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If the signature-permutation of a Catalan automorphism SP satisfies the condition A129593(SP(n)) = A129593(n) for all n, then it is called a Łukasiewicz-word permuting automorphism. In addition to all the automorphisms whose signature permutation satisfies the more restricted condition A127301(SP(n)) = A127301(n) for all n, this includes also certain automorphisms like *A072797 that do not preserve the non-oriented form of the general tree. A000041(n) distinct values occur in each range [A014137(n-1)..A014138(n-1)]. All natural numbers occur. Cf. A129599.

a(n) is the binary expansion of A014486(n). - Joerg Arndt, Feb 27 2013

Replacing "1" by "(" and "0" by ")" yields well-formed bracket expressions (the first term being the empty string)

, (), ()(), (()), ()()(), ()(()), (())(), (()()), ((())), ()()()(), ()()(()), ()(())(), ()(()()), ()((())), (())()(), (())(()), (()())(), (()()()), (()(())), ((()))(), ((())()), ((()())), (((()))), ()()()()(), ()()()(()), ()()(())(), ()()(()()), ()()((())), ()(())()(), ()(())(()), ()(()())(), ()(()()()), ()(()(())), ()((()))(), ()((())()), ()((()())), ()(((()))), (())()()(), (())()(()), (())(())(), (())(()()), (())((())), (()())()(), (()())(()), (()()())(), (()()()()), (()()(())), (()(()))(), (()(())()), (()(()())), (()((()))), ((()))()(), ((()))(()), ((())())(), ((())()()), ((())(())), ((()()))(), ((()())()), ((()()())), ((()(()))), (((())))(), (((()))()), (((())())), (((()()))), ((((()))))

The term a(0)=0 stands for the empty string. - Joerg Arndt, Feb 27 2013

This sequence is produced when the function 'length' (present in programming languages like Lisp, Scheme, Prolog and Haskell) acts on symbolless S-expressions encoded by A014486/A063171.

We write the nonnegative integers as restricted growth strings (so called by J. Arndt in his book fxtbook.pdf, p. 325) in such a way that the Catalan numbers (cf. A000108) are expressed: 1=1, 10=2, 100=5, 1000=14, etc., 10...0 (with k zeros) = the k-th Catalan number. Once the entries of a restricted-growth string grow above 9, one would need commas or parentheses, say, to separate those entries. See Dejter (2017) for the precise definition.

In the paper "A system of numeration for middle-levels", restricted growth strings (RGSs) are defined as sequences that begin with either 0 or 1, with each successive number to the right being at least zero and at most one greater than its immediate left neighbor. Moreover, apart from case a(0), the RGSs are finite integer sequences of restricted growth which always start with 1 as their first element b_1 in position 1, and from then on, each successive element b_{i+1} in the sequence is restricted to be in range [0,(b_i)+1].

This sequence gives all such finite sequences in size-wise and lexicographic order, represented as decimal numbers by concatenating the integers of such finite sequences (e.g., from [1,2,0,1] we get 1201). The 58784th such sequence is [1, 2, 3, 4, 5, 6, 7, 8, 9, 9], thus a(58784) = 1234567899, after which comes the first RGS, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], where an element larger than 9 is present, which means that the decimal system employed here is unambiguous only up to n=58784. Note that 58785 = A000108(11)-1.

Also, if one considers Stanley's interpretation (u) of Catalan numbers, "sequences of a_1, a_2, ..., a_n of integers such that a_1 = 0 and 0 <= a_{i+1} <= a_{i} + 1" (e.g., 000, 001, 010, 011, 012 for C_3), and discards their initial zero, then one has a bijective correspondence with Dejter's RGSs of one element shorter length, which in turn are in bijective correspondence with the first C_n terms of this sequence (by discarding any leading zeros), from a(0) to a(C_n - 1). From this follows that the k-th Catalan number, A000108(k) (k>0), is represented in this system as 1 followed by k-1 zeros: a(1)=1, a(2)=10, a(5)=100, a(14)=1000, etc., and also that there exist exactly A000245(k) RGSs of length k.

Note how this differs from other number representations utilizing Catalan numbers, A014418 and A244159, in that while the latter are base-systems, where a simple weighted Sum_{k} digit(k)*C(k) recovers the natural number n (which the n-th numeral of such system represents), in contrast here it is the sum of appropriate terms in Catalan's Triangle (A009766, A030237), obtained by unranking a unique instance of a certain combinatorial structure (one of the Catalan interpretations), that gives a correspondence with a unique natural number. (Cf. also A014486.)

This sequence differs from "Semigreedy Catalan Representation", A244159, for the first time at n=10, where a(10) = 120, while A244159(10) = 121. That is also the first position where A244158(a(n)) <> n.

Please see Dejter's preprint for a more formal mathematical definition and how this number system is applied in relation to Havel's Conjecture on the existence of Hamiltonian cycles in the middle-levels graphs.

a(n) is given by the concatenation (with leading zeros removed) of the terms of row n + 23714 of A370222. - Paolo Xausa, Feb 17 2024

Sum_{i=A014137(n)..(A014137(n+1)-1)} a(i) = A001700(n), i.e., A001700(n) gives the total number of leaves in all ordered trees with n + 1 edges.

The initial portion of this sequence (up to the 6917th term) is equal to A071153 (Łukasiewicz words for rooted plane trees) sorted in ascending order.

Note: Here the last leaf is explicit, i.e. the terms are obtained from those of A071153 by multiplying them by 10.

Note: this finite decimal representation works only up to the 6917th term, as the 6918th such word is already "x0000000000" (where x stands for digit "ten").

Apart from the initial term (0, which encodes the null tree), if we scan the digits from the right (the least significant digit which is always 1) to the left (the most significant), then each successive digit to the left is at most one greater than the previous and never less than one.

Note: this finite decimal representation works only up to the 23712nd term, as the 23713rd such walk is already (10,9,8,7,6,5,4,3,2,1). The sequence A071158 shows the initial portion of this sequence sorted.