A note on 2-distant noncrossing partitions and weighted Motzkin paths

Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths

Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored Motzkin path in the sense that there are three types of level steps and two types of down steps. A large (3,2)-Motzkin path is defined to be a (3,2)-Motzkin path for which there are only two types of level steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of [n+1] and the set of large (3,2)-Motzkin p…

Bijections on two variations of noncrossing partitions

We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schr{ö}der paths and Schr{ö}der paths without peaks at even height. We also give a direct bijection between 2-distant noncrossing partitions and 12312-avoiding partitions.

增强、经典和2-距离k-非交叉分区的简单双射

A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions

本文提出一个简单的扩展映射，证明了增强、经典和2-距离k-非交叉分区的数量通过二项式变换依次关联，解决了Zhicong Lin的猜想并推广了早期归约恒等式。

A combinatorial bijection on $k$-noncrossing partitions

For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of weights, $t^\text{number of blocks}$, of partitions of $\{1,\ldots,m\}$ without $k$-crossings (resp.~enhanced $k$-crossings). The special $k=2$ and $t=1$ case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for $k=3$ and $t=1$, arising naturally in a recent study of pattern avoidance in …

$k$-distant crossings and nestings of matchings and partitions

We define and consider k-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of k-distant crossings and nestings is symmetric. We also study the numbers of k-distant noncrossing matchings and partitions for small k, which are counted by well-known sequences, as well as the orthogonal polynomials related to k-distant noncrossing matchings and par…

理查森表与莫茨金路径

Richardson tableaux and Motzkin paths

本文从莫茨金路径视角研究理查森表，引入独立于RS算法的形状算法，构造局部双射并重新证明q计数公式，还证明了郭关于特定理查森表生成函数的猜想。