An extension of the Lindstr\"om-Gessel-Viennot theorem

TL;DR

This paper extends the Lindström-Gessel-Viennot theorem by providing a determinant formula for counting non-intersecting paths with specified start and end points in weighted directed acyclic graphs with upward planar drawings.

Contribution

It introduces a new formula that counts non-intersecting paths directly as a determinant, improving upon the signed enumeration approach of prior theorems.

Findings

Provides a determinant formula for total weight of non-intersecting paths

Expresses path counts as signed counts of lattice paths

Applicable to graphs with upward planar drawings

Abstract

Consider a weighted directed acyclic graph $G$ having an upward planar drawing. We give a formula for the total weight of the families of non-intersecting paths on $G$ with any given starting and ending points. While the Lindstr\"om-Gessel-Viennot theorem gives the signed enumeration of these weights (according to the connection type), our result provides the straight count, expressing it as a determinant whose entries are signed counts of lattice paths with given starting and ending points.