1,2

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).

A sequence of words is uniform if they all have the same length.

Conjecture: Also the number of compositions of n that are either weakly increasing or weakly decreasing. Hence a(n) = 2 * A000041(n) - A000005(n). - Gus Wiseman, Mar 05 2020

Table of n, a(n) for n=1..25.

The a(1) = 1 through a(6) = 18 compositions:

(1) (2) (3) (4) (5) (6)

(11) (12) (13) (14) (15)

(21) (22) (23) (24)

(111) (31) (32) (33)

(112) (41) (42)

(211) (113) (51)

(1111) (122) (114)

(221) (123)

(311) (222)

(1112) (321)

(2111) (411)

(11111) (1113)

(1122)

(2211)

(3111)

(11112)

(21111)

(111111)

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], SameQ@@Length/@lynfac[#]&&SameQ@@Length/@colynfac[#]&]], {n, 10}]

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

Lyndon compositions that are not weakly increasing are A329141.

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

Cf. A000740, A001037, A001523, A008965, A059204, A060223, A211100, A328596, A329312, A329318, A329396, A329397, A329399, A332578, A332669, A332834.

Sequence in context: A343657 A033500 A003318 * A353150 A035300 A035296

Adjacent sequences: A329395 A329396 A329397 * A329399 A329400 A329401

nonn,more

Gus Wiseman, Nov 13 2019

a(19)-a(25) from Robert Price, Jun 20 2021

approved