cp's OEIS Frontend

A329326 Length of the co-Lyndon factorization of the reversed binary expansion of n.

%I A329326 #12 Nov 15 2019 09:35:23
%S A329326 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 A329326 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 A329326 2,3,2,5,3,3,2,4,3,3,2,6,3,4,2,5,4,3,2
%N A329326 Length of the co-Lyndon factorization of the reversed binary expansion of n.
%C A329326 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 A329326 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 A329326 The reversed binary expansion of each positive integer together with their co-Lyndon factorizations begins:
%e A329326    1:     (1) = (1)
%e A329326    2:    (01) = (0)(1)
%e A329326    3:    (11) = (1)(1)
%e A329326    4:   (001) = (0)(0)(1)
%e A329326    5:   (101) = (10)(1)
%e A329326    6:   (011) = (0)(1)(1)
%e A329326    7:   (111) = (1)(1)(1)
%e A329326    8:  (0001) = (0)(0)(0)(1)
%e A329326    9:  (1001) = (100)(1)
%e A329326   10:  (0101) = (0)(10)(1)
%e A329326   11:  (1101) = (110)(1)
%e A329326   12:  (0011) = (0)(0)(1)(1)
%e A329326   13:  (1011) = (10)(1)(1)
%e A329326   14:  (0111) = (0)(1)(1)(1)
%e A329326   15:  (1111) = (1)(1)(1)(1)
%e A329326   16: (00001) = (0)(0)(0)(0)(1)
%e A329326   17: (10001) = (1000)(1)
%e A329326   18: (01001) = (0)(100)(1)
%e A329326   19: (11001) = (1100)(1)
%e A329326   20: (00101) = (0)(0)(10)(1)
%t A329326 colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A329326 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 A329326 Table[Length[colynfac[Reverse[IntegerDigits[n,2]]]],{n,100}]
%Y A329326 The non-"co" version is A211100.
%Y A329326 Positions of 2's are A329357.
%Y A329326 Numbers whose binary expansion is co-Lyndon are A275692.
%Y A329326 Length of the co-Lyndon factorization of the binary expansion is A329312.
%Y A329326 Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A328596, A329131, A329314, A329318, A329324, A329325.
%K A329326 nonn
%O A329326 1,2
%A A329326 _Gus Wiseman_, Nov 11 2019