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A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of graded reverse-lexicographic, 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.

- Combinatory separations are A334030.

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Binary necklaces are counted by A000031.

Necklace compositions are counted by A008965.

Necklaces covering an initial interval are counted by A019536.

allocated for Gus WisemanNumber of co-Lyndon factorizations of the k-th composition in standard order.

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 2, 4, 4, 7, 7, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 5, 2, 5, 2, 4, 4, 9, 4, 7, 7, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 4, 1

0,4

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 factorization of a composition c is a multiset of compositions whose co-Lyndon product is c.

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.

Also the number of multiset partitions of the co-Lyndon-word factorization of the n-th composition in standard order.

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A034691(n).

The a(54) = 5, a(61) = 7, and a(237) = 9 factorizations:

((1,2,1,2)) ((1,1,1,2,1)) ((1,1,2,1,2,1))

((1),(2,1,2)) ((1),(1,1,2,1)) ((1),(1,2,1,2,1))

((1,2),(2,1)) ((1,1),(1,2,1)) ((1,1),(2,1,2,1))

((2),(1,2,1)) ((2,1),(1,1,1)) ((1,2,1),(1,2,1))

((1),(2),(2,1)) ((1),(1),(1,2,1)) ((2,1),(1,1,2,1))

((1),(1,1),(2,1)) ((1),(1),(2,1,2,1))

((1),(1),(1),(2,1)) ((1,1),(2,1),(2,1))

((1),(2,1),(1,2,1))

((1),(1),(2,1),(2,1))

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

colynprod[]:={}; colynprod[{}, b_List]:=b; colynprod[a_List, {}]:=a; colynprod[a_List]:=a;

colynprod[{x_, a___}, {y_, b___}]:=Switch[Ordering[If[x=!=y, {x, y}, {colynprod[{a}, {x, b}], colynprod[{x, a}, {b}]}]], {1, 2}, Prepend[colynprod[{a}, {y, b}], x], {2, 1}, Prepend[colynprod[{x, a}, {b}], y]];

colynprod[a_List, b_List, c__List]:=colynprod[a, colynprod[b, c]];

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

dealings[q_]:=Union[Function[ptn, Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];

Table[Length[Select[dealings[stc[n]], colynprod@@#==stc[n]&]], {n, 0, 100}]

The dual version is A333940.

Binary necklaces are A000031.

Necklace compositions are A008965.

Necklaces covering an initial interval are A019536.

Lyndon compositions are counted by A059966.

Numbers whose reversed binary expansion is a necklace are A328595.

Numbers whose prime signature is a necklace are A329138.

Length of Lyndon factorization of binary expansion is A211100.

Length of co-Lyndon factorization of binary expansion is A329312.

Length of co-Lyndon factorization of reversed binary expansion is A329326.

Length of Lyndon factorization of reversed binary expansion is A329313.

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.

- Length of Lyndon factorization is A329312.

- Rotational period is A333632.

- Co-necklaces are A333764.

- Dealings are counted by A333939.

- Reversed necklaces are A333943.

- Length of co-Lyndon factorization is A334029.

Cf. A000740, A001037, A034691, A060223, A211100, A269134, A292884, A328596.

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Gus Wiseman, Apr 13 2020

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