ON THE NUMBER OF FULLY PACKED LOOP CONFIGURATIONS WITH A FIXED ASSOCIATED MATCHING

ON THE NUMBER OF FULLY PACKED LOOP CONFIGURATIONS WITH A FIXED ASSOCIATED MATCHING F. Caselli_, C. Krattenthaler, B. Lass and P. Nadeauy *Institut Camille Jordan, Universite Claude Bernard Lyon-I, 21, avenue Claude Bernard, F-69622 Villeurbanne Cedex, France. E-mail: (caselli,kratt,lass)@euler.univ-lyon1.fr yLaboratoire de Recherche en Informatique, Universite Paris-Sud 91405 Orsay Cedex, France E-mail: Submitted: Feb 17, 2005; Accepted: Mar 14, 2005; Published: Apr 6, 2005 Dedicated to Richard Stanley Abstract. We show that the number of fully packed loop congurations correspond- ing 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), Arti- cle _R13]. 1. Introduction In this paper we continue the enumerative study of fully packed loop congurations corresponding to a prescribed matching begun by the rst two authors in [2], where we proved two conjectures by Zuber [22] on this subject matter. (See also [6, 7, 8, 9] for related results.) The interest in this study originates in conjectures by Razumov and Stroganov [18], and by Mitra, Nienhuis, de Gier and Batchelor [17], which predict that the coordinates of the groundstate vectors of certain Hamiltonians in the dense O(

• ferrers diagram


• diagram representation


• arches between


• fully packed


• associated matching


• see figure


• congurations corresponding


• packed loop