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%I #39 Apr 16 2023 03:11:30
%S 0,0,1,5,31,239,2233,24725,318631,4707359,78691633,1471482725,
%T 30469552111,692488851599,17141242421353,459033875802485,
%U 13221994489388791,407574126219013439,13386292717807416673,466636446695213384645,17205919477720642772671,669019022588385113932079,27357684052927560953626393
%N Number of decomposable fixed-point free involutions, also the number of disconnected chord diagrams with 2n nodes on an open string.
%C Line up 2n distinguishable nodes sequentially on an open string. Connect each two nodes with only one chord, there will be a (2n-1)!! variety of chord diagrams. Amongst this variety, we can classify a diagram as disconnected when it is possible to find a node index 2s with all nodes <=2s in group A and the rest in group B where none of the chords connect nodes between group A and B.
%C The subsequence of primes begins 5, 31, 239, 4707359, 78691633, 17141242421353, no more through a(22). - _Jonathan Vos Post_, Jan 31 2011
%H Alois P. Heinz, <a href="/A177797/b177797.txt">Table of n, a(n) for n = 0..200</a>
%H A. King, <a href="http://dx.doi.org/10.1016/j.disc.2006.01.005">Generating indecomposable permutations</a>, Discrete Math., 306 (2006), 508-518.
%H F. Kuehnel, L. P. Pryadko, M. I. Dykman, <a href="http://dx.doi.org/10.1103/PhysRevB.63.165326">Single-electron magnetoconductivity of nondegenerate two-dimensional electron system in a quantizing magnetic field</a>, Phys. Rev. B Vol. 63, 16 (2001).
%H Frank Kuehnel, Leonid P. Pryadko and M. I. Dykman, <a href="http://arxiv.org/abs/cond-mat/0008416">Single electron magneto-conductivity of a nondegenerate 2D electron system in a quantizing magnetic field</a> (See diagrams on page 6), arXiv:cond-mat/0008416 [cond-mat.str-el], 2000.
%t (* derived from _Joerg Arndt_'s PARI code *)
%t f[n_] := f[n] = (2n-1)!!
%t s[n_] := s[n] = f[n] - Sum[f[k] s[n - k], {k, 1, n - 1}]
%t Table[f[k] - s[k], {k, 0, 22}]
%t (* original brute force method *)
%t GenerateDiagramsOfOrder[n_Integer /; n >= 0] := Diagrams[Range[2 n]]
%t Diagrams[pool_List] := Block[{n = Count[pool, _]}, If[n <= 2, {{pool}},
%t Flatten[Map[
%t Flatten[
%t Outer[Join, {{{pool[[1]], pool[[#]]}}},
%t Diagrams[
%t Function[{poolset, droppos},
%t Drop[poolset, {droppos}] // Rest][pool, #]], 1], 1] &,
%t Range[2, n]], 1]]]
%t SelectDisconnected[diagrams_List] := Select[diagrams, IsDisconnected]
%t IsDisconnected[{{}}] = False;
%t IsDisconnected[pairs_List] :=
%t Block[{newPairs=Map[#~Append~(#[[2]] - #[[1]]) &, pairs],
%t distanceList},
%t distanceList = Fold[
%t ReplacePart[#1, {#2[[1]] -> #2[[3]], #2[[2]] -> -#2[[3]]}] &,
%t Range[2 Length[pairs]],
%t newPairs];
%t Return[Length[Select[Drop[Accumulate[distanceList], -1], #<1 &]] > 0]
%t ]
%t Map[Length[SelectDisconnected[GenerateDiagramsOfOrder[#]]]&, Range[0,7]]
%o (PARI)
%o f(n)=(2*n)!/n!/2^n; \\ == (2n-1)!!
%o s(n)=f(n) - sum(k=1, n-1, f(k)*s(n-k) )
%o a(n)=f(n)-s(n)
%o \\ _Joerg Arndt_
%o (Python)
%o from functools import cache
%o def a(n):
%o @cache
%o def h(n):
%o if n <= 1: return 1
%o return h(n - 1) * (2 * n - 1)
%o @cache
%o def c(n):
%o if n == 0: return 1
%o return h(n) - sum(h(k) * c(n - k) for k in range(1, n))
%o return h(n) - c(n)
%o print([a(n) for n in range(19)]) # _Peter Luschny_, Apr 16 2023
%Y Chord Diagrams: A054499, A007769.
%Y Permutations: A001147, A000698, A003319. - _Joerg Arndt_
%Y Cf. A000637. - _Jonathan Vos Post_
%K nonn,easy
%O 0,4
%A _Frank Kuehnel_, Dec 27 2010
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