2_(0)-Dyck路与三元树之间的双射  

Bijection between 2_(0)-Dyck path and ternary tree

作　　者：王佳合 窦裕杰 庞兴梅 吕仑 WANG Jiahe;DOU Yujie;PANG Xingmei;LYU Lun(School of Sciences,Hebei University of Science and Technology,Shijiazhuang,Hebei 050018,China;College of Statistics and Mathematics,Hebei University of Economics and Business,Shijiazhuang,Hebei 050062,China)

机构地区：[1]河北科技大学理学院,河北石家庄050018 [2]河北经贸大学统计与数学学院,河北石家庄050062

出　　处：《河北科技大学学报》2025年第2期161-166,共6页Journal of Hebei University of Science and Technology

基　　金：国家自然科学基金(11001073);河北省自然科学基金(A2023208006);河北省引进留学人员资助项目(C20230357)。

摘　　要：为了拓展Dyck路和三元树的基本理论,研究了2_(0)-Dyck路和具有n个内点的三元树之间的双射与计数问题。首先,通过构造全新的组合结构L-R-T序列,建立了2_(0)-Dyck路与L-R-T序列之间的双射;其次,通过分析法、归纳法构建了L-R-T序列与n个内点的三元树之间的转化关系,从而给出了其计数公式的一种组合证明;最后,研究了2_(0)-Dyck路的峰谷与L-R-T序列和三元树的对应关系。结果表明,在研究2个组合结构之间的双射问题时,可借助与其他结构的关系来完成证明。研究结果丰富了Dyck路和三元树的相关理论,为这2个结构在组合数学领域的应用提供了理论参考。In order to expand the basic theory of Dyck path and ternary tree,the bijection and counting problems between the 2_(0)-Dyck path and the ternary tree with n inliers were studied.Firstly,the bijection between the 2_(0)-Dyck path and the L-R-T sequence was established by constructing a novel combinatorial L-R-T sequence.Secondly,the transformation relationship between the L-R-T sequence and the ternary tree with n inliers was constructed by means of analysis and induction,thus providing a combined proof of its counting formula.Finally,the relationship between the peaks and valleys of the 2_(0)-Dyck Path and the L-R-T sequence and the ternary tree was studied.The results show that the bijection problem between two combined structures can be proved by the relationship with other structures.The research results enrich the related theories of Dyck path and ternary tree,and provide theoretical reference value for the application of these two structures in the field of combinatorics.

关 键 词：计数组合学 2_(0)-Dyck路 k_(t)-Dyck路 三元树 CATALAN数 L-R-T序列 

分 类 号：O157[理学—数学]