On some problems about ternary paths: a linear algebra approach

Abstract

Ternary paths consist of an upstep of one unit, a downstep of two units, never go below the $x$-axis, and return to the $x$-axis. This paper addresses the enumeration of partial ternary paths, ending at a given level $i$, reading the path either from left-to-right or from right-to-left. Since the paths are not symmetric with respect to left vs. right, as classical Dyck paths, this leads to different results. The right-to-left enumeration is quite challenging, but leads at the end to very satisfying results. The methods are elementary (solving systems of linear equations). In this way, several conjectures left open in Naiomi Cameron’s Ph.D. thesis could be successfully settled.