scholarly journals A Generalization of Parking Functions Allowing Backward Movement

10.37236/8948 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Alex Christensen ◽  
Pamela E. Harris ◽  
Zakiya Jones ◽  
Marissa Loving ◽  
Andrés Ramos Rodríguez ◽  
...  
Keyword(s):  
Recursive Formula ◽  
Parking Space ◽  
Parking Functions ◽  
Parking Function ◽  
Dyck Paths ◽  
Positive Integers

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function.   In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to $k$ spaces west of their preferred spot to park before proceeding east if all of those $k$ spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule $k$-Naples parking functions of length $n$. This generalization gives a natural interpolation between classical parking functions, the case when $k=0$, and all $n$-tuples of positive integers $1$ to $n$, the case when $k\geq n-1$. Our main result provides a recursive formula for counting $k$-Naples parking functions of length $n$. We also give a characterization for the $k=1$ case by introducing a new function that maps $1$-Naples parking functions to classical parking functions, i.e. $0$-Naples parking functions. Lastly, we present a bijection between $k$-Naples parking functions of length $n$ whose entries are in weakly decreasing order and a family of signature Dyck paths. 

scholarly journals A further correspondence between $(bc,\bar{b})$-parking functions and $(bc,\bar{b})$-forests

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Heesung Shin ◽  
Jiang Zeng
Keyword(s):  
The Other ◽  
Parking Functions ◽  
Fixed Sequence ◽  
Parking Function ◽  
International Audience ◽  
Positive Integers ◽  
Appl Math

International audience For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions. Soit $(a,\bar{b}) := (a, b, b,\ldots, b)$ une suite d'entiers positifs. Une $(a,\bar{b})$-fonction de parking est une suite $(p_1, p_2, \ldots, p_n)$ d'entiers positives telle que son réarrangement non décroissant $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfait $q_i \leq a+(i-1)b$ pour tout $i=1,\ldots, n$. Une $(a,\bar{b})$-forêt enracinée sur un $n$-ensemble est une forêt enracinée dont les racines sont colorées avec les couleurs $0, 1, \ldots, a-1$ et les autres sommets sont colorés avec les couleurs $0, 1, \ldots, b-1$. Dans cet article, on construit une bijection entre $(bc,\bar{b})$-fonctions de parking et $(bc,\bar{b})$-forêts avec des des propriétés intéressantes. Comme applications, on obtient une généralisation d'un résultat de Gessel-Seo sur $(c,\bar{1})$-fonctions de parking [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] et une extension de l'identité de Yan [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] entre l'énumérateur d'inversion de $(bc,\bar{b})$-forêts et l'énumérateur complémentaire de $(bc,\bar{b})$-fonctions de parking.

scholarly journals Parking Functions and Noncrossing Partitions

10.37236/1335 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Symmetric Function ◽  
Local Action ◽  
Parking Functions ◽  
Noncrossing Partitions ◽  
Parking Function ◽  
Noncrossing Partition ◽  
Increasing Rearrangement ◽  
Positive Integers

A parking function is a sequence $(a_1,\dots,a_n)$ of positive integers such that, if $b_1\leq b_2\leq \cdots\leq b_n$ is the increasing rearrangement of the sequence $(a_1,\dots, a_n),$ then $b_i\leq i$. A noncrossing partition of the set $[n]=\{1,2,\dots,n\}$ is a partition $\pi$ of the set $[n]$ with the property that if $a < b < c < d$ and some block $B$ of $\pi$ contains both $a$ and $c$, while some block $B'$ of $\pi$ contains both $b$ and $d$, then $B=B'$. We establish some connections between parking functions and noncrossing partitions. A generating function for the flag $f$-vector of the lattice NC$_{n+1}$ of noncrossing partitions of $[{\scriptstyle n+1}]$ is shown to coincide (up to the involution $\omega$ on symmetric function) with Haiman's parking function symmetric function. We construct an edge labeling of NC$_{n+1}$ whose chain labels are the set of all parking functions of length $n$. This leads to a local action of the symmetric group ${S}_n$ on NC$_{n+1}$.

scholarly journals Counting Defective Parking Functions

10.37236/816 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Peter J Cameron ◽  
Daniel Johannsen ◽  
Thomas Prellberg ◽  
Pascal Schweitzer
Keyword(s):  
Kernel Method ◽  
Rayleigh Distribution ◽  
The Other ◽  
Partial Sums ◽  
Square Root ◽  
Parking Space ◽  
Parking Functions ◽  
Parking Function ◽  
Car Park ◽  
Binomial Identity

Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a defective parking function of defect $k$. Let ${\rm cp}(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers ${\rm cp}(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of ${\rm cp}(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in the case $m=\omega(n)$, the probability that all spaces are occupied tends asymptotically to one.

scholarly journals Extending the parking space

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Andrew Berget ◽  
Brendon Rhoades
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Algebraic Combinatorics ◽  
Parking Space ◽  
Parking Functions ◽  
Dyck Paths ◽  
International Audience ◽  
Special Case

International audience The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. L’action du groupe symétrique $S_n$ sur l’ensemble $\mathrm{Park}_n$ des fonctions de stationnement de longueur $n$ a reçu beaucoup d’attention dans la combinatoire algébrique. Nous démontrons que l’action de $S_n$ sur $\mathrm{Park}_n$ s’étend à une action de $S_{n+1}$. Plus précisément, nous construisons un gradué $S_{n+1}$-module $V_n$ telles que la restriction de $S_n$ est isomorphe à $\mathrm{Park}_n$. Nous décrivons la $S_n$-Frobenius caractères des modules $V_n$ à tous les degrés et décrivent le $S_{n+1}$-Frobenius caractères de $V_n$ en degrés extrêmes. Nous donnons une généralisation bivariée $V_n^{(\ell, m)}$ de notre module $V_n$ dont la représentation théorie est régie par une généralisation bivariée des chemins de Dyck. Une généralisation Fuss de nos résultats est un cas particulier de cette généralisation bivariée.

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals Rational Catalan Combinatorics: The Associahedron

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Catalan Number ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
International Audience ◽  
Positive Integers ◽  
Coprime Positive Integers ◽  
Product Formulas

International audience Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .

scholarly journals Affine permutations and rational slope parking functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Eugene Gorsky ◽  
Mikhail Mazin ◽  
Monica Vazirani
Keyword(s):  
Hyperplane Arrangements ◽  
Parking Functions ◽  
New Approach ◽  
Dyck Paths ◽  
International Audience ◽  
Affine Permutations ◽  
Combinatorial Constructions

International audience We introduce a new approach to the enumeration of rational slope parking functions with respect to the <mathrm>area</mathrm> and a generalized <mathrm>dinv</mathrm> statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (<mathrm>area</mathrm>,<mathrm>dinv</mathrm>) and (<mathrm>bounce</mathrm>, <mathrm>area</mathrm>) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions.

scholarly journals On Parking Functions and the Zeta Map in Types B, C and D

10.37236/6714 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Hilbert Series ◽  
Root Systems ◽  
The Other ◽  
Lattice Paths ◽  
Square Lattice ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Coxeter Number ◽  
Dyck Paths ◽  
Diagonal Harmonics

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

scholarly journals An alternative recursive formula for the sums of powers of integers

2016 ◽  
Vol 100 (548) ◽  
pp. 233-238
Author(s):  
José Luis Cereceda
Keyword(s):  
Stirling Numbers ◽  
Recursive Formula ◽  
Stirling Number ◽  
Harmonic Numbers ◽  
Binomial Theorem ◽  
Fundamental Identity ◽  
Stirling Cycle ◽  
Sums Of Powers ◽  
Image Position ◽  
Positive Integers

The sums of powers of the first n positive integers Sp(n) = 1p + 2p + …+np, (p = 0, 1, 2, … )satisfy the fundamental identity(1)from which we can successively compute S0 (n), S1 (n), S2 (n), etc. Identity (1) can easily be proved by using the binomial theorem; see e.g. [1, 2]. Several variations of (1) are also well known [3, 4, 5].In this note, we derive the following lesser-known recursive formula for Sp (n):(2)where denote the (unsigned) Stirling numbers of the first kind, also known as the Stirling cycle numbers (see e.g. [6, Chapter 6]). Table 1 shows the first few rows of the Stirling number triangle. Although the recursive formula (2) is by no means new, our purpose in dealing with recurrence (2) in this note is two-fold. On one hand, we aim to provide a new algebraic proof of (2) by making use of two related identities involving the harmonic numbers.

scholarly journals Generalized Small Schröder Numbers

10.37236/4827 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
JiSun Huh ◽  
SeungKyung Park
Keyword(s):  
Lattice Path ◽  
Dyck Paths ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Positive Integers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of  $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0.  We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.

Sign in / Sign up

Export Citation Format

Share Document