Skew Motzkin paths

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U =(1, 1),down steps D =(1,-1), horizontal steps H =(1, 0), and left steps L =(-1,-1), and such that up steps never overlap with left steps. Let S_n be the set of all skew Motzkin paths of length n and let 8_n = |S_n|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence{8_n}n≥0. Then we present several involutions on S_n and consider the number of their fixed points.Finally we consider the enumeration of some statistics on S_n.     

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

We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1,2,…,n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.     

Enumeration of Łukasiewicz paths modulo some patterns

For any pattern $\alpha$ of length at most two, we enumerate equivalence classes of ?ukasiewicz paths of length $n\ge 0$ where two paths are equivalent whenever the occurrence positions of $\alpha$ are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of ?ukasiewicz paths.     

Bijections on two variations of noncrossing partitions

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

A partial order on Motzkin paths

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a possibility by defining and studying an analogy of the Tamari lattice on Motzkin paths. While our generalization is not a lattice, each of its connected components is isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also extend the structural and enumerative results to Schröder paths. We conclude by a discussion on the relation between our work and that of Baril and Pallo (2014).     

Inversion polynomials for 321-avoiding permutations

We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formula for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.     

A new statistic on Dyck paths for counting 3-dimensional Catalan words

A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.     

一些与Dyck路有关的数的恒等式

本文通过Cauchy留数定理和算子方法导出了一些形如∑_i=0ⁿ (-1)^n-i(n i)U_{m+k+i, k+i} =f(n) 和∑_i=0²ⁿ(-1 )ⁱ(2n i) U_{m+k+i, k+i} = g(n)的差分恒等式,这里U_{n, κ}表示Dyck路在不同条件下的计数公式,f(n),g(n)与m(n)只和n有关的函数.     

On the combinatorics of the Pfaff identity

Recently, there has been a revival of interest in the Pfaff identity on hypergeometric series because of the specialization of Simons and a generalization of Munarini. We present combinatorial settings and interpretations of the specialization and the generalization; one is based on free Dyck paths and free Schröder paths, and the other relies on a correspondence of Foata and Labelle between the Meixner endofunctions and bicolored permutations, and an extension of the technique developed by Labelle and Yeh for the Pfaff identity. Applying the involution on weighted Schröder paths, we derive a formula for the Narayana numbers as an alternating sum of the Catalan numbers.     

Riordan 矩阵在广义 Motzkin 路计数中的应用

用Riordan矩阵的方法研究了具有4种步型的加权格路(广义Motzkin路)的计数问题,引入了一类新的计数矩阵,即广义Motzkin矩阵.同时给出了这类矩阵的Riordan表示,也得到了广义Motzkin路的计数公式.Catalan矩阵,Schrder矩阵和Motzkin矩阵都是广义Motzkin矩阵的特殊情形.