Proof of two conjectures of Zuber on fully packed loop configurations

F. Caselli (1) , C. Krattenthaler (1)

1 IGD - Institut Girard Desargues

Two conjectures of Zuber [``On the counting of fully packed loops configurations. Some new conjectures,'' preprint] 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.

Domaines: Combinatoire [math.CO]; Physique mathématique [math-ph]; Mécanique statistique [cond-mat.stat-mech]

F. Caselli, C. Krattenthaler. Proof of two conjectures of Zuber on fully packed loop configurations. J. Combin. Theory Ser. A, 2004, 108, pp.123-146. ⟨hal-00129282⟩