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 (2004), Article #R13]. 
Fabrizio Caselli, Christian Krattenthaler, Bodo Lass (Universit\'e   Lyon-I), Philippe Nadeau (Universit\'e Paris Sud)