|
%I #6 Nov 12 2019 04:19:44
%S 1,2,21,211,221,2111,2211,2221,21111,21211,22111,22121,22211,22221,
%T 211111,212111,221111,221121,221211,222111,222121,222211,222221,
%U 2111111,2112111,2121111,2121211,2211111,2211121,2211211,2212111,2212121,2212211,2221111,2221121
%N List of co-Lyndon words on {1,2} sorted first by length and then lexicographically.
%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).
%t colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t Join@@Table[FromDigits/@Select[Tuples[{1,2},n],colynQ],{n,5}]
%Y The non-"co" version is A102659.
%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, A027375, A059966, A060223, A211100, A328596, A329324.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 11 2019
|
