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A000112 Number of partially ordered sets ("posets") with n unlabeled elements.
(Formerly M1495 N0588)
1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231, 2567284, 46749427, 1104891746, 33823827452, 1338193159771, 68275077901156, 4483130665195087 (list; graph; refs; listen; history; text; internal format)



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


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.

Davison, J. L. 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E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.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, 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.

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'énumeration des posets, C. R. Acad. Sci. Paris 314 serie I (1992) 691-694. [See Chaunier letter]

S. R. Finch, Transitive relations, topologies and partial orders

R. Fraisse and N. Lygeros, Petits posets: dénombrement, représentabilité par cercles et compenseurs C. R. Acad. Sci. Paris, 313, I, 417-420, 1991.

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, 2013

Ann Marie Hess, Mixed Models Site

C. Joslyn, E. Hogan, A. Pogel, Conjugacy and Iteration of Standard Interval Rank in Finite Ordered Sets, arXiv preprint arXiv:1409.6684, 2014

Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550, 2012. - From N. J. A. Sloane, Nov 09 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 elements, 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

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, Classic Sequences

Index entries for sequences related to posets

Index entries for "core" sequences


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.


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

Cf. A079263, A079265.

Sequence in context: A111004 A079566 A059685 * A205498 A178119 A185998

Adjacent sequences:  A000109 A000110 A000111 * A000113 A000114 A000115




N. J. A. Sloane


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



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Last modified July 30 23:18 EDT 2015. Contains 260145 sequences.