A329312 - OEIS

A329312

Length of the co-Lyndon factorization of the binary expansion of n.

%I #13 Nov 15 2019 09:34:37

%S 1,1,2,1,2,1,3,1,2,2,3,1,2,1,4,1,2,2,3,1,3,2,4,1,2,1,3,1,2,1,5,1,2,2,

%T 3,2,3,2,4,1,2,3,4,2,3,2,5,1,2,1,3,1,2,2,4,1,2,1,3,1,2,1,6,1,2,2,3,2,

%U 3,2,4,1,3,3,4,2,3,2,5,1,2,2,3,1,4,3

%N Length of the co-Lyndon factorization of the binary expansion of n.

%C 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).

%C Also the length of the Lyndon factorization of the inverted binary expansion of n, where the inverted digits are 1 minus the binary digits.

%e The binary indices of 1..20 together with their co-Lyndon factorizations are:

%e 1: (1) = (1)

%e 2: (10) = (10)

%e 3: (11) = (1)(1)

%e 4: (100) = (100)

%e 5: (101) = (10)(1)

%e 6: (110) = (110)

%e 7: (111) = (1)(1)(1)

%e 8: (1000) = (1000)

%e 9: (1001) = (100)(1)

%e 10: (1010) = (10)(10)

%e 11: (1011) = (10)(1)(1)

%e 12: (1100) = (1100)

%e 13: (1101) = (110)(1)

%e 14: (1110) = (1110)

%e 15: (1111) = (1)(1)(1)(1)

%e 16: (10000) = (10000)

%e 17: (10001) = (1000)(1)

%e 18: (10010) = (100)(10)

%e 19: (10011) = (100)(1)(1)

%e 20: (10100) = (10100)

%t colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];

%t colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];

%t Table[Length[colynfac[IntegerDigits[n,2]]],{n,100}]

%Y The non-"co" version is A211100.

%Y Positions of 1's are A275692.

%Y The reversed version is A329326.

%Y Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A329131, A329314, A329318, A329324, A329325.

%K nonn

%O 1,3

%A _Gus Wiseman_, Nov 10 2019