Schützenbergerʹs Jeu de Taquin and Plane Partitions

We modify Schützenbergerʹs “jeu de taquin” and Knuthʹs generalization DELETE of the Robinson–Schensted correspondence to apply to unrestricted rather than just column-strict plane partitions. The “jeu de taquin,” DELETE, their modifications, and the Hillman–Grassl mapping are essentially equivalent. We extend the combinatorial methods of Bender and Knuth to give an extension of an elegant, unpublished result of Stanley. Our main result is equivalent to the evaluation of the generating function for column-strict plane partitions of fixed shape with parts less than or equal tom. We prove MacMahonʹs “box” theorem and give a generating function for plane partitions with parts less than or equal tomand the parts below rowrform a column-strict plane partition with at mostccolumns.