Search: id:A001035
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%I A001035 M3068 N1244
%S A001035 1,1,3,19,219,4231,130023,6129859,431723379,44511042511,6611065248783,
%T A001035 1396281677105899,414864951055853499,171850728381587059351,
%U A001035 98484324257128207032183,77567171020440688353049939,83480529785490157813844256579,
               122152541250295322862941281269151,241939392597201176602897820148085023
%N A001035 Number of partially ordered sets ("posets") with n labeled elements (or 
               labeled acyclic transitive digraphs).
%D A001035 G. Birkhoff, Lattice Theory, Amer. Math. Soc., 1961, p. 4.
%D A001035 Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 Points, Order 
               19 (2002), 147-179.
%D A001035 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.
%D A001035 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.
%D A001035 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.
%D A001035 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 60, 229.
%D A001035 M. Erne, Struktur- und Anzahlformeln fuer Topologien auf endlichen Mengen, 
               PhD dissertation, Westfaelische Wilhelms-Universitaet zu Muenster, 
               1972.
%D A001035 M. Erne and K. Stege, Counting Finite Posets and Topologies, Order, 8 
               (1991), 247-265.
%D A001035 M. Erne and K. Stege, The number of labeled orders on fifteen elements, 
               personal communication.
%D A001035 J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of 
               finite topologies, Commun. ACM, 10 (1967), 295-297, 313.
%D A001035 J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen 
               elements, Order 17 (2000) no. 4, 333-341.
%D A001035 D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial 
               orders on a finite set, Trans. Amer. Math. Soc., 205 (1975) 205-220.
%D A001035 G. Kreweras, Denombrement des ordres etages, Discrete Math., 53 (1985), 
               147-149.
%D A001035 A. Shafaat, On the number of topologies definable for a finite set, J. 
               Austral. Math. Soc., 8 (1968), 194-198.
%D A001035 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001035 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001035 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, 
               pages 96ff; Vol. 2, Problem 5.39, p. 88.
%H A001035 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001035 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Transitive relations, 
               topologies and partial orders</a>
%H A001035 Institut f. Mathematik, Univ. Hanover, <a href="http://www-ifm.math.uni-hannover.de/
               html/preprints.phtml">Erne/Heitzig/Reinhold papers</a>
%H A001035 N. Lygeros and P. Zimmermann, <a href="http://www.lygeros.org/Math/poset.html">
               Computation of P(14), the number of posets with 14 elements: 1.338.193.159.771</
               a>
%H A001035 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting 
               Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), 
               Article 04.3.2.
%H A001035 Bob Proctor, <a href="http://www.unc.edu/~rap/Posets/">Chapel Hill Poset 
               Atlas</a>
%H A001035 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A001035 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/97/finite.top">
               Number of topologies</a>
%H A001035 N. J. A. Sloane, <a href="classic.html#LOSS">Classic Sequences</a>
%H A001035 <a href="Sindx_Pos.html#posets">Index entries for sequences related to 
               posets</a>
%F A001035 Related to A000798 by A000798(n) = Sum Stirling2(n, k)*A001035(k).
%F A001035 Related to A000112 by Erne's formulae: A001035(n+1)=-s(n, 1), A001035(n+2)=n*A001035(n+1)+s(n, 
               2), A001035(n+3)=binomial(n+4, 2)*A001035(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).
%e A001035 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 3, 
               page 98, Fig. 3-1 shows the unlabeled posets with <= 4 points.
%Y A001035 Cf. A000798 (labeled topologies), A001930 (unlabeled topologies), A000112 
               (unlabeled posets), A006057.
%Y A001035 Sequence in context: A005647 A158876 A001833 this_sequence A166380 A136652 
               A136504
%Y A001035 Adjacent sequences: A001032 A001033 A001034 this_sequence A001036 A001037 
               A001038
%K A001035 nonn,nice
%O A001035 0,3
%A A001035 N. J. A. Sloane (njas(AT)research.att.com).
%E A001035 Terms for n=15 and 16 from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), 
               Jul 03 2000
%E A001035 a(17) and a(18) from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 
               2008

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