  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.
