On Parking Functions and The Tower of Hanoi
Dylan Alvarenga2022
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length n with displacement one is in bijection with the set of ideal states in the famous Tower of Hanoi game with n + 1 disks and n + 1 pegs, both sets being enumerated by the Lah numbers.
Related papers
Two-Player Tower of Hanoi
The Tower of Hanoi game is a classical puzzle in recreational mathematics, which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is 2n--1, to transfer a tower of n disks. But there are also other variations to the game, involving for example move edges weighted by real numbers. This gives rise to a similar type of problem, but where the final score seeks to be optimized. We study extensions of the one-player setting to two players, invoking classical winning conditions in combinatorial game theory such as the player who moves last wins, or the highest score wins. Here we solve both these winning conditions on three heaps.
The Tower of Hanoi problem on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mstyle mathvariant="monospace"><mml:mi>Path</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub></mml:...
Discrete Applied Mathematics, 2012
The generalized Tower of Hanoi problem with h ≥ 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h, n and the source and destination pegs.
On increasing and invariant parking sequences
Australas. J Comb., 2021
The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the cars to have different sizes and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. A preference sequence in which all the cars are able to park is called a parking sequence. In this paper, we study increasing parking sequences and count them via bijections to lattice paths with right boundaries. Then we study two notions of invariance in parking sequences and present various characterizations and enumerative results.
On the Outcome Map of MVP Parking Functions: Permutations Avoiding 321 and 3412, and Motzkin Paths
2022
We introduce a new parking procedure called MVP parking in which n cars sequentially enter a one-way street with a preferred parking spot from the n parking spots on the street. If their preferred spot is empty they park there, otherwise a later car bumps a previous cars out of their preferred spot sending that car forward in the street to find the first available spot on the street. If all cars can park under this parking procedure, we say the list of preferences of the n cars is an MVP parking function of length n. We show that the set of (classical) parking functions is exactly the set of MVP parking functions although the parking outcome (order in which the cars park) is different under each parking process. Motivating the question: Given a permutation describing the outcome of the MPV parking process, what is the number of MVP parking functions resulting in that given outcome? Our main result establishes a bound for this count which is tight precisely when the permutation describing the parking outcome avoids the patterns 321 and 3412. We then consider special cases of permutations and give closed formulas for the number of MVP parking functions with those outcomes. In particular, we show that the number of MVP parking functions which park in reverse order (that is the permutation describing the outcome is the longest word in Sn, which does not avoid the pattern 321) is given by the nth Motzkin number. We also give families of permutations describing the parking outcome for which the cardinality of the set of cars parking in that order is exponential and others in which it is linear.
A Generalization of Parking Functions Allowing Backward Movement
The Electronic Journal of Combinatorics, 2020
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 p...
Optimal Algorithms for Tower of Hanoi Problems with Relaxed Placement Rules
Lecture Notes in Computer Science, 2006
We study generalizations of the Tower of Hanoi (ToH) puzzle with relaxed placement rules. In 1981, D. Wood suggested a variant, where a bigger disk may be placed higher than a smaller one if their size difference is less than k. In 1992, D. Poole suggested a natural diskmoving strategy, and computed the length of the shortest move sequence (algorithm) under its framework. However, other strategies were not considered, so the lower bound/optimality question remained open. In 1998, Beneditkis, Berend, and Safro were able to prove the optimality of Poole's algorithm for the first non-trivial case k = 2 only. We prove it be optimal in the general case. Besides, we prove a tight bound for the diameter of the configuration graph of the problem suggested by Wood. Further, we consider a generalized setting, where the disk sizes should not form a continuous interval of integers. To this end, we describe a finite family of potentially optimal algorithms and prove that for any set of disk sizes, the best one among those algorithms is optimal. Finally, a setting with the ultimate relaxed placement rule (suggested by D. Berend) is defined. We show that it is not more general, by finding a reduction to the second setting.
Coding Parking Functions by Pairs of Permutations
The electronic journal of combinatorics
Abstract We introduce a new class of admissible pairs of triangular sequences and prove a bijection between the set of admissible pairs of triangular sequences of length n and the set of parking functions of length n For all u and v = 0 , 1 , 2 , 3 and all n = 7 we describe in terms of admissible pairs the dimensions of the bi - graded com - ponents hu,v of diagonal harmonics C[x1 , , xn; y1 , , yn]/Sn , i e , polynomials in two groups of n variables modulo the diagonal action of symmetric group Sn
Cycle Lemma, Parking Functions and Related Multigraphs
Graphs and Combinatorics, 2010
For positive integers a and b, an (a, b)-parking function of length n is a sequence (p 1 ,. .. , p n) of nonnegative integers whose weakly increasing order q 1 ≤ • • • ≤ q n satisfies the condition q i < a + (i − 1)b. In this paper, we give a new proof of the enumeration formula for (a, b)-parking functions by using of the cycle lemma for words, which leads to some enumerative results for the (a, b)-parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between (a, b)-parking functions and rooted forests, we enumerate combinatorially the (a, b)-parking functions with identical initial terms and symmetric (a, b)-parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to (a, b)-parking functions.
Parking Functions: Choose Your Own Adventure
The College Mathematics Journal, 2021
The reading of this paper will send you down many winding roads toward new and exciting research topics enumerating generalized parking functions. Buckle up!
The Tower of Hanoi problem on Pathh graphs
Discrete Applied Mathematics, 2012
The generalized Tower of Hanoi problem with h ≥ 4 pegs is known to require a sub-exponentially fast growing number of moves in order to transfer a pile of n disks from one peg to another. In this paper we study the Path h variant, where the pegs are placed along a line, and disks can be moved from a peg to its nearest neighbor(s) only. Whereas in the simple variant there are h(h−1)/2 possible bi-directional interconnections among pegs, here there are only h − 1 of them. Despite the significant reduction in the number of interconnections, the number of moves needed to transfer a pile of n disks between any two pegs also grows sub-exponentially as a function of n. We study these graphs, identify sets of mutually recursive tasks, and obtain a relatively tight upper bound for the number of moves, depending on h, n and the source and destination pegs.
Related topics
Related papers
On a question of Leiss regarding the Hanoi Tower problem
Theoretical Computer Science, 2006
The Tower of Hanoi problem is generalized in such a way that the pegs are located at the vertices of a directed graph G, and moves of disks may be made only along edges of G. Leiss obtained a complete characterization of graphs in which arbitrarily many disks can be moved from the source vertex S to the destination vertex D. Here we consider graphs which do not satisfy this characterization; hence, there is a bound on the number of disks which can be handled. Denote by g n the maximal such number as G varies over all such graphs with n vertices and S, D vary over the vertices.
Enumeration of (p,q)-parking functions
Discrete Mathematics, 2002
Parking functions are central in many aspects of combinatorics. We deÿne in this communication a generalization of parking functions which we call (p1; : : : ; p k )-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent conÿgurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p1; : : : ; p k )-parking functions as well as increasing ones. Les suites de parking se sont rà evà elà ees être au centre de di à erents probl emes combinatoires. Nous introduisons ici des k-uplets de suites qui les gà enà eralisent, et dont nous montrons qu'ils peuvent être interprà età es comme les conÿgurations rà ecurrentes de l'automate du tas de sable sur certains graphes. Nous à etablissons à egalement une correspondance avec un langage de Lukasiewicz, ce qui nous permet d'obtenir des rà esultats d'à enumà eration.
A Note on the Tower of Hanoi Problem with Evildoers
Engineering International
Recent literature considers the variant of the classical Tower of Hanoi problem with n (³ 1) discs, where r (1 £ r < n) discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Letting E(n, r) be the minimum number of moves required to solve the new variant, an explicit form of E(n, r) is available which depends on a positive integer constant N. This study investigates the properties of N.
The Tower of Hanoi Problem
Introduction This lecture is about the well-known Tower of Hanoi problem. The problem is discussed in many mathematical texts, and is often used in computing science and articial intelligence as an illustration of ecursion" as a problem-solving strategy. The discussion of the problem in The Magical Maze" by Ian Stewart is similar (although less deep) than that given here, so if there is anything you don't understand you might try looking there. The goal of this lecture is to bridge some of what you have learnt in MC1, in particular the use of induction but also graphs, with what you have done in MC2. I will use the Tower of Hanoi problem to explain the dierence between the WHAT", the HOW" and the WHY" of algorithm development and show how proof by induction is used to relate the WHAT" to the HOW", thus providing the WHY" in the context of this particular problem. 2 Problem Specication | the WHAT" The Tower of Hanoi problem comes from ...
Parking Functions, Shi Arrangements, and Mixed Graphs
2016
Abstract. The Shi arrangement is the set of all hyperplanes in Rn of the form xj − xk = 0 or 1 for 1 ≤ j < k ≤ n. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is (n+ 1)n−1. An unrelated combinatorial concept is that of a parking function, i.e., a sequence (x1, x2,..., xn) of positive integers that, when rearranged from smallest to largest, satisfies xk ≤ k. (There is an illustrative reason for the term parking function.) It turns out that the number of parking functions of length n also equals (n + 1)n−1, a result due to Konheim and Weiss from 1966. A natural problem consists of finding a bijection between the n-dimensional Shi arragnement and the parking functions of length n. Stanley and Pak (1996) and Athanasiadis and Linusson 1999) gave such (quite different) bijections. We will shed new light on the former bijection by taking a scenic route through certain mixed graphs. 1.
Pattern Avoidance in Parking Functions
arXiv (Cornell University), 2022
In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first considered by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more patterns of length 3, and we obtain a number of well-known combinatorial sequences as a result. Along the way, we find bijections between specific sets of pattern-avoiding parking functions and a number of combinatorial objects such as partitions of polygons and trees with certain restrictions.
On Optimal Solutions for the Bottleneck Tower of Hanoi Problem
Lecture Notes in Computer Science, 2007
We study two aspects of a generalization of the Tower of Hanoi puzzle. In 1981, D. Wood suggested its variant, where a bigger disk may be placed higher than a smaller one if their size difference is less than k. In 1992, D. Poole suggested a natural disk-moving strategy for this problem, but only in 2005, the authors proved it be optimal in the general case. We describe the family of all optimal solutions to this problem and present a closed formula for their number, as a function of the number of disks and k. Besides, we prove a tight bound for the diameter of the configuration graph of the problem suggested by Wood. Finally, we prove that the average length of shortest sequence of moves, over all pairs of initial and final configurations, is the same as the above diameter, up to a constant factor.
Orientations, Semiorders, Arrangements, and Parking Functions
The Electronic Journal of Combinatorics, 2012
It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph $G$, we define the $G$-semiorder arrangement and show that the Pak-Stanley labeling of its regions produces all $G$-parking functions.
Disks on a Tree: Analysis of a Combinatorial Game
2005
Anderson et al. [1] studied a combinatorial game on an infinite path that is started with n disks at a vertex and ends with the disks spread between k = bn=2c vertices to the left and to the right of the initial vertex. They showed that the number of steps the game takes to converge to the final configuration is ck 2 + o(k 2) for some constant c. We
Book Review of The Tower of Hanoi: Myths and Maths (Birkhäuser)
The Mathematics Enthusiast, 2014
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.