论文标题
有界的堤防路径，有限的交替序列，正交多项式和互惠
Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity
论文作者
论文摘要
本文的主题是有限的上向路径和有限交替序列之间的“互惠”。粗略地说，这种``互惠''表现出了以下事实，即长度$ n $的长度路径的序列的扩展，包括对角线的上下阶段，并被局限于有界宽度的条纹，限制为负面$ n $的条件，从上方和范围内的整体序列范围内。我们称之为交替的序列的路径和交替的序列。我们提供了这些结果的加权版本。最后，我们展示了出现的交替tableaux与带状形状平面隔板的关系。
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-Mélou 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.
