On the Refinement of Certain Statistics on Alternating Words

TL;DR

This paper explores the combinatorial statistics of paths representing reflections in layered glass, establishing recursive formulas and conditions for their existence, with specific solutions for three-layer cases.

Contribution

It introduces recursive methods and closed-form solutions for counting and characterizing reflection paths in layered media, linking physical reflection paths to combinatorial structures.

Findings

Derived recursion formulas for counting paths with given reflection vectors

Established a closed-form solution for three-layer cases

Provided conditions for the existence of paths for given vectors

Abstract

In this paper, we investigate statistics on alternating words under correspondence between ``possible reflection paths within several layers of glass'' and ``alternating words''. For $v=(v_1,v_2,\cdots,v_n)\in\mathbb{Z}^{n}$, we say $P$ is a path within $n$ glass plates corresponding to $v$, if $P$ has exactly $v_i$ reflections occurring at the $i^{\rm{th}}$ plate for all $i\in\{1,2,\cdots,n\}$. We give a recursion for the number of paths corresponding to $v$ satisfying $v \in \mathbb{Z}^n$ and $\sum_{i\geq 1} v_i=m$. Also, we establish recursions for statistics around the number of paths corresponding to a given vector $v\in\mathbb{Z}^n$ and a closed form for $n=3$. Finally, we give a equivalent condition for the existence of path corresponding to a given vector $v$.