On the Refinement of Certain Statistics on Alternating Words

In this paper, we investigate statistics on alternating words under correspondence between ``possible reflection paths within several layers of glass and ``alternating words. For $v=(v_1,v_2,cdots,v_n)inmathbb{Z}^{n}$, 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 $iin{1,2,cdots,n}$. We give a recursion for the number of paths corresponding to $v$ satisfying $v in mathbb{Z}^n$ and $sum_{igeq 1} v_i=m$. Also, we establish recursions for statistics around the number of paths corresponding to a given vector $vinmathbb{Z}^n$ and a closed form for $n=3$. Finally, we give a equivalent condition for the existence of path corresponding to a given vector $v$.