CAJ | 학술논문

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、Schr(o)der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.
본문고필료유최고봉적고도위m,병차봉적고도연착Dyck로엄격체증적소유Dyck로조성적집합,즉집합Dm적자집적계수문제.이용쌍사、생성수이급Riordan진적방법래대집합Dm적일사자집진행계수,득도료일사이경전적서렬여Catalan수、Narayana수、Motzkin수、Fibonacci수、Schr(o)der수이급제일류무부호Stirling수래계수적조합결구.특별지,아문급출료량개신적Catalan결구,타문병몰유명현지출현재Stanley관우Catalan결구적렬표중.
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., Schr(o)der 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.