On the Counting of Fully Packed Loop Configurations: Some New Conjectures

On the Number of Fully Packed Loop Configurations with a Fixed Associated Matching

We show that the number of fully packed loop configurations corresponding to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11(1) (2004), Article #R13].

Refined Counting of Fully Packed Loop Configurations

Abstract. We give a generalisation of a conjecture by Propp on a summation formula for fully packed loop configurations. The original conjecture states that the number of configurations in which each external edge is connected to its neighbour is equal to the total number of configurations of size one less. This conjecture was later generalised by Zuber to include more types of configurations. ...

Proof of two conjectures of Zuber on fully packed loop configurations

On the link pattern distribution of quarter-turn symmetric FPL configurations

We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, ...

Wieland Drift for Triangular Fully Packed Loop Configurations

Triangular fully packed loop configurations (TFPLs) emerged as auxiliary objects in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers Aπ of FPLs corresponding to a given link pattern π. The focus of this article is...