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

1, 1, 2, 3, 5, 7, 12, 17, 28, 43, 70, 111, 184, 303, 510, 865, 1482, 2573, 4480, 7915, 14008

Offset: 0

Author

Gus Wiseman, Apr 25 2020

Keywords

• nonn
• more

Comments

Also the number of compositions of n that are both a necklace and a reversed co-necklace.

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.

Examples

			The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (21)   (22)    (32)     (33)
             (111)  (31)    (41)     (42)
                    (211)   (221)    (51)
                    (1111)  (311)    (222)
                            (2111)   (321)
                            (11111)  (411)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (21111)
                                     (111111)
		

Crossrefs

Normal sequences of this type are counted by A334272.

The aperiodic case is A334269.

These compositions are ranked by A334273.

Binary (or reversed binary) necklaces are counted by A000031.

Normal sequences are counted by A000670.

Necklace compositions are counted by A008965.

Lyndon compositions are counted by A059966.

Normal Lyndon words are counted by A060223.

Normal necklaces are counted by A019536.

Normal aperiodic words are counted by A296975.

All of the following pertain to compositions in standard order (A066099):

- Necklaces are A065609.

- Reversed necklaces are A333943.

- Co-necklaces are A333764.

- Reversed co-necklaces are A328595.

- Lyndon words are A275692.

- Co-Lyndon words are A326774.

- Reversed Lyndon words are A334265.

- Reversed co-Lyndon words are A328596.

- Aperiodic compositions are A328594.

Cf. A000740, A001037, A034691, A329138, A329324, A334266, A334267, A334274.

Programs

Mathematica

neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,15}]