Proof of Razumov-Stroganov conjecture for some infinite families of link patterns

We prove the Razumov--Stroganov conjecture relating ground state of the O(1) loop model and counting of Fully Packed Loops in the case of certain types of link patterns. The main focus is on link patterns with three series of nested arches, for which we use as key ingredient of the proof a generalization of the Mac Mahon formula for the number of plane partitions which includes three series of parameters.