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A272135 Numbers of ways of placing the numbers 1, ..., n on a circle (not counting rotations and reflections) such that for each s in {1, ..., n(n+1)/2}, there exists a connected subset S of the circle such that the numbers covered by S add up to s.

%I A272135 #16 Aug 12 2022 19:23:35
%S A272135 1,1,1,1,2,10,41,126,537,3956,19776,76340,388047,2775155,15013424,
%T A272135 54188455,272147013
%N A272135 Numbers of ways of placing the numbers 1, ..., n on a circle (not counting rotations and reflections) such that for each s in {1, ..., n(n+1)/2}, there exists a connected subset S of the circle such that the numbers covered by S add up to s.
%H A272135 Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/1603906/">Placing the integers {1,2,...,n} on a circle (for n>1) in some special order</a>
%e A272135 Out of the 3 essentially different arrangements (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4) of four points on a circle, only (1, 2, 3, 4) and (1, 3, 2, 4) yield all sums in {1, ..., 10}, so a(4) = 2.
%t A272135 ok[w_] := Block[{v = Join[w,w], n = Length@w}, n(n+1)/2 == Length@ Union@ Flatten@ Table[ Total@ Take[v, {i, i+k}], {i,n}, {k, 0, n-1}]]; a[n_] := If[n<3, 1, Sum[ Length@ Select[ Permutations@ Complement[ Range@n, e], ok@ Join[e, #] &], {e, Flatten[ Table[{a,1,b}, {a,2,n}, {b,a+1,n}], 1]}]]; a /@ Range[0, 9] (* _Giovanni Resta_, Apr 21 2016 *)
%K A272135 nonn,more
%O A272135 0,5
%A A272135 _Jens Voß_, Apr 21 2016
%E A272135 a(15) from _Giovanni Resta_, Apr 21 2016
%E A272135 a(16) from _Giovanni Resta_, Apr 22 2016