Combinatorial reciprocity for non-intersecting paths

TL;DR

This paper establishes a combinatorial reciprocity theorem for counting non-intersecting paths in acyclic planar networks, with applications to Dyck paths and Schur function evaluations.

Contribution

It introduces a new reciprocity theorem for non-intersecting paths and demonstrates its applications in combinatorics and algebraic functions.

Findings

Proves a reciprocity theorem for non-intersecting paths in acyclic planar networks.

Derives reciprocity results for fans of bounded Dyck paths.

Establishes reciprocity for Schur function evaluations with repeated values.

Abstract

We prove a combinatorial reciprocity theorem for the enumeration of non-intersecting paths in a linearly growing sequence of acyclic planar networks. We explain two applications of this theorem: reciprocity for fans of bounded Dyck paths, and reciprocity for Schur function evaluations with repeated values.