We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of (m,n)-parking functions, and associated (q,t)-analogs.
This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schröder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over $m$-Schröder paths with a fix number of diagonal steps into Schur functions in the variables $q$ and $t$. Some results have been generalized to parking functions, and some bijections were made with standard Young tableaux giving way to partial combinatorial formulas in the basis $s_μ(q,t)s_λ(X)$ for $\nabla(e_n)$ (respectively, $\nabla^m(e_n)$), when $μ$ and $λ…
We study $ν$-Schröder paths, which are Schröder paths which stay weakly above a given lattice path $ν$. Some classical bijective and enumerative results are extended to the $ν$-setting, including the relationship between small and large Schröder paths. We introduce two posets of $ν$-Schröder objects, namely $ν$-Schröder paths and trees, and show that they are isomorphic to the face poset of the $ν$-associahedron $A_ν$ introduced by Ceballos, Padrol and Sarmiento. A consequence of our results is that the $i$-dimensional faces of $A_ν$ are indexed by $ν$-Schröder paths with $i$ diagonal steps, a…
Cyclic Sieving of Increasing Tableaux and small Schröder Paths
递增表是具有严格递增的行和列的半标准表。众所周知，卡塔兰数既能枚举两行的矩形标准杨氏图，也能枚举戴克路径。我们将其推广到矩形2行增加tableaux和小施罗德路径之间的双射。我们证明了由H. Thomas和A.杨与热带中楣图案的组合学。然后，我们使用这个jeu de taquin提出新的情况下，循环筛分现象的V。Stanton和D. White的结果，推广了D.怀特和J·斯坦布里奇。
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$; we study here combinatorial and probabilistic aspects of this generalized case. We construct a family of bijections between parking functions $\text{PF}(m, n)$ with $m$ cars and $n$ spots and spanning forests $\mathscr{…
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size $n$ and $n$…
We apply the concept of parking functions to rooted labelled trees and functional digraphs of mappings (i.e., functions $f : [n] \to [n]$) by considering the nodes as parking spaces and the directed edges as one-way streets: Each driver has a preferred parking space and starting with this node he follows the edges in the graph until he either finds a free parking space or all reachable parking spaces are occupied. If all drivers are successful we speak about a parking function for the tree or mapping. We transfer well-known characterizations of parking functions to trees and mappings. Especial…
The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$ and $s$. Classical notions such as Dyck paths and parking functions are naturally generalized by considering the set of partitions included in a given triangular partition. One of our striking results is that the restriction of the Young lattice to triangular partition has a planar Hasse diagram, wi…
We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper bound on the total variation distance between the joint distribution of cycle counts and independent Poisson random variables using a multivariate version of Stein's method via exchangeable pairs. Under a mild condition, the process of cycle counts converges in distribution to a process of independent Poisson random variables.
We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of…
