A329394 Number of compositions of n whose Lyndon and co-Lyndon factorizations both have the same length.

1, 2, 2, 4, 4, 10, 13, 28, 46, 99, 175, 359, 672, 1358, 2627, 5238, 10262, 20438, 40320, 80137

Offset: 1

Author

Gus Wiseman, Nov 13 2019

Keywords

• nonn
• more

Comments

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).

Examples

			The a(1) = 1 through a(7) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (131)    (33)      (151)
                    (121)   (212)    (141)     (214)
                    (1111)  (11111)  (213)     (232)
                                     (222)     (241)
                                     (231)     (313)
                                     (1221)    (1312)
                                     (2112)    (1321)
                                     (11211)   (2113)
                                     (111111)  (11311)
                                               (12121)
                                               (21112)
                                               (1111111)
		

Crossrefs

Lyndon and co-Lyndon compositions are (both) counted by A059966.

Lyndon compositions that are not weakly increasing are A329141.

Lyndon compositions of n whose reverse is not co-Lyndon are A329324.

Cf. A000740, A001037, A008965, A060223, A102659, A211100, A275692, A328596, A329312, A329318, A329395, A329398.

Programs

Mathematica

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[lynfac[#]]==Length[colynfac[#]]&]],{n,10}]