Matrix correspondences of plane partitions

Abstract

Three correspondences between nonnegative integer arrays and plane partitions, due to Burge, Knuth, and Hillman and Grassl, are investigated. A variety of parallel and complementary properties are derived for the first two. In particular, the images of an array under the correspondences are characterized in terms of certain sets of in the arrays. The correspondences are also related to each other through the action of the dihedral group on rectangular arrays. Finally, the Hillman-Grassl correspondence is similarly characterized in terms of sets of paths and is shown to be an extension of Surge's correspondence. One of the most elegant developments in the study of plane partitions has been the invention of constructive correspondences between nonnegative integer arrays and certain forms of plane partitions. Besides providing potent tools for investigating these partitions, they also serve to connect the theory of plane partitions with those of symmetric functions and group representations (cf. [23]), among others. Here we limit our attention to three correspondences, four if we include the correspondence of Schensted [20] between permutations and pairs of standard Young tableaux. This bijection was extended by Knuth [15] to a correspondence K between matrices and pairs of generalized Young tableaux or, as we shall call them, column-strict plane partitions. In turn, Burge [4] defined a new correspondence B based on a variation of the original construction that was also investigated by Schensted and Knuth. The final correspondence to be considered, due to Hillman and Grassl [13], is entirely different in definition, yet, as will be shown, is very much related to B and K. Indeed, a presentation of these correspondences in a fairly unified setting is the main purpose of this paper, in which their interconnections and parallel developments are emphasized. The first three sections are devoted to preliminaries and the definitions of B and K. Schensted's correspondence is introduced in § 4 as a restriction of K. However, it is also noted that both B and K factor naturally through Schensted's map, implying that the properties of the latter correspondence will occur in some form as properties of B and K. This fact has been common knowledge for some time; here we present an explicit factorization.