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A Relation Between Schröder Paths and Motzkin Paths
Graphs and Combinatorics ( IF 0.7 ) Pub Date : 2020-05-24 , DOI: 10.1007/s00373-020-02185-6
Lin Yang , Sheng-Liang Yang

A small q-Schröder path of semilength n is a lattice path from (0, 0) to (2n, 0) using up steps \(U = (1, 1)\), horizontal steps \(H = (2, 0)\), and down steps \(D = (1,-1)\) such that it stays weakly above the x-axis, has no horizontal steps on the x-axis, and each horizontal step comes in q colors. In this paper, we provide a bijection between the set of small q-Schröder paths of semilength \(n+1\) and the set of \((q+2, q+1)\)-Motzkin paths of length n. Furthermore, a one-to-one correspondence between the set of small 3-Schröder paths of semilength n and the set of Catalan rook paths of semilength n is obtained, and a bijection between small 4-Schröder paths and Catalan queen paths is also presented.



中文翻译:

Schröder路径与Motzkin路径之间的关系

一条半长为n的q- Schröder路径是从(0,0)到(2 n,0)的晶格路径,使用向上步长\(U =(1,1)\),水平步长\(H =(2, 0)\)和向下步\(d =(1,-1)\) ,使得它保持弱上面的X轴,对没有水平步骤X轴,并且每个水平台阶进来q颜色。在本文中,我们提供了一套小的之间的双射q -Schrödersemilength路径\(N + 1 \)和该组的\((Q + 2,Q + 1)\)长度的-Motzkin路径Ñ。此外,该组semilength的小3-施罗德路径之间的一对一的对应Ñ和该组semilength加泰罗尼亚车路径Ñ被获得的,并且小4-施罗德路径和加泰罗尼亚王后路径之间的双射还提出。

更新日期:2020-05-24
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