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 A001035 Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs). (Formerly M3068 N1244) 43
 1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Altug Alkan, Dec 22 2015: (Start) a(p^k) == 1 mod p and a(n + p) == a(n + 1) mod p for all primes p. a(0+19) == a(0+1) mod 19 or a(19^1) == 1 mod 19, that is, a(19) mod 19 = 1. a(2+17) == a(2+1) mod 17. So a(19) == 19 mod 17, that is, a(19) mod 17 = 2. a(6+13) == a(6+1) mod 13. So a(19) == 6129859 mod 13, that is, a(19) mod 13 = 8. a(8+11) == a(8+1) mod 11. So a(19) == 44511042511 mod 11, that is, a(19) mod 11 = 1. a(12+7) == a(12+1) mod 7. So a(19) == 171850728381587059351 mod 7, that is, a(19) mod 7 = 1. a(14+5) == a(14+1) mod 5. So a(19) == 77567171020440688353049939 mod 5, that is, a(19) mod 5 = 4. a(16+3) == a(16+1) mod 3. So a(19) == 122152541250295322862941281269151 mod 3, that is, a(19) mod 3 = 1. a(17+2) == a(17+1) mod 2. So a(19) mod 2 = 1. In conclusion, a(19) is a number of the form 2*3*5*7*11*13*17*19*n - 1615151, that is, 9699690*n - 1615151. Additionally, for n > 0, note that the last digit of a(n) has the simple periodic pattern: 1,3,9,9,1,3,9,9,1,3,9,9,... (End) Number of rank n sublattices of the Boolean algebra B_n. - Kevin Long, Nov 20 2018 REFERENCES G. Birkhoff, Lattice Theory, Amer. Math. Soc., 1961, p. 4. Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 427. Butler, Kim Ki-Hang. "The number of partially ordered sets." Journal of Combinatorial Theory, Series B 13.3 (1972): 276-289. K. K.-H. Butler, A Moore-Penrose inverse for Boolean relation matrices, pp. 18-28 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974. 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. Butler, K. K. H., and G. Markowsky. "The number of partially ordered sets. I." Journal of Korean Mathematical Society 11.1 (1974). Butler, Kim Ki-Hang, and Gaoacs Markowsky. "The number of partially ordered sets. II." J. Korean Math. Soc 11 (1974): 7-17. S. D. Chatterji, The number of topologies on n points, Manuscript, 1966. L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 60, 229. M. Erné, Struktur- und Anzahlformeln für Topologien auf endlichen Mengen, PhD dissertation, Westfälische Wilhelms-Universität zu Münster, 1972. M. Erné and K. Stege, The number of labeled orders on fifteen elements, personal communication. 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. 2, Problem 5.39, p. 88. LINKS Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017. G. Brinkmann,  B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39. 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] P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. S. D. Chatterji, The number of topologies on n points, Manuscript, 1966 [Annotated scanned copy] Narendrakumar R. Dasre, Pritam Gujarathi, Approximating the Bounds for Number of Partially Ordered Sets with n Labeled Elements, Computing in Engineering and Technology, Advances in Intelligent Systems and Computing, Vol. 1025, Springer (Singapore 2019), 349-356. 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, The number of partially ordered (labeled) sets, Preprint, 1989. (Annotated scanned copy) M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265. J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313. J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313. [Annotated scanned copy] S. R. Finch, Transitive relations, topologies and partial orders S. R. Finch, Transitive relations, topologies and partialorders, June 5, 2003. [Cached copy, with permission of the author] Joël Gay, Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018. J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341. Dongseok Kim, Young Soo Kwon, Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550 [math.CO], 2012. - From N. J. A. Sloane, Nov 09 2012 M. Y. Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015. D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220. G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149. Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers 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 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. Rosenberg, Ivo; The number of maximal closed classes in the set of functions over a finite domain, J. Combinatorial Theory Ser. A 14 (1973), 1-7. Ivo Rosenberg and N. J. A. Sloane, Correspondence, 1971 D. Rusin, Further information and references [Broken link] D. Rusin, Further information and references [Cached copy] A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198. N. J. A. Sloane, List of sequences related to partial orders, circa 1972 N. J. A. Sloane, List of sequences related to partial orders, circa 1972 N. J. A. Sloane, Classic Sequences Gus Wiseman, Hasse diagrams of the a(4) = 219 posets. J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy] FORMULA A000798(n) = Sum Stirling2(n, k)*a(k). Related to A000112 by Erné's formulas: a(n+1)=-s(n, 1), a(n+2)=n*a(n+1)+s(n, 2), a(n+3)=binomial(n+4, 2)*a(n+2)-s(n, 3), where s(n, k)=sum(binomial(n+k-1-m, k-1)*binomial(n+k, m)*sum((m!)/(number of automorphisms of P)*(-(number of antichains of P))^k, P an unlabeled poset with m elements), m=0..n). From Altug Alkan, Dec 22 2015: (Start) a(p^k) == 1 mod p for all primes p and for all nonnegative integers k. a(n + p) == a(n + 1) mod p for all primes p and for all nonnegative integers n. If n = 1, then a(1 + p) == a(2) mod p, that is, a(p + 1) == 3 mod p. If n = p, then a(p + p) == a(p + 1) mod p, that is, a(2*p) == a(p + 1) mod p. In conclusion, a(2*p) == 3 mod p for all primes p. (End) EXAMPLE R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points. From Gus Wiseman, Aug 14 2019: (Start) Also the number of T_0 topologies with n points. For example, the a(0) = 1 through a(3) = 19 topologies are:   {}  {}{1}  {}{1}{12}     {}{1}{12}{123}              {}{2}{12}     {}{1}{13}{123}              {}{1}{2}{12}  {}{2}{12}{123}                            {}{2}{23}{123}                            {}{3}{13}{123}                            {}{3}{23}{123}                            {}{1}{2}{12}{123}                            {}{1}{3}{13}{123}                            {}{2}{3}{23}{123}                            {}{1}{12}{13}{123}                            {}{2}{12}{23}{123}                            {}{3}{13}{23}{123}                            {}{1}{2}{12}{13}{123}                            {}{1}{2}{12}{23}{123}                            {}{1}{3}{12}{13}{123}                            {}{1}{3}{13}{23}{123}                            {}{2}{3}{12}{23}{123}                            {}{2}{3}{13}{23}{123}                            {}{1}{2}{3}{12}{13}{23}{123} (End) MATHEMATICA dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}]; Table[Length[Select[Subsets[Subsets[Range[n]]], MemberQ[#, {}]&&MemberQ[#, Range[n]]&&UnsameQ@@dual[#]&&SubsetQ[#, Union@@@Tuples[#, 2]]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Aug 14 2019 *) CROSSREFS Cf. A000798 (labeled topologies), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057. Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110. Cf. A316978, A319564, A326876, A326906, A326939, A326943, A326944, A326947. Sequence in context: A005647 A158876 A001833 * A267634 A277407 A271587 Adjacent sequences:  A001032 A001033 A001034 * A001036 A001037 A001038 KEYWORD nonn,nice AUTHOR EXTENSIONS a(15)-a(16) from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000 a(17)-a(18) from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008 STATUS approved

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Last modified October 26 14:30 EDT 2020. Contains 338027 sequences. (Running on oeis4.)