On some problems about ternary paths: a linear algebra approach

Abstract

Ternary paths consist of an up-step of one unit, a down-step 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 w.r.t.\ 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.