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A177797 Number of decomposable fixed-point free involutions, also the number of disconnected chord diagrams with 2n nodes on an open string. 1
0, 0, 1, 5, 31, 239, 2233, 24725, 318631, 4707359, 78691633, 1471482725, 30469552111, 692488851599, 17141242421353, 459033875802485, 13221994489388791, 407574126219013439, 13386292717807416673, 466636446695213384645, 17205919477720642772671, 669019022588385113932079, 27357684052927560953626393 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

The subsequence of primes begins 5, 31, 239, 4707359, 78691633, 17141242421353, no more through a(22). - Jonathan Vos Post, Jan 31 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

A. King, Generating indecomposable permutations, Discrete Math., 306 (2006), 508-518.

F. Kuehnel, L. P. Pryadko, M. I. Dykman, Single-electron magnetoconductivity of nondegenerate two-dimensional electron system in a quantizing magnetic field, Phys. Rev. B Vol. 63, 16 (2001).

Frank Kuehnel, Leonid P. Pryadko and M. I. Dykman, Single electron magneto-conductivity of a nondegenerate 2D electron system in a quantizing magnetic field (See diagrams on page 6), arXiv:cond-mat/0008416 [cond-mat.str-el], 2000.

MATHEMATICA

(* derived from Joerg Arndt's PARI code *)

f[n_] := f[n] = (2n-1)!!

s[n_] := s[n] = f[n] - Sum[f[k] s[n - k], {k, 1, n - 1}]

Table[f[k] - s[k], {k, 0, 22}]

(* original brute force method *)

GenerateDiagramsOfOrder[n_Integer /; n >= 0] := Diagrams[Range[2 n]]

Diagrams[pool_List] := Block[{n = Count[pool, _]}, If[n <= 2, {{pool}},

  Flatten[Map[

    Flatten[

      Outer[Join, {{{pool[[1]], pool[[#]]}}},

        Diagrams[

         Function[{poolset, droppos},

           Drop[poolset, {droppos}] // Rest][pool, #]], 1], 1] &,

     Range[2, n]], 1]]]

SelectDisconnected[diagrams_List] := Select[diagrams, IsDisconnected]

IsDisconnected[{{}}] = False;

IsDisconnected[pairs_List] :=

  Block[{newPairs=Map[#~Append~(#[[2]] - #[[1]]) &, pairs],

         distanceList},

    distanceList = Fold[

      ReplacePart[#1, {#2[[1]] -> #2[[3]], #2[[2]] -> -#2[[3]]}] &,

      Range[2 Length[pairs]],

      newPairs];

    Return[Length[Select[Drop[Accumulate[distanceList], -1], #<1 &]] > 0]

  ]

Map[Length[SelectDisconnected[GenerateDiagramsOfOrder[#]]]&, Range[0, 7]]

PROG

(PARI)

f(n)=(2*n)!/n!/2^n;  \\ == (2n-1)!!

s(n)=f(n) - sum(k=1, n-1, f(k)*s(n-k) )

a(n)=f(n)-s(n)

\\ Joerg Arndt

CROSSREFS

Chord Diagrams: A054499, A007769.

Permutations: A001147, A000698, A003319. - Joerg Arndt

Cf. A000637. - Jonathan Vos Post

Sequence in context: A213048 A069321 A211179 * A186859 A082579 A261498

Adjacent sequences:  A177794 A177795 A177796 * A177798 A177799 A177800

KEYWORD

nonn,easy

AUTHOR

Frank Kuehnel, Dec 27 2010

STATUS

approved

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Last modified March 25 13:37 EDT 2017. Contains 284081 sequences.