Séminaire Lotharingien de Combinatoire, B29a (1992), 12
pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1992, 1993/033, p.
51-63.]
Christian Krattenthaler
The Major Counting of Nonintersecting Lattice
Paths and Generating Functions for Tableaux;
Summary
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.
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.
The following version are available: