A001929 - OEIS

%I M3070 N1245 #43 Aug 30 2018 15:45:01

%S 1,1,3,19,233,4851,158175,7724333,550898367,56536880923,8267519506789,

%T 1709320029453719,496139872875425839,200807248677750187825,

%U 112602879608997769049739,86955243134629606109442219,91962123875462441868790125305,132524871920295877733718959290203,259048612476248175744581063815546423

%N Number of connected topologies on 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 and  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 IV 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="http://dx.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>

%F a(n) = Sum_{k=0..n} Stirling2(n,k)*A001927(k). - _Vladeta Jovovic_, Apr 10 2006

%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 A001927 = CoefficientList[ A[x] + O[x]^(max-1), x]*Range[0, max-2]!;

%t a[n_] := Sum[StirlingS2[n, k] *A001927[[k+1]], {k, 0, n}];

%t Table[a[n], {n, 0, max -2}] (* _Jean-François Alcover_, Aug 30 2018, after _Vladeta Jovovic_ *)

%Y Cf. A001928, A001930.

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

%K nonn,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Apr 10 2006

%E a(17)-a(18) using data from A001035 from _Alois P. Heinz_, Aug 30 2018