峰严格递增的Dyck路的计数

本文考虑了由最高峰的高度为m,并且峰的高度沿着Dyck路严格递增的所有Dyck路组成的集合,即集合Dm的子集的计数问题．利用双射、生成树以及Riordan阵的方法来对集合Dm的一些子集进行计数,得到了一些以经典的序列如Catalan数、Narayana数、Motzkin数、Fibonacci数、Schroder数以及第一类无符号Stirling数来计数的组合结构．特别地,我们给出了两个新的Catalan结构,它们并没有明显地出现在Stanley关于Catalan结构的列表中．

Counting Dyck Paths with Strictly Increasing Peak Sequences

In this paper we consider the enumeration of subsets of the set, say Dm, of those Dyck paths of arbitrary length with maximum peak height equal to m and having a strictly increasing sequence of peak height (as one goes along the path). Bijections and the methods of generating trees together with those of Riordan arrays are used to enumerate these subsets, resulting in many combinatorial structures counted by such well-known sequences as the Catalan nos., Narayana nos., Motzkin nos., Fibonacci nos., Schroder nos., and the unsigned Stirling numbers of the first kind. In particular, we give two configurations which do not appear in Stanley's well-known list of Catalan structures.