cp's OEIS Frontend

A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

%I A334269 #10 Apr 26 2020 09:10:45
%S A334269 1,1,2,3,6,8,16,23,40,62,110,169,302,492,856,1454,2572,4428,7914,
%T A334269 13935,25036,44842,81298,147149,268952,491746,904594,1667091,3085950,
%U A334269 5723367,10652544,19865887,37150314,69608939,130723184,245935633,463590444,875306913,1655451592,3135613649,5948011978,11298215516
%N A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.
%C A334269 Also the number of compositions of n that are both a Lyndon word and a reversed co-Lyndon word.
%C A334269 A composition of n is a finite sequence of positive integers summing to n.
%C A334269 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 A334269 The a(1) = 1 through a(7) = 16 compositions:
%e A334269   (1)  (2)  (3)   (4)    (5)     (6)      (7)
%e A334269             (21)  (31)   (32)    (42)     (43)
%e A334269                   (211)  (41)    (51)     (52)
%e A334269                          (221)   (321)    (61)
%e A334269                          (311)   (411)    (322)
%e A334269                          (2111)  (2211)   (331)
%e A334269                                  (3111)   (421)
%e A334269                                  (21111)  (511)
%e A334269                                           (2221)
%e A334269                                           (3121)
%e A334269                                           (3211)
%e A334269                                           (4111)
%e A334269                                           (21211)
%e A334269                                           (22111)
%e A334269                                           (31111)
%e A334269                                           (211111)
%t A334269 lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
%t A334269 colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A334269 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[Reverse[#]]&&colynQ[#]&]],{n,0,15}]
%Y A334269 The version for binary expansion is A334267.
%Y A334269 Compositions of this type are ranked by A334266.
%Y A334269 Normal sequences of this type are counted by A334270.
%Y A334269 Necklace compositions of this type are counted by A334271.
%Y A334269 Aperiodic compositions are counted by A000740.
%Y A334269 Binary Lyndon words are counted by A001037.
%Y A334269 Necklace compositions are counted by A008965.
%Y A334269 Normal Lyndon words are counted by A060223.
%Y A334269 Lyndon compositions are counted by A059966.
%Y A334269 All of the following pertain to compositions in standard order (A066099):
%Y A334269 - Lyndon words are A275692.
%Y A334269 - Co-Lyndon words are A326774.
%Y A334269 - Reversed Lyndon words are A334265.
%Y A334269 - Reversed co-Lyndon words are A328596.
%Y A334269 - Length of Lyndon factorization is A329312.
%Y A334269 - Length of co-Lyndon factorization is A334029.
%Y A334269 - Length of Lyndon factorization of reverse is A334297.
%Y A334269 - Length of co-Lyndon factorization of reverse is A329313.
%Y A334269 - Lyndon factorizations are counted by A333940.
%Y A334269 - Co-Lyndon factorizations are counted by A333765.
%Y A334269 - Aperiodic compositions are A328594.
%Y A334269 - Distinct rotations are counted by A333632.
%Y A334269 Cf. A034691, A065609, A275692, A328596, A329141, A329324, A329326, A334266, A334272, A334273, A334274.
%K A334269 nonn
%O A334269 1,3
%A A334269 _Gus Wiseman_, Apr 24 2020
%E A334269 Offset corrected and a(21)-a(42) from _Bert Dobbelaere_, Apr 26 2020