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A306386
Number of chord diagrams with n chords all having arc length at least 3.
6
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.
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}]
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