On the Refinement of Certain Statistics on Alternating Words

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 $i^{\mathrm{th}}$ 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 ${\sum }_{i\ge 1}{v}_i=m$. 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 $v$.

13 pages, 17 figures