A001929 Number of connected topologies on n labeled points.

1, 1, 3, 19, 233, 4851, 158175, 7724333, 550898367, 56536880923, 8267519506789, 1709320029453719, 496139872875425839, 200807248677750187825, 112602879608997769049739, 86955243134629606109442219, 91962123875462441868790125305, 132524871920295877733718959290203, 259048612476248175744581063815546423

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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.
• 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

Crossrefs

Cf. A001928, A001930.

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

Programs

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 *)

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k)*A001927(k). - Vladeta Jovovic, Apr 10 2006

Extensions

More terms from Vladeta Jovovic, Apr 10 2006

a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018