THE MAJOR COUNTING OF NONINTERSECTING LATTICE PATHS AND GENERATING-FUNCTIONS FOR TABLEAUX

A theory of counting nonintersecting lattice paths by the major index and generalizations of it is developed. We obtain determinantal expres sions for the corresponding generating functions for families of nonin tersecting lattice paths with given starting points and given final po ints, where the starting points lie on a line parallel to x + y = 0. I n some cases these determinants can be evaluated to result in simple p roducts. As applications we compute the generating function for tablea ux with p odd rows, with at most c columns, and with parts between 1 a nd 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 partition s with at most n rows, with parts between 1 and c, and with p odd entr ies 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-Knut h and MacMahon (ex-)Conjectures, which were first proved by Andrews, G ordon, and Macdonald. The link between nonintersecting lattice paths a nd tableaux is given by variations of the Knuth correspondence.