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%I #12 Nov 15 2019 09:35:23
%S 1,2,2,3,2,3,3,4,2,3,2,4,3,4,4,5,2,3,2,4,3,3,2,5,3,4,3,5,4,5,5,6,2,3,
%T 2,4,2,3,2,5,3,4,2,4,3,3,2,6,3,4,3,5,4,4,3,6,4,5,4,6,5,6,6,7,2,3,2,4,
%U 2,3,2,5,3,3,2,4,3,3,2,6,3,4,2,5,4,3,2
%N Length of the co-Lyndon factorization of the reversed binary expansion of n.
%C First differs from A211100 at a(77) = 3, A211100(77) = 2. The reversed binary expansion of 77 is (1011001), with co-Lyndon factorization (10)(1100)(1), while the binary expansion is (1001101), with Lyndon factorization of (1)(001101).
%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 certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).
%e The reversed binary expansion of each positive integer together with their co-Lyndon factorizations begins:
%e 1: (1) = (1)
%e 2: (01) = (0)(1)
%e 3: (11) = (1)(1)
%e 4: (001) = (0)(0)(1)
%e 5: (101) = (10)(1)
%e 6: (011) = (0)(1)(1)
%e 7: (111) = (1)(1)(1)
%e 8: (0001) = (0)(0)(0)(1)
%e 9: (1001) = (100)(1)
%e 10: (0101) = (0)(10)(1)
%e 11: (1101) = (110)(1)
%e 12: (0011) = (0)(0)(1)(1)
%e 13: (1011) = (10)(1)(1)
%e 14: (0111) = (0)(1)(1)(1)
%e 15: (1111) = (1)(1)(1)(1)
%e 16: (00001) = (0)(0)(0)(0)(1)
%e 17: (10001) = (1000)(1)
%e 18: (01001) = (0)(100)(1)
%e 19: (11001) = (1100)(1)
%e 20: (00101) = (0)(0)(10)(1)
%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[Reverse[IntegerDigits[n,2]]]],{n,100}]
%Y The non-"co" version is A211100.
%Y Positions of 2's are A329357.
%Y Numbers whose binary expansion is co-Lyndon are A275692.
%Y Length of the co-Lyndon factorization of the binary expansion is A329312.
%Y Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A328596, A329131, A329314, A329318, A329324, A329325.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 11 2019
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