Enumeration of lattice paths and generating functions for skew plane partitions

Abstract

n-dimensional lattice paths not touching the hyperplanesX i−X i+1=−1,i=1,2,...,n, are counted by four different statistics, one of which is MacMahon's major index. By a reflection-like proof, heavily relying on Zeilberger's (Discrete Math. 44(1983), 325–326) solution of then-candidate ballot problem, determinantal expressions are obtained. As corollaries the generating functions for skew plane partitions, column-strict skew plane partitions, reverse skew plane plane partitions and column-strict reverse skew plane partitions, respectively, are evaluated, thus establishing partly new results, partly new proofs for known theorems in the theory of plane partitions.