A306386 Number of chord diagrams with n chords all having arc length at least 3.

1, 0, 0, 1, 7, 68, 837, 11863, 189503, 3377341, 66564396, 1439304777, 33902511983, 864514417843, 23735220814661, 698226455579492, 21914096529153695, 731009183350476805, 25829581529376423945, 963786767538027630275, 37871891147795243899204, 1563295398737378236910447

Offset

0,5

Comments

A cyclical form of A190823.

Also the number of 2-uniform set partitions of {1...2n} such that, when the vertices are arranged uniformly around a circle, no block has its two vertices separated by an arc length of less than 3.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..404

Gus Wiseman, The a(5) = 68 chord diagrams with all arc lengths at least 3.

Formula

a(n) is even <=> n in { A135042 }. - Alois P. Heinz, Feb 27 2019

Example

The a(8) = 7 2-uniform set partitions with all arc lengths at least 3:
  {{1,4},{2,6},{3,7},{5,8}}
  {{1,4},{2,7},{3,6},{5,8}}
  {{1,5},{2,6},{3,7},{4,8}}
  {{1,5},{2,6},{3,8},{4,7}}
  {{1,5},{2,7},{3,6},{4,8}}
  {{1,6},{2,5},{3,7},{4,8}}
  {{1,6},{2,5},{3,8},{4,7}}

Maple

a:= proc(n) option remember; `if`(n<8, [1, 0$2, 1, 7, 68, 837, 11863][n+1],
      ((8*n^4-64*n^3+142*n^2-66*n+109)    *a(n-1)
      -(24*n^4-248*n^3+870*n^2-1106*n+241)*a(n-2)
      +(24*n^4-264*n^3+982*n^2-1270*n+145)*a(n-3)
      -(8*n^4-96*n^3+374*n^2-486*n+33)    *a(n-4)
      -(4*n^3-24*n^2+39*n-2)              *a(n-5))/(4*n^3-36*n^2+99*n-69))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 27 2019

Mathematica

dtui[{}, _]:={{}}; dtui[set:{i_, ___}, n_]:=Join@@Function[s, Prepend[#, s]&/@dtui[Complement[set, s], n]]/@Table[{i, j}, {j, Switch[i, 1, Select[set, 3<#<n-1&], 2, Select[set, 4<#<n&], _, Select[set, #>i+2&]]}];
Table[Length[dtui[Range[n], n]], {n, 0, 12, 2}]

Crossrefs

Cf. A000296, A000699, A001006, A001147, A001610, A003436, A038041, A054726, A135042, A170941, A190823, A278990, A306419, A322402, A324011, A324169.

Column k=3 of A324428.

Sequence in context: A087567 A328046 A371392 * A136629 A197525 A133697

Adjacent sequences: A306383 A306384 A306385 * A306387 A306388 A306389

Keyword

nonn

Author

Gus Wiseman, Feb 26 2019

Extensions

a(10)-a(16) from Alois P. Heinz, Feb 26 2019

a(17)-a(21) from Alois P. Heinz, Feb 27 2019

Status

approved