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The k-th composition in standard order (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.

The k-th composition in standard order (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.

The k-th composition in standard order (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.

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).

A sequence of words is uniform if they all have the same length.

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

A sequence of words is uniform if they all have the same length.

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).

A co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations.

Every positive integer occurs exactly once, so that, as a sequence, a(n) is a permutation of the positive integers.

Descending from the root node 1, generate tree by outer composition of L(n) = n + F(Finv(n)) and R(n) = n + F(Finv(n) + 1), respectively, according to left or right branching, where F(n) = A000045(n) are the Fibonacci numbers and Finv(n) = A130233(n) is the 'lower' Fibonacci inverse. This produces each number by maximal Fibonacci expansion (cf. example below of Method 2, entry A343152, and links).

(Level of tree): The number of terms in this expansion of n is the level of the tree on which n appears, A112310(n-1) + 1 = A200648(n+1). The number of terms in the expansion of a(n) is floor(log_2(n)) + 1 = A113473(n) = A070939(n) = A029837(n+1).

"Maximal Fibonacci expansion" maximizes the sum of coefficients over all Fibonacci numbers (of positive index), allowing both F(1) = 1 and F(2) = 1. Thus, it is just an expansion and not a representation (like "greedy" and "lazy"), as it "breaks the rule" by using two bits that correspond to elements of equal value, rather than using distinct basis elements (link). This reveals connections to the cf. sequences: Binary strings that emerge in lexicographic order from "maximal Fibonacci gaps" (example), binary trees of the positive integers, and I-D arrays "harvested" from the trees. To define the expansion uniquely, always include F(1), so that the expansion of positive integer n equals F(1) for n = 1 and F(1) prepended to the lazy Fibonacci representation of n-1 for n > 1. Hence, a(1) = 1, and for n > 1, a(n) = A095903(n-1) + 1. The "redundant" expansion arranges the positive integers in the single binary tree {T(n,k)}, rather than the two trees at A255773 and A255774 that result from representation (see link).

(Left-to-right order in tree): Each F(t)-sized block (F(t+1), ..., F(t+2) - 1) of successive positive integers ("Fibonacci cohort" t) appears in right-to-left order in the tree as reordered in A343152, where elements of each cohort appear consecutively (see link).

Descending from the root node 1, generate tree by the inner composition of A026351 and A026352, that is, one plus the sequences of lower and upper Wythoff numbers, A000201 and A001950, respectively, according to left or right branching (see example below of Method 1 and links).

Generate tree from (one plus) the number of (initial) zeros on the positive integers for the outer composition of sequences, A060143 and A060144, respectively, according to left or right branching descending from the identity (c.f example below of Method 3 and links).

The lower Wythoff numbers, A000201, appear exclusively in the 1st, 3rd, 5th, ... right clades of the tree, while the upper Wythoff numbers A001950, appear exclusively in the 2nd, 4th, 6th, ... right clades of the tree. Here, the k-th right clade comprises the nodes at positions 2^(k+1) and 2^k + 1, together with all descendants of the latter (link).

(Duality with tree A232560, and related arrays): Consider the labeled binary trees a(n) = A232560(A059893(n)) and A232560(n) = a(A059893(n)). Labels along maximal straight paths that always branch left in a(n) give rows of array A345252, while labels along maximal straight paths that always branch left in A232560 give rows of array A083047.

Sorting the labels from each successive right clade of the binary tree a(n) gives the successive columns of A083047, while sorting labels from each successive right clade of A232560 gives each successive column of A345252. This makes the trees a(n) and A232560 "blade-duals," blade being a contraction of branch-clade (see entry for A345254 and link). A200648(n)+1 gives the level of the tree on which elements of array first-columns A345252(n,1) and A083047(n,1) appear.

(Palindromes and coincidence of elements): Trees a(n) and A232560 coincide when the sequence of left and right branching is a palindrome: a(A329395(n)) = A232560(A329395(n)). As Kimberling notes (cf. A059893), this happens at fixed points of A059893(n) or, equivalently, at n for which A081242(n) is a palindrome.

The inverse permutation of a(n) as a sequence can be read from a "tetrangle" or "irregular triangle" tableau with F(t) (Fibonacci number) entries on each row t, for t = 1, 2, 3, ..., in which an entry on row t is 2*x the entry x immediately above it on row t-1, if such exists, or otherwise 2*x + 1 the entry x in the corresponding position on row t-2 (thus generating new rows as in A243571 but without sorting the numbers into increasing order, linked reference):

1,

2,

3, 4,

5, 6, 8,

7, 9, 10, 12, 16,

11, 13, 17, 14, 18, 20, 24, 32,

...

With the right-justified tableau substituted by a left-justified tableau, the same procedure yields the inverse permutation for the "minimal Fibonacci tree," A048680(A059893(n)), the "cohort-dual" tree of a(n), where "cohort" t is the F(t)-sized block of successive entries in the tableau (see entry for A345252, linked reference).

(Coincidence of elements): a(A020988(n)) = A048680(A059893(A020988(n))) = A099919(n) and a(A020989(n)) = A048680(A059893(A020989(n))) = A049651(n). Collectively, a(A061547(n)) = A048680(A059893(A061547(n))) = union(A049651(n), A099919(n)).

With two types of duality, the tree forms a quartet of binary-tree arrangements of the positive integers, together with its blade dual A232560, its cohort dual A048680(A059893), and blade dual A048680 of the latter.

Order in the tree is "memory-less": Let a(n) and a(m) label nodes at positions n and m, respectively. Let d1 and d2 be two descending paths, i.e., sequences branching left or right from a starting node. (Nodal positions for the left and right children of the node at position p are given by 2*p and 2*p + 1, resp., and d1 and d2 are compositions of these.) Then a(d1(n)) < a(d2(n)) if and only if a(d1(m)) < a(d2(m)) (linked reference).

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

The k-th composition in standard order (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.