A334029 Length of the co-Lyndon factorization of the k-th composition in standard order.

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 5, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2

Offset: 0

Author

Gus Wiseman, Apr 14 2020

Keywords

• nonn

Comments

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). 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, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The 441st composition in standard order is (1,2,1,1,3,1), with co-Lyndon factorization {(1),(3,1),(2,1,1)}, so a(441) = 3.
		

Crossrefs

The dual version is A329312.

The version for binary expansion is (also) A329312.

The version for reversed binary expansion is A329326.

Binary Lyndon/co-Lyndon words are counted by A001037.

Necklaces covering an initial interval are A019536.

Lyndon/co-Lyndon compositions are counted by A059966

Length of Lyndon factorization of binomial expansion is A211100.

Numbers whose prime signature is a necklace are A329138.

Length of Lyndon factorization of reversed binary expansion is A329313.

A list of all binary co-Lyndon words is A329318.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Necklaces are A065609.

- Sum is A070939.

- Runs are counted by A124767.

- Rotational symmetries are counted by A138904.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Lyndon compositions are A275692.

- Co-Lyndon compositions are A326774.

- Aperiodic compositions are A328594.

- Reversed co-necklaces are A328595.

- Rotational period is A333632.

- Co-necklaces are A333764.

- Co-Lyndon factorizations are counted by A333765.

- Lyndon factorizations are counted by A333940.

- Reversed necklaces are A333943.

- Co-necklaces are A334028.

Cf. A034691, A060223, A102659, A211097, A292884, A296372, A328596, A329358, A329359, A329362, A329400, A329401, A333939.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#1],q}]=={RotateRight[q,#1],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,#1]]&]]]]
Table[Length[colynfac[stc[n]]],{n,0,100}]