Johann Cigler and
Christian Krattenthaler
Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials,
and reciprocity
(70 pages)
Abstract.
The theme of this article is a "reciprocity" between bounded up-down
paths and bounded alternating sequences. Roughly speaking, this
"reciprocity" manifests itself by the fact
that the extension of the sequence of numbers
of paths of length n,
consisting of diagonal up- and down-steps and being confined
to a strip of bounded width, to negative n produces numbers
of alternating sequences of integers that are bounded from below and
from above. We show that
this reciprocity extends to families of non-intersecting
bounded up-down paths and certain arrays of alternating sequences
which we call alternating tableaux. We provide as well weighted
versions of these results. Our proofs are based on Viennot's
theory of heaps of pieces and on the combinatorics of non-intersecting
lattice paths.
Finally, we exhibit the relation of
the arising alternating tableaux to plane partitions of strip shapes.
The following versions are available:
Back to Christian Krattenthaler's
home page.