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Abstract : |
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. Summary of results and sketch of proofs We announce the proof of the following refinements of the MacMahon (ex-)Conjecture and the Bender?Knuth (ex-)Conjecture. (All the definitions can be found in the Appendix.) Theorem 1 (Refinement of the MacMahon (ex-)Conjecture) The generating function for tableaux with p odd rows (i.e. exactly p rows have odd length), with at 1 most c columns, and with only odd parts which lie between 1 and 2n ? 1, is given by q p2 [2r + 2p] q2 [r] q2 ? ? n [2r + p] q2 [r + p] q2 p q2 ? n + 2r, |
