Proof of two conjectures of Zuber on fully packed loop configurations

Abstract

Two conjectures of Zuber (On the counting of fully packed loops configurations. Some new conjectures, Electronic J. Combin 11 (2004)) on the enumeration of configurations in the fully packed loop model on the square grid with periodic boundary conditions, which have a prescribed linkage pattern, are proved. Following an idea of de Gier (Loops, matchings and alternating-sign matrices, Discrete Math., to appear), the proofs are based on bijections between such fully packed loop configurations and rhombus tilings, and the hook-content formula for semistandard tableaux.