cp's OEIS Frontend

A334270 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed Lyndon word and a co-Lyndon word.

%I A334270 #5 Apr 25 2020 08:41:12
%S A334270 1,1,1,3,10,42,224,1505,12380,120439
%N A334270 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed Lyndon word and a co-Lyndon word.
%C A334270 Also the number of sequences of length n that cover an initial interval of positive integers and are both a Lyndon word and a reversed co-Lyndon word.
%C A334270 A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations. Co-Lyndon is defined similarly, except with strictly greater instead of strictly less.
%e A334270 The a(1) = 1 through a(4) = 10 normal sequences:
%e A334270   (1)  (2,1)  (2,1,1)  (2,1,1,1)
%e A334270               (2,2,1)  (2,2,1,1)
%e A334270               (3,2,1)  (2,2,2,1)
%e A334270                        (3,1,2,1)
%e A334270                        (3,2,1,1)
%e A334270                        (3,2,2,1)
%e A334270                        (3,2,3,1)
%e A334270                        (3,3,2,1)
%e A334270                        (4,2,3,1)
%e A334270                        (4,3,2,1)
%t A334270 lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
%t A334270 colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A334270 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A334270 Table[Length[Select[Join@@Permutations/@allnorm[n],lynQ[Reverse[#]]&&colynQ[#]&]],{n,0,6}]
%Y A334270 These compositions are ranked by A334266 (standard) and A334267 (binary).
%Y A334270 Compositions of this type are counted by A334269.
%Y A334270 Necklace compositions of this type are counted by A334271.
%Y A334270 Dominated by A334272 (the necklace version).
%Y A334270 Normal sequences are counted by A000670.
%Y A334270 Binary (or reversed binary) Lyndon words are counted by A001037.
%Y A334270 Lyndon compositions are counted by A059966.
%Y A334270 Normal Lyndon words are counted by A060223.
%Y A334270 Normal sequences by length and Lyndon factorization length are A296372.
%Y A334270 All of the following pertain to compositions in standard order (A066099):
%Y A334270 - Lyndon words are A275692.
%Y A334270 - Co-Lyndon words are A326774.
%Y A334270 - Reversed Lyndon words are A334265.
%Y A334270 - Reversed co-Lyndon words are A328596.
%Y A334270 - Length of Lyndon factorization is A329312.
%Y A334270 - Length of co-Lyndon factorization is A334029.
%Y A334270 - Length of Lyndon factorization of reverse is A334297.
%Y A334270 - Length of co-Lyndon factorization of reverse is A329313.
%Y A334270 Cf. A008965, A034691, A329324, A329398, A334273.
%K A334270 nonn,more
%O A334270 0,4
%A A334270 _Gus Wiseman_, Apr 24 2020