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A001927 Number of connected partially ordered sets with n labeled points.
(Formerly M2043 N0809)
16
1, 1, 2, 12, 146, 3060, 101642, 5106612, 377403266, 40299722580, 6138497261882, 1320327172853172, 397571105288091506, 166330355795371103700, 96036130723851671469482, 76070282980382554147600692, 82226869197428315925408327266, 120722306604121583767045993825620, 239727397782668638856762574296226842 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
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).
LINKS
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.
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
FORMULA
E.g.f. A(x)=log(B(x)) where B(x) is e.g.f. of A001035.
MATHEMATICA
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 *)
CROSSREFS
Cf. A000112, A001035, A000608, A066303, A342501 (refined by rank).
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Sequence in context: A296137 A086928 A228551 * A289987 A105558 A322399
KEYWORD
nonn,nice,hard
AUTHOR
EXTENSIONS
More terms from Christian G. Bower, Dec 12 2001
a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)