the chromatic symmetric functions of (3+1)-free posets, arXiv preprint arXiv:1306.2400 [math.CO], 2013.
Ann Marie Hess, Mixed Models Site
C. Joslyn, E. Hogan, and A. Pogel, Conjugacy and Iteration of Standard Interval Rank in Finite Ordered Sets, arXiv preprint arXiv:1409.6684 [math.CO], 2014.
Dongseok Kim, Young Soo Kwon, and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550 [math.CO], 2012.
D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
N. Lygeros, Calculs exhaustifs sur les posets d'au plus 7 éléments, SINGULARITE, vol. 2 n4 p. 10-24, avril 1991.
N. Lygeros and P. Zimmermann, Computation of P(14), the number of posets with 14 elements: 1.338.193.159.771
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Bob Proctor, Chapel Hill Poset Atlas
D. Rusin, Further information and references [Broken link]
D. Rusin, Further information and references [Cached copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Szilárd Szalay, The classification of multipartite quantum correlation, arXiv:1806.04392 [quant-ph], 2018.
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Two related abstracts, 1970 and 1972 [Annotated scanned copies]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Stav Zalel, Covariant Growth Dynamics, arXiv:2302.10582 [gr-qc], 2023.
Index entries for sequences related to posets
Index entries for "core" sequences

Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A006057.
Cf. A079263, A079265, A065066 (refined by maximal elements), A342447 (refined by number of arcs).
Row sums of A263859. Euler transform of A000608.
Cf. A316978, A319559, A326939, A326943, A326944, A326947.

a(15)-a(16) are from Brinkmann's and McKay's paper. - Vladeta Jovovic, Jan 04 2006

A001927 Number of connected partially ordered sets with n labeled points.

Original entry on oeis.org

1, 1, 2, 12, 146, 3060, 101642, 5106612, 377403266, 40299722580, 6138497261882, 1320327172853172, 397571105288091506, 166330355795371103700, 96036130723851671469482, 76070282980382554147600692, 82226869197428315925408327266, 120722306604121583767045993825620, 239727397782668638856762574296226842

Offset: 0

N. J. A. Sloane

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table II, up to 18 points)
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Cf. A000112, A001035, A000608, A066303, A342501 (refined by rank).

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

A001035 = {1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023};
max = Length[A001035]-1;
B[x_] = Sum[A001035[[k+1]]*x^k/k!, {k, 0, max}];
A[x_] = 1 + Log[B[x]];
CoefficientList[A[x] + O[x]^(max-1), x]*Range[0, max-2]! (* Jean-François Alcover, Apr 17 2014, updated Aug 30 2018 *)

E.g.f. A(x)=log(B(x)) where B(x) is e.g.f. of A001035.

More terms from Christian G. Bower, Dec 12 2001

a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018

A263864 Triangle read by rows: T(n,k) (n>=1, k>=1) is the number of posets with n elements whose Hasse diagram has k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 10, 4, 1, 1, 44, 13, 4, 1, 1, 238, 60, 14, 4, 1, 1, 1650, 312, 63, 14, 4, 1, 1, 14512, 2075, 328, 64, 14, 4, 1, 1, 163341, 17316, 2159, 331, 64, 14, 4, 1, 1, 2360719, 186173, 17801, 2175, 332, 64, 14, 4, 1, 1, 43944974, 2594568, 189406, 17885, 2178, 332, 64, 14, 4, 1, 1, 1055019099, 47041877

Offset: 1

Christian Stump, Oct 28 2015

Multiset transformation of A000608.

			Triangle begins:
   1;
   1,  1;
   3,  1,  1;
  10,  4,  1,  1;
  44, 13,  4,  1,  1;
  ...

Alois P. Heinz, Rows n = 1..16, flattened
FindStat - Combinatorial Statistic Finder, The number of connected components of the Hasse diagram for the poset.
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.
Wikipedia, Hasse diagram
Index entries for sequences related to posets

Cf. A000112 (row sums), A000608 (k=1).

More terms from R. J. Mathar, Jul 12 2020

A342500 T(n,k) is the number of connected unlabeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 10, 24, 9, 1, 0, 27, 123, 73, 14, 1, 0, 88, 734, 638, 169, 20, 1, 0, 328, 5184, 6460, 2178, 334, 27, 1, 0, 1460, 44518, 78385, 32468, 5880, 594, 35, 1, 0, 7799, 472859, 1164966, 581533, 118933, 13605, 979, 44, 1

Offset: 1

R. J. Mathar, Mar 14 2021

This is a variant of A263859 admitting only connected posets.

			The table starts in row n=1 shows ranks k>=0:
1: 1
2: 0 1
3: 0 2 1
4: 0 4 5 1
5: 0 10 24 9 1
6: 0 27 123 73 14 1
7: 0 88 734 638 169 20 1
8: 0 328 5184 6460 2178 334 27 1
9: 0 1460 44518 78385 32468 5880 594 35 1
10: 0 7799 472859 1164966 581533 118933 13605 979 44 1

Index entries for sequences related to posets
R. J. Mathar, Poset Illustrations

Cf. A000608 (row sums), A007776 (rank 1), A263859, A000096 (subdiagonal), A342501 (labeled).

T(n,0) = 0 for k>0; due to the connectivity constraint.

T(n,n-1) = 1; the poset with elements in a single chain.

A022017 Number of connected partially ordered sets with n "lines": pairs (a,b) where a < b and there is no c with a < c < b. The lines form the minimal basis for the partial ordering.

Original entry on oeis.org

1, 3, 8, 29, 103, 442, 1953, 9502, 48533, 262634, 1485764, 8777397, 53869119

Offset: 1

Franklin T. Adams-Watters, Mar 30 2002

The points are unlabeled.

			See Sloane's link.

See A000112 for references and links about partially ordered sets.

N. J. A. Sloane, Illustration of initial terms of A022016 and A022017.
Index entries for sequences related to posets

Cf. A000112, A000608, A022016. Column sums of A342590.

a(6)-a(9) from A342590. - R. J. Mathar, Mar 21 2021

a(10)-a(13) from Rico Zöllner and Konrad Handrich, Nov 19 2024

A342590 T(n,k) is the number of connected posets of n unlabeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 0, 8, 2, 0, 0, 0, 0, 27, 12, 5, 0, 0, 0, 0, 0, 91, 87, 45, 12, 3, 0, 0, 0, 0, 0, 0, 350, 532, 475, 201, 71, 14, 7, 0, 0, 0, 0, 0, 0, 0, 1376, 3272, 4298, 3197, 1565, 554, 186, 44, 16, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5743, 19396, 36664, 41706, 31931, 16972

Offset: 1

R. J. Mathar, Mar 16 2021

			The table starts
1: 1
2: 0 1
3: 0 0 3
4: 0 0 0 8 2
5: 0 0 0 0 27 12   5
6: 0 0 0 0 0  91  87   45   12    3
7: 0 0 0 0 0   0 350  532  475  201   71   14   7
8: 0 0 0 0 0   0   0 1376 3272 4298 3197 1565 554 186 44 16 4

R. J. Mathar, Table of T(n,k) for n = 1..10
Index entries for sequences related to posets

Cf. A000608 (row sums), A022017 (column sums), A342447 (not necess. connected), A342588 (labeled).

A361955 Number of unlabeled connected weakly graded (ranked) posets with n elements.

Original entry on oeis.org

1, 1, 1, 3, 10, 42, 202, 1146, 7493, 56996, 508609, 5414635, 70214227, 1134439731, 23331152887, 621768153861, 21761221300058, 1009759125475973, 62534859409597022, 5193886959561972984, 580677490292990902682, 87649885799470898359728, 17907726747155924589913398

Offset: 0

Andrew Howroyd, Mar 31 2023

nonn

Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.

Andrew Howroyd, Table of n, a(n) for n = 0..40
Wikipedia, Graded poset.

Row sums of A361954.

Cf. A000608, A361920, A361953.

\\ See PARI link in A361953 for program code.
A361955seq(20)

Inverse Euler transform of A361920.

A066305 Number of connected reduced partially ordered sets (posets) with n unlabeled elements.

Original entry on oeis.org

1, 0, 2, 5, 22, 117

Offset: 1

Christian G. Bower, Dec 12 2001

N. J. A. Sloane, Transforms
Index entries for sequences related to posets

Cf. A000608. Inverse Euler transform of A066304.

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).

cp's OEIS Frontend

A174118 Partial sums of A000608.

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A000112 Number of partially ordered sets ("posets") with n unlabeled elements.

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A001927 Number of connected partially ordered sets with n labeled points.

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A263864 Triangle read by rows: T(n,k) (n>=1, k>=1) is the number of posets with n elements whose Hasse diagram has k connected components.

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A342500 T(n,k) is the number of connected unlabeled posets with n elements and rank k: triangle read by rows.

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A022017 Number of connected partially ordered sets with n "lines": pairs (a,b) where a < b and there is no c with a < c < b. The lines form the minimal basis for the partial ordering.

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A342590 T(n,k) is the number of connected posets of n unlabeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.

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A361955 Number of unlabeled connected weakly graded (ranked) posets with n elements.

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Programs

PARI

Formula

A066305 Number of connected reduced partially ordered sets (posets) with n unlabeled elements.

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A349401 Number of unlabeled disconnected posets with n elements.