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%I #21 Apr 30 2014 01:32:53
%S 1,1,4,19,112,811,7024,70939,818752,10630891,153371344,2433948859,
%T 42137351392,790287522571,15962014455664,345424786466779,
%U 7973482022972032,195556150543703851,5078301994885267984
%N Number of topologies on n labeled elements in which no element belongs to any pair of noncomparable members of the topology.
%C The number of topologies on n labeled elements is a fundamental sequence (A000798), which many mathematicians believe is impossible to completely determine.
%C The present sequence is an elegant recursion that enumerates the topologies on n labeled elements that can be "drawn" (as, for example, on page 76 of Munkres) in such a way that the boundaries of the subsets do not "cross" one another. Thus I recommend that topologies be classified as "planar" if their members can be drawn without crossings and "non-planar" otherwise.
%C This is analogous to the way in which subgroup lattices are called planar or non-planar. Using this terminology, the above sequence gives the number of planar topologies on n labeled elements. If the number of non-planar topologies on n labeled elements (see A122836) could be enumerated, then so could the total number of topologies on n labeled elements.
%C Another way to state the definition is that any two members of the topology are comparable or disjoint. - Rainer Rosenthal, Jan 02 2011
%C Conjectural closed form for n>0: 3*2^(k-3)(LerchPhi[1/4, -k, 1/2] + 2 PolyLog[-k, 1/4]) - 1/2. - _Vladimir Reshetnikov_, Jan 07 2011
%D J. Munkres, Topology, Prentice Hall, (2000), p. 76.
%F a(n) = 2^(n-1) - 1 + Sum{C(n,k)*a(n-k), k = 1 ... n}
%F E.g.f.: (3/4) / (1 - exp(x)/2) - exp(x)/2. - _Michael Somos_, Jan 07 2011
%F a(n) = (A000629(n) + 0^n) * (3/4) - 1/2. - _Michael Somos_, Jan 07 2011
%p a122835:=proc(n) option remember; if n=0 then 1 else 2^(n-1) - 1 + add(a122835(n-k)*binomial(n,k),k=1..n); fi; end;
%t a[n_]:=a[n]=2^(n-1)-1+Sum[a[n-k]*Binomial[n,k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,25}]
%t a[ n_] := (3/4) * (PolyLog[ -n, 1/2] + Boole[n==0]) - 1/2 (* _Michael Somos_, Jan 07 2011 *)
%o (PARI) {a(n) = local(A); if( n<1, n==0, A = exp(x + x * O(x^n)) / 2; n! * polcoeff( (3/4) / (1 - A) - A, n))} /* _Michael Somos_, Jan 07 2011 */
%Y Cf. A000798, A122836.
%K easy,nice,nonn
%O 0,3
%A Nathan K. McGregor (mcgregnk(AT)ese.wustl.edu), Sep 15 2006
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