The Major Counting of Nonintersecting Lattice Paths and Generating Functions for Tableaux; Summary

Christian Krattenthaler

Séminaire lotharingien de combinatoire, Tome 29 (1992)

Résumé

A theory of counting nonintersecting lattice paths by the major index and generalizations of it is developed. We obtain determinantal expressions for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given final points, where the starting points lie on a line parallel to x+y=0. In some cases these determinants can be evaluated to result into simple products. As applications we compute the generating function for tableaux with p odd rows, with at most c columns, and with parts between 1 and n. Besides, we compute the generating function for the same kind of tableaux which in addition have only odd parts. We thus also obtain a closed form for the generating function for symmetric plane partitions with at most n rows, with parts between 1 and c, and with p odd entries on the main diagonal. In each case the result is a simple product. By summing with respect to p we provide new proofs of the Bender-Knuth and MacMahon ex-Conjectures, which were first proved by Andrews, Gordon, and Macdonald. The link between nonintersecting lattice paths and tableaux is given by variations of the Knuth correspondence.

A full account of this work appeared as "The major counting of nonintersecting lattice paths and generating functions for tableaux", Mem. Amer. Math. Soc. 115, no. 552, Providence, R. I., 1995.