On some problems about ternary paths -- a linear algebra approach

TL;DR

This paper studies the enumeration of partial ternary paths with specific up and down steps, using linear algebra to resolve conjectures and analyze path counts from both directions, revealing asymmetries and new results.

Contribution

It introduces a linear algebra approach to enumerate partial ternary paths from both directions, solving open conjectures and highlighting asymmetries unlike classical Dyck paths.

Findings

Enumeration formulas for paths ending at various levels

Asymmetry between left-to-right and right-to-left counts

Resolution of open conjectures from Naomi Cameron's thesis

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.