Dyck paths with coloured ascents

Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.

Short note on the number of 1-ascents in dispersed dyck paths

A dispersed Dyck path (DDP) of length n is a lattice path on $N\times N$ from (0,0) to (n,0) in which the following steps are allowed: "up" (x, y) $\to$ (x+1, y+1); "down" (x, y) $\to$ (x+1, y-1); and "right" (x,0) $\to$ (x+1,0). An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length n, A191386 in Sloane's OEIS. Previously, only implicit generating function relations and asymptotics were known.

Refinements of Lattice paths with flaws

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. F…

一个与 $k$ 叉树相关的斜 Dyck 路径子族

A subfamily of skew Dyck paths related to $k$-ary trees

本文定义盒路径及更一般的 $k$-盒路径，建立其与多元 $(k+2)$ 叉树、$k_t$-Dyck 路径和阈值序列子族的双射，并给出按返回与长上升的精细化枚举。

Dyck路径上的上升格

The ascent lattice on Dyck paths

本文研究Dyck路径上Stanley格的贪心变体，证明其仍为格（上升格$L_n$），通过递归分解和催化变量计数区间，并推广到$m$-Dyck路径情形，揭示与sylvester同余及象限行走的联系，给出渐近估计。

On rational Dyck paths and the enumeration of factor-free Dyck words

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alterna…