cp's OEIS Frontend

A334272 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.

%I A334272 #6 Apr 25 2020 08:41:29
%S A334272 1,1,2,4,12,43,229,1506,12392,120443
%N A334272 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 A334272 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 A334272 The a(1) = 1 through a(4) = 12 normal sequences:
%e A334272   (1)  (1,1)  (1,1,1)  (1,1,1,1)
%e A334272        (2,1)  (2,1,1)  (2,1,1,1)
%e A334272               (2,2,1)  (2,1,2,1)
%e A334272               (3,2,1)  (2,2,1,1)
%e A334272                        (2,2,2,1)
%e A334272                        (3,1,2,1)
%e A334272                        (3,2,1,1)
%e A334272                        (3,2,2,1)
%e A334272                        (3,2,3,1)
%e A334272                        (3,3,2,1)
%e A334272                        (4,2,3,1)
%e A334272                        (4,3,2,1)
%t A334272 neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A334272 coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
%t A334272 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A334272 Table[Length[Select[Join@@Permutations/@allnorm[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,8}]
%Y A334272 Dominates A334270 (the aperiodic case).
%Y A334272 Compositions of this type are counted by A334271.
%Y A334272 These compositions are ranked by A334273 (standard) and A334274 (binary).
%Y A334272 Binary (or reversed binary) necklaces are counted by A000031.
%Y A334272 Normal sequences are counted by A000670.
%Y A334272 Necklace compositions are counted by A008965.
%Y A334272 Normal Lyndon words are counted by A060223.
%Y A334272 Normal necklaces are counted by A019536.
%Y A334272 All of the following pertain to compositions in standard order (A066099):
%Y A334272 - Necklaces are A065609.
%Y A334272 - Reversed necklaces are A333943.
%Y A334272 - Co-necklaces are A333764.
%Y A334272 - Reversed co-necklaces are A328595.
%Y A334272 - Lyndon words are A275692.
%Y A334272 - Co-Lyndon words are A326774.
%Y A334272 - Reversed Lyndon words are A334265.
%Y A334272 - Reversed co-Lyndon words are A328596.
%Y A334272 - Reversed Lyndon co-Lyndon compositions are A334266.
%Y A334272 - Aperiodic compositions are A328594.
%Y A334272 Cf. A034691, A059966, A296372, A296975, A329138, A334269.
%K A334272 nonn,more
%O A334272 0,3
%A A334272 _Gus Wiseman_, Apr 25 2020