Alternating paths of fully packed loops and inversion number

TL;DR

This paper explores the structure of alternating paths in fully packed loops, introduces Dyck islands as nested osculating loops, and provides a graphical method to compute the inversion number of alternating sign matrices.

Contribution

It establishes a bijection between alternating paths and fully packed loops, and introduces Dyck islands to simplify inversion number calculations.

Findings

Dyck islands are nested osculating loops constructed from lattice Dyck paths.

A graphical formula for calculating the inversion number of an alternating sign matrix.

The set of alternating paths is in bijection with fully packed loops of the same size.

Abstract

We consider the set of alternating paths on a fixed fully packed loop of size n. This set is in bijection with the set of fully packed loops of size n. Furthermore, for a special choice of fully packed loop, we demonstrate that the set of alternating paths are nested osculating loops, which we call Dyck islands. Dyck islands can be constructed as a union of lattice Dyck paths, and we use this structure to give a simple graphical formula for the calculation of the inversion number of an alternating sign matrix.