cp's OEIS Frontend

A334271 Number of compositions of n that are both a reversed necklace and a co-necklace.

%I A334271 #5 Apr 25 2020 08:41:20
%S A334271 1,1,2,3,5,7,12,17,28,43,70,111,184,303,510,865,1482,2573,4480,7915,
%T A334271 14008
%N A334271 Number of compositions of n that are both a reversed necklace and a co-necklace.
%C A334271 Also the number of compositions of n that are both a necklace and a reversed co-necklace.
%C A334271 A necklace is a finite sequence of positive integers that is lexicographically less than or equal to any cyclic rotation. Co-necklace is defined similarly, except with greater instead of less.
%e A334271 The a(1) = 1 through a(6) = 12 compositions:
%e A334271   (1)  (2)   (3)    (4)     (5)      (6)
%e A334271        (11)  (21)   (22)    (32)     (33)
%e A334271              (111)  (31)    (41)     (42)
%e A334271                     (211)   (221)    (51)
%e A334271                     (1111)  (311)    (222)
%e A334271                             (2111)   (321)
%e A334271                             (11111)  (411)
%e A334271                                      (2121)
%e A334271                                      (2211)
%e A334271                                      (3111)
%e A334271                                      (21111)
%e A334271                                      (111111)
%t A334271 neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A334271 coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
%t A334271 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,15}]
%Y A334271 Normal sequences of this type are counted by A334272.
%Y A334271 The aperiodic case is A334269.
%Y A334271 These compositions are ranked by A334273.
%Y A334271 Binary (or reversed binary) necklaces are counted by A000031.
%Y A334271 Normal sequences are counted by A000670.
%Y A334271 Necklace compositions are counted by A008965.
%Y A334271 Lyndon compositions are counted by A059966.
%Y A334271 Normal Lyndon words are counted by A060223.
%Y A334271 Normal necklaces are counted by A019536.
%Y A334271 Normal aperiodic words are counted by A296975.
%Y A334271 All of the following pertain to compositions in standard order (A066099):
%Y A334271 - Necklaces are A065609.
%Y A334271 - Reversed necklaces are A333943.
%Y A334271 - Co-necklaces are A333764.
%Y A334271 - Reversed co-necklaces are A328595.
%Y A334271 - Lyndon words are A275692.
%Y A334271 - Co-Lyndon words are A326774.
%Y A334271 - Reversed Lyndon words are A334265.
%Y A334271 - Reversed co-Lyndon words are A328596.
%Y A334271 - Aperiodic compositions are A328594.
%Y A334271 Cf. A000740, A001037, A034691, A329138, A329324, A334266, A334267, A334274.
%K A334271 nonn,more
%O A334271 0,3
%A A334271 _Gus Wiseman_, Apr 25 2020