Schützenberger's Jeu de Taquin and Plane Partitions

✍ Scribed by Kevin W.J. Kadell

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 670 KB
• Volume: 77
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.2738

✦ Synopsis

We modify Schu tzenberger's jeu de taquin'' and Knuth's 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 We extend the combinatorial methods of Bender and Knuth to give an extension of an elegant, unpublished result of Stanley.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 to m. We prove MacMahon's ``box'' theorem and give a generating function for plane partitions with parts less than or equal to m and the parts below row r form a column-strict plane partition with at most c columns.

1997 Academic Press for plane partitions with parts less than or equal to m, at most r rows, and at most c columns. Here, q is fixed with |q| <1 and (x) n => n&1 i=0 (1&xq i ), n 1.