A001929 - OEIS

A001929

Number of connected topologies on n labeled points.
(Formerly M3070 N1245)

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

Table of n, a(n) for n=0..18.

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.

N. J. A. Sloane, List of sequences related to partial orders, circa 1972

J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]

J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

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

Cf. A001928, A001930.

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

Sequence in context: A295812 A228229 A323875 * A349962 A230316 A157675

Adjacent sequences: A001926 A001927 A001928 * A001930 A001931 A001932

KEYWORD