MacMahon's partition analysis XII: Plane partitions

Abstract

MacMahon developed partition analysis as a calculational and analytic method to produce the generating function for plane partitions. His efforts did not turn out as he had hoped, and he had to spend nearly twenty years finding an alternative treatment. This paper provides a detailed account of our retrieval of MacMahon’s original project. One of the key results obtained with partition analysis is an extension of a theorem of Gansner which generalizes Stanley’s famous trace theorem. This is the twelfth paper in this series on MacMahon’s partition analysis. It has been our belief from the beginning that MacMahon’s ideas could be best exploited by computer implementation, and that was the genesis of our partition analysis project. Our algorithmic version of MacMahon’s method has been implemented in the form of the Mathematica package Omega which is freely available via the web; see [20]. In the back of our minds was always MacMahon’s melodramatic experience with his own invention. He created partition analysis solely to treat the generating functions associated with various classes of plane partitions. Plane partitions are two-dimensional arrays of non-negative integers which are weakly decreasing in rows and columns; for a formal definition, including the notion of k-trace (1.3), see below. This specific project failed, and in this paper we shall retrieve MacMahon’s original project and obtain, using only partition analysis, an extension, namely Theorem 5.4, of a general plane partition theorem originally due to Gansner [13, Theorem 4.2]. For further remarks on plane partition history, in particular, on how partition analysis has led us to a rediscovery and to an alternative proof of Gansner’s theorem, we refer the interested reader to [4]. The initial stage of MacMahon’s investigations is chronicled by him in [16], where he refines the study of partitions and compositions of multipartite numbers into the theory of plane partitions. MacMahon [16, p. 658] first stated as an unproven assertion that the generating function for plane partitions is, in fact,