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%I M2043 N0809 #52 Mar 14 2021 11:41:27
%S 1,1,2,12,146,3060,101642,5106612,377403266,40299722580,6138497261882,
%T 1320327172853172,397571105288091506,166330355795371103700,
%U 96036130723851671469482,76070282980382554147600692,82226869197428315925408327266,120722306604121583767045993825620,239727397782668638856762574296226842
%N Number of connected partially ordered sets with n labeled points.
%D 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H C. M. Bender et al., <a href="http://arxiv.org/abs/quant-ph/0604164">Combinatorics and field theory</a>, arXiv:quant-ph/0604164, 2006.
%H G. Brinkmann, B. D. McKay, <a href="http://dx.doi.org/10.1023/A:1016543307592">Posets on up to 16 Points</a>, Order 19 (2) (2002) 147-179 (Table II, up to 18 points)
%H K. K.-H. Butler and G. Markowsky, <a href="http://www.laptop.maine.edu/Enumeration.pdf">Enumeration of finite topologies</a>, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
%H K. K.-H. Butler and G. Markowsky, <a href="/A000798/a000798_1.pdf">Enumeration of finite topologies</a>, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
%H M. Erné, <a href="https://doi.org/10.1007/BF01173716">Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259.
%H M. Erné, <a href="/A006056/a006056.pdf">Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen</a>, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
%H M. Erné and K. Stege, <a href="http://dx.doi.org/10.1007/BF00383446">Counting Finite Posets and Topologies</a>, Order, 8 (1991), 247-265.
%H N. J. A. Sloane, <a href="/A000112/a000112_2.pdf">List of sequences related to partial orders, circa 1972</a>
%H J. A. Wright, <a href="/A000798/a000798_3.pdf">There are 718 6-point topologies, quasiorderings and transgraphs</a>, Preprint, 1970 [Annotated scanned copy]
%H J. A. Wright, <a href="/A000798/a000798_4.pdf">Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences</a>
%H <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F E.g.f. A(x)=log(B(x)) where B(x) is e.g.f. of A001035.
%t A001035 = {1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023};
%t max = Length[A001035]-1;
%t B[x_] = Sum[A001035[[k+1]]*x^k/k!, {k, 0, max}];
%t A[x_] = 1 + Log[B[x]];
%t CoefficientList[A[x] + O[x]^(max-1), x]*Range[0, max-2]! (* _Jean-François Alcover_, Apr 17 2014, updated Aug 30 2018 *)
%Y Cf. A000112, A001035, A000608, A066303, A342501 (refined by rank).
%Y Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
%K nonn,nice,hard
%O 0,3
%A _N. J. A. Sloane_.
%E More terms from _Christian G. Bower_, Dec 12 2001
%E a(17)-a(18) using data from A001035 from _Alois P. Heinz_, Aug 30 2018
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