OFFSET
1,2
COMMENTS
The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).
A sequence of words is uniform if they all have the same length.
EXAMPLE
The sequence of terms together with their co-Lyndon factorizations begins:
1: (1) = (1)
2: (10) = (10)
3: (11) = (1)(1)
4: (100) = (100)
6: (110) = (110)
7: (111) = (1)(1)(1)
8: (1000) = (1000)
10: (1010) = (10)(10)
12: (1100) = (1100)
14: (1110) = (1110)
15: (1111) = (1)(1)(1)(1)
16: (10000) = (10000)
20: (10100) = (10100)
24: (11000) = (11000)
26: (11010) = (11010)
28: (11100) = (11100)
30: (11110) = (11110)
31: (11111) = (1)(1)(1)(1)(1)
32: (100000) = (100000)
36: (100100) = (100)(100)
38: (100110) = (100)(110)
40: (101000) = (101000)
42: (101010) = (10)(10)(10)
MATHEMATICA
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, #]]&]]];
Select[Range[100], SameQ@@Length/@colynfac[IntegerDigits[#, 2]]&]
CROSSREFS
Numbers whose binary expansion has uniform Lyndon factorization are A023758.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 13 2019
STATUS
approved