# Fully packed loop configurations: polynomiality and nested arches

## TL;DR

This paper proves Zuber's conjecture that the enumeration of fully packed loops with specific link patterns is polynomial in the number of nested arches, using wheel polynomial theory and a new basis.

## Contribution

It introduces a new basis for wheel polynomials and a generalized polynomiality theorem, completing the proof of Zuber's conjecture on FPL enumeration.

## Key findings

- Confirmed polynomiality of FPL counts with nested arches
- Established the degree and leading coefficient of the polynomial
- Extended the polynomiality result to a broader setting

## Abstract

This article proves a conjecture by Zuber about the enumeration of fully packed loops (FPLs). The conjecture states that the number of FPLs whose link pattern consists of two noncrossing matchings which are separated by $m$ nested arches is a polynomial function in $m$ of certain degree and with certain leading coefficient. Contrary to the approach of Caselli, Krattenthaler, Lass and Nadeau (who proved a partial result) we make use of the theory of wheel polynomials developed by Di Francesco, Fonseca and Zinn-Justin. We present a new basis for the vector space of wheel polynomials and a polynomiality theorem in a more general setting. This allows us to finish the proof of Zubers conjecture.