Skew Dyck paths with catastrophes

Partial skew Dyck paths -- a kernel method approach

Skew Dyck are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end with decorated Dyck paths. We analyze partial versions of them where the path ends on a fixed level $j$, not necessarily at level 0. We exclusively use generating functions and derive them with the celebrated kernel method. In the second part of the paper, a dual version is studied, where the paths are read from right to left. In this way, we have tw…

具有两种下降步变体的斜戴克路径的高度

The height of skew Dyck paths with two variants of downsteps

研究具有两种下降步变体的斜戴克路径的高度，利用生成函数、核方法和线性系统，对其平均高度等计数做出贡献，得出平均高度为$\sqrt{n}$阶的结论。

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。

x轴回归次数为奇数的Stanley Catalan路径的前缀——标准版本与斜Catalan-Stanley路径

Prefixes of Stanley's Catalan paths with odd returns to the $x$-axis -- standard version and skew Catalan-Stanley paths

本文研究满足x轴奇回归的Stanley Dyck路径及其前缀计数，用核方法推导前缀计数公式，并拓展研究带Stanley限制的斜Dyck路径问题。

Dyck paths and a bijection for multisets of hook numbers

We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture, by means of a construction involving Dyck paths, a particular type of lattice path.

Combinatorics of the zeta map on rational Dyck paths

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $¶$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $χ$ on the set of $(a,b)$-Dyck paths when $ζ$ is a bijection, as well as a new method for calculating $ζ^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck path…

Partial Skew Motzkin Paths

Motzkin paths consist of up-steps, down-steps, level-steps, and never go below the $x$-axis. They return to the $x$-axis at the end. The concept of skew Dyck path \cite{Deutsch-italy} is transferred to skew Motzkin paths, namely, a left step $(-1,-1)$ is additionally allowed, but the path is not allowed to intersect itself. The enumeration of these combinatorial objects was known \cite{Qing}; here, using the kernel method, we extend the results by allowing them to end at a prescribed level $j$. The approach is completely based on generating functions. Asymptotics of the total number of objects…

用于格路径的片段堆

Heaps of pieces for lattice paths

本文引入格路径的I型与II型片段堆，建立对偶、双射及高维对应，并给出生成函数的行列式与显式公式。

Lattice paths with a first return decomposition constrained by the maximal height of a pattern

We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply this general result to the enumeration of certain subsets of Dyck, Motzkin, skew Dyck, and skew Motzkin paths, defined recursively according to the first return decomposition with a monotonically non-increasing condition relative to the maximal ordinate reached by an occurrence of a given pattern $π$.

带灾变的格路径

Lattice paths with catastrophes

本文研究排队论中带“重置”的格路径（即带灾变的格路径），通过双射、连分数及核方法，给出其枚举、极限定律和均匀随机生成算法。