# Generating functions for plane partitions of a given shape

Abstract. For fixed integers \alpha and \beta, planar arrays of integers of a given shape, in which the entries decrease at least by \alpha along rows and at least by \beta along columns, are considered. For various classes of these (\alpha,\beta)-plane partitions we compute three different kinds of generating functions. By a combinatorial method, determinantal expressions are obtained for these generating functions. In special cases these determinants may be evaluated by a simple determinant lemma. All known results concerning plane partitions of a given shape are included. Thus our approach to computation of generating functions for plane partitions of a given shape provides a uniform method and yields numerous generalizations of known results.

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