Maximal right smooth extension chains

TL;DR

This paper studies maximal right smooth extension chains (MRSE) in smooth words, proving they partition the set of smooth words at each height and deriving formulas for their counts, aiding bounds estimation.

Contribution

It introduces the concept of MRSE chains, proves they form a partition of smooth words at each height, and provides formulas for their enumeration, simplifying bounds analysis.

Findings

MRSE chains partition smooth words of a fixed height.

Formulas for counting MRSE chains at each height.

MRSE chains at specific heights help estimate bounds of smooth words.

Abstract

If $w=u\alpha$ for $\alpha\in \Sigma=\{1,2\}$ and $u\in \Sigma^*$, then $w$ is said to be a \textit{simple right extension}of $u$ and denoted by $u\prec w$. Let $k$ be a positive integer and $P^k(\epsilon)$ denote the set of all $C^\infty$-words of height $k$. Set $u_{1},\,u_{2},..., u_{m}\in P^{k}(\epsilon)$, if $u_{1}\prec u_{2}\prec ...\prec u_{m}$ and there is no element $v$ of $P^{k}(\epsilon)$ such that $v\prec u_{1}\text{or} u_{m}\prec v$, then $u_{1}\prec u_{2}\prec...\prec u_{m}$ is said to be a \textit{maximal right smooth extension (MRSE) chains}of height $k$. In this paper, we show that \textit{MRSE} chains of height $k$ constitutes a partition of smooth words of height $k$ and give the formula of the number of \textit{MRSE} chains of height $k$ for each positive integer $k$. Moreover, since there exist the minimal height $h_1$ and maximal height $h_2$ of smooth words of length $n$ for each positive integer $n$, we find that \textit{MRSE} chains of heights $h_1-1$ and $h_2+1$ are good candidates to be used to establish the lower and upper bounds of the number of smooth words of length $n$ respectively, which is simpler and more intuitionistic than the previous methods.