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A001929 Number of connected topologies on n labeled points.
(Formerly M3070 N1245)
13
1, 1, 3, 19, 233, 4851, 158175, 7724333, 550898367, 56536880923, 8267519506789, 1709320029453719, 496139872875425839, 200807248677750187825, 112602879608997769049739, 86955243134629606109442219, 91962123875462441868790125305, 132524871920295877733718959290203, 259048612476248175744581063815546423 (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 and  B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179, Table IV 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
a(n) = Sum_{k=0..n} Stirling2(n,k)*A001927(k). - Vladeta Jovovic, Apr 10 2006
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]];
A001927 = CoefficientList[ A[x] + O[x]^(max-1), x]*Range[0, max-2]!;
a[n_] := Sum[StirlingS2[n, k] *A001927[[k+1]], {k, 0, n}];
Table[a[n], {n, 0, max -2}] (* Jean-François Alcover, Aug 30 2018, after Vladeta Jovovic *)
CROSSREFS
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Sequence in context: A295812 A228229 A323875 * A349962 A230316 A157675
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 10 2006
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 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)