%I #6 Apr 25 2020 08:41:29
%S 1,1,2,4,12,43,229,1506,12392,120443
%N Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.
%C A necklace is a finite sequence of positive integers that is lexicographically strictly less than or equal to any cyclic rotation. Co-necklace is defined similarly, except with strictly greater instead of strictly less.
%e The a(1) = 1 through a(4) = 12 normal sequences:
%e (1) (1,1) (1,1,1) (1,1,1,1)
%e (2,1) (2,1,1) (2,1,1,1)
%e (2,2,1) (2,1,2,1)
%e (3,2,1) (2,2,1,1)
%e (2,2,2,1)
%e (3,1,2,1)
%e (3,2,1,1)
%e (3,2,2,1)
%e (3,2,3,1)
%e (3,3,2,1)
%e (4,2,3,1)
%e (4,3,2,1)
%t neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Length[Select[Join@@Permutations/@allnorm[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,8}]
%Y Dominates A334270 (the aperiodic case).
%Y Compositions of this type are counted by A334271.
%Y These compositions are ranked by A334273 (standard) and A334274 (binary).
%Y Binary (or reversed binary) necklaces are counted by A000031.
%Y Normal sequences are counted by A000670.
%Y Necklace compositions are counted by A008965.
%Y Normal Lyndon words are counted by A060223.
%Y Normal necklaces are counted by A019536.
%Y All of the following pertain to compositions in standard order (A066099):
%Y - Necklaces are A065609.
%Y - Reversed necklaces are A333943.
%Y - Co-necklaces are A333764.
%Y - Reversed co-necklaces are A328595.
%Y - Lyndon words are A275692.
%Y - Co-Lyndon words are A326774.
%Y - Reversed Lyndon words are A334265.
%Y - Reversed co-Lyndon words are A328596.
%Y - Reversed Lyndon co-Lyndon compositions are A334266.
%Y - Aperiodic compositions are A328594.
%Y Cf. A034691, A059966, A296372, A296975, A329138, A334269.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Apr 25 2020