Generating function for Naturalized Series: The case of Ordered Motzkin Words

Naturalized bracket row and Motzkin triangle

The paper considers the sequence of the Motzkin words, which is constructed according to formal features of natural numbers. We investigate the decomposition of well-formed parentheses into the matched pairs of parentheses (analogous to Prime Numbers). The point triangle of bracket pairs is constructed and studied, as well as the numerical Motzkin triangle (similar to Pascal's triangle). To calculate the Motzkin triangle, the graphical method of stars and bars is used.

大 Motzkin 路径与 {0,1,2}-树——一个简单双射

Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection

本文构造大 Motzkin 路径与奇数棵 {0,1,2}-树有序列表之间的简单双射，推广经典 Motzkin 路径对应关系，并为已有工作提供替代证明。

Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words

We establish a total lexicographical order on the set of Motzkin words. Elements are ordered similarly to Natural Numbers in accordance with known rules (axioms). As a result, we were able to obtain arithmetic and logical operations on the elements of the ordered sequence, Motzkin Row. This sequence consists of balanced brackets without leading zeros, with the exception of the initial word "0". It is the word "0" as well as the alphabetical symbol "0" that are analogues of numeric zero in the corresponding operations. Logical operations allow you to navigate Motzkin Row. Operations on words ar…

Mathematics of Balanced Parentheses: The case of Ordered Motzkin Words

It is known that the core of mathematics is natural numbers. And everything related to the natural number is interesting to mathematicians. In this paper, we draw parallels between natural numbers and elements of a non-numeric lexicographic sequence, Motzkin words (well-formed strings of parentheses and zeros). We will also talk about the decomposition of Motzkin words into pairs of parentheses (analogue of Prime Numbers). Finally, we will try to interest the reader in the elements of mathematical analysis on bracket expressions. The nesting procedure of parentheses is described by the author …

Raney numbers, threshold sequences and Motzkin-like paths

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up and down steps. Given three integers n, k, l such that n >= 1, k >= 2 and 0 <= l <= k-2, a (k,l)-threshold sequence of length n is any strictly increasing sequence S=(s_1 s_2 ... s_n) of integers such that ki <= s_i <= kn+l. These sequences are in bijection with ordered (l+1)-tuples of k-ary trees. We prove this result and identify the Raney numbers that count the (k,l)-threshold sequences. As a consequence, when k=2 and k=3, we deduce combinatorial identities involvi…