Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity

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. An unexpected application leads to a refinement of a result of Bousquet-M\'elou and Viennot on the width-height-area generating function of parallelogram polyominoes. Finally, we exhibit the relation of the arising alternating tableaux to plane partitions of strip shapes.