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J. Carlos Martínez Mori

Publications and source records attributed to J. Carlos Martínez Mori.

17 recordsLinked to original sources
arXiv2026

Parking Cascades: From the Simplest Sequence to Motzkin and Catalan

We introduce $k$-cascading parking functions, a parametrized variant of parking functions in which cars form bumping cascades of up to $k \geq 0$ cars. Setting $k = 0$ recovers classical parking functions, whereas $k = 1$ recovers MVP parking functions. Although parking functions and cascading parking functions are equivalent as sets, they are generally distinct as maps. Therefore, in this paper we consider the enumeration of the fibers of their outcomes. Our main result is a recursive, permutation pattern-based formula for the size of the fiber of any given permutation, for any given $k \geq 0$. When specialized to the longest word, the formula reduces to a family of integer sequences that interpolate between the simplest sequence ($k=0$), the Motzkin numbers ($k = 1$), and the Catalan numbers ($k\geq n-1$). When specialized to the set of layered permutations, the formula gives new combinatorial interpretations for the row sums of certain convolution triangles, including Motzkin and Catalan convolution triangles.

math.CO↗
arXiv2026

Linear Production Games with Non-transferable Utilities

We introduce non-transferable utility linear production (NTU LP) games, a non-transferable utility analogue of classical linear production games, as a framework for the study of cooperative behavior in the production or establishment of public goods with pooled resources. NTU LP games combine the game-theoretic tensions inherent in public decision-making with the modeling flexibility of linear programming. We derive structural properties regarding the non-emptiness, representability and complexity of the core, a solution concept that models the viability of cooperation. In particular, we provide fairly general sufficient conditions under which the core of an NTU LP game is guaranteed to be non-empty, prove that determining membership in the core is co-NP-complete, and develop a cutting plane algorithm to optimize various social welfare objectives subject to core membership. We apply these results in a data-driven case study on service plan optimization for the Chicago bus system. As our study illustrates, cooperation is necessary for the successful deployment of transportation service plans and similar public goods, but it may also have adverse or counterintuitive distributive implications.

cs.GT↗
arXiv2025

A Probabilistic Parking Process and Labeled IDLA

In 1966, Konheim and Weiss [33] introduced a now classical parking protocol. The deterministic process and its resultant objects, known as parking functions, have since become a favorite object of study in enumerative combinatorics. In our work, we introduce and study a probabilistic variant of the classical parking protocol, which is closely related to Internal Diffusion Limited Aggregation, or IDLA, introduced in 1991 by Diaconis and Fulton [19]. In particular, we compute the stationary distribution of this process when initiated with a particular class of initial preferences, of which weakly increasing parking functions are a subset. Furthermore, we compute the expected time it takes for the protocol to complete assuming all of the cars park, and prove that, in some cases, the parking process is negatively correlated. In addition, we study statistics of uniformly random weakly increasing parking functions such as the distribution of the last entry, the probability that a specific set of cars is lucky, and the expected number of lucky cars.

math.PR↗
arXiv2024

Who's the GOAT? Sports Rankings and Data-Driven Random Walks on the Symmetric Group

Given a collection of historical sports rankings, can one tell which player is the greatest of all time (i.e., the GOAT)? In this work, we design a data-driven random walk on the symmetric group to obtain a stationary distribution over player rankings, spanning across different time periods in sports history. We combine this distribution with a notion of stochastic dominance to obtain a partial order over the players. Compared to existing methods, our approach is distinct in that i) using historical rankings captures the evolution of value systems and facilitates player comparisons when head-to-head data is unavailable, and i) aggregating into a partial order formally comes to terms with the possibility that, while some player comparisons can be established conclusively, others are "too close to call." We implement our methods using publicly available data from the Association of Tennis Professionals (ATP) and the Women's Tennis Association (WTA). Our main findings indicate that Steffi Graf and Serena Williams are the ones that come ahead as the GOATs of the WTA. Likewise, the "Big Three," that is Novak Djokovic, Roger Federer, and Rafael Nadal, are the ones that come ahead as the GOATs of the ATP. As a secondary finding, we note major differences in terms of career and dominance longevity for top players across the associations. While initially motivated by this application in sports analytics, our methods can also be applied to other practical domains where deriving rankings from historical data can inform operational decisions, such as route planning logistics.

stat.AP↗
arXiv2024

Interval and $\ell$-interval Rational Parking Functions

Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.

math.CO↗
arXiv2024

Cost-sharing in Parking Games

In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from the total displacement statistic. We show that parking games are supermodular cost-sharing games, indicating that cooperation is difficult (i.e., their core is empty). Next, we study their Shapley value, which formalizes a notion of "fair" cost-sharing and amounts to charging each car for its expected marginal displacement under a random arrival order. Our main contribution is a polynomial-time algorithm to compute the Shapley value of parking games, in contrast with known hardness results on computing the Shapley value of arbitrary games. The algorithm leverages the permutation-invariance of total displacement, combinatorial enumeration, and dynamic programming. We conclude with open questions around an alternative solution concept for supermodular cost-sharing games and connections to other areas in combinatorics.

math.CO↗
arXiv2024

Parking functions, Fubini rankings, and Boolean intervals in the weak order of $\mathfrak{S}_n$

Let $\mathfrak{S}_n$ denote the symmetric group and let $W(\mathfrak{S}_n)$ denote the weak order of $\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W(\mathfrak{S}_n)$ and the total number of Boolean intervals of rank $k$ in $W(\mathfrak{S}_n)$. Furthermore, for any $π\in\mathfrak{S}_n$, we establish that the number of Boolean intervals in $W(\mathfrak{S}_n)$ with minimal element $π$ is a product of Fibonacci numbers. We conclude with some directions for further study.

math.CO↗
arXiv2024

What is a Parking Function?

In this expository article I describe classical results in the combinatorics of parking functions. Its English-Spanish translation is included. -- -- En este artículo de difusión matemática describo resultados clásicos en la combinatoria de funciones de parqueo. Su traducción español-inglés está incluida.

math.HO↗
arXiv2024

Boolean intervals in the weak Bruhat order of a finite Coxeter group

Given a Coxeter group $W$ with Coxeter system $(W,S)$, where $S$ is finite. We provide a complete characterization of Boolean intervals in the weak order of $W$ uniformly for all Coxeter groups in terms of independent sets of the Coxeter graph. Moreover, we establish that the number of Boolean intervals of rank $k$ in the weak order of $W$ is ${i_k(Γ_W)\cdot|W|}\,/\,2^{k}$, where $Γ_W$ is the Coxeter graph of $W$ and $i_k(Γ_W)$ is the number of independent sets of size $k$ of $Γ_W$ when $W$ is finite. Specializing to $A_n$, we recover the characterizations and enumerations of Boolean intervals in the weak order of $A_n$ given in arXiv:2306.14734. We provide the analogous results for types $C_n$ and $D_n$, including the related generating functions and additional connections to well-known integer sequences.

math.CO↗
arXiv2024

Modeling Processes of Neighborhood Change

An urban planner might design the spatial layout of transportation amenities so as to improve accessibility for underserved communities -- a fairness objective. However, implementing such a design might trigger processes of neighborhood change that change who benefits from these amenities in the long term. If so, has the planner really achieved their fairness objective? Can algorithmic decision-making anticipate second order effects? In this paper, we take a step in this direction by formulating processes of neighborhood change as instances of no-regret dynamics; a collective learning process in which a set of strategic agents rapidly reach a state of approximate equilibrium. We mathematize concepts of neighborhood change to model the incentive structures impacting individual dwelling-site decision-making. Our model accounts for affordability, access to relevant transit amenities, community ties, and site upkeep. We showcase our model with computational experiments that provide semi-quantitative insights on the spatial economics of neighborhood change, particularly on the influence of residential zoning policy and the placement of transit amenities.

cs.MA↗
arXiv2023

Sparse Graphical Designs via Linear Programming

Graphical designs are a framework for sampling and numerical integration of functions on graphs. In this note, we introduce a method to address the trade-off between graphical design sparsity and accuracy. We show how to obtain sparse graphical designs via linear programming and design objective functions that aim to maximize their accuracy. We showcase our approach using yellow taxicab data from New York City.

math.OC↗
arXiv2023

Permutation Invariant Parking Assortments

We introduce parking assortments, a generalization of parking functions with cars of assorted lengths. In this setting, there are $n\in\mathbb{N}$ cars of lengths $\mathbf{y}=(y_1,y_2,\ldots,y_n)\in\mathbb{N}^n$ entering a one-way street with $m=\sum_{i=1}^ny_i$ parking spots. The cars have parking preferences $\mathbf{x}=(x_1,x_2,\ldots,x_n)\in[m]^n$, where $[m]:=\{1,2,\ldots,m\}$, and enter the street in order. Each car $i \in [n]$, with length $y_i$ and preference $x_i$, follows a natural extension of the classical parking rule: it begins looking for parking at its preferred spot $x_i$ and parks in the first $y_i$ contiguously available spots thereafter, if there are any. If all cars are able to park under the preference list $\mathbf{x}$, we say $\mathbf{x}$ is a parking assortment for $\mathbf{y}$. Parking assortments also generalize parking sequences, introduced by Ehrenborg and Happ, since each car seeks for the first contiguously available spots it fits in past its preference. Given a parking assortment $\mathbf{x}$ for $\mathbf{y}$, we say it is permutation invariant if all rearrangements of $\mathbf{x}$ are also parking assortments for $\mathbf{y}$. While all parking functions are permutation invariant, this is not the case for parking assortments in general, motivating the need for characterization of this property. Although obtaining a full characterization for arbitrary $n\in\mathbb{N}$ and $\mathbf{y}\in\mathbb{N}^n$ remains elusive, we do so for $n=2,3$. Given the technicality of these results, we introduce the notion of minimally invariant car lengths, for which the only invariant parking assortment is the all ones preference list. We provide a concise, oracle-based characterization of minimally invariant car lengths for any $n\in\mathbb{N}$. Our results around minimally invariant car lengths also hold for parking sequences.

math.CO↗
arXiv2023

Lucky Cars and the Quicksort Algorithm

Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of $2(n+1)H_n - 4n$ comparisons on an array of size $n$ ordered uniformly at random, where $H_n = \sum_{i=1}^n\frac{1}{i}$ is the $n$th harmonic number. Therefore, it makes $n!\left[2(n+1)H_n - 4n\right]$ comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of $n$ cars parking on a one-way street with $n$ parking spots resulting in exactly $n-1$ lucky cars (i.e., cars that park in their preferred spot). For $n\geq 2$, both counts satisfy the second order recurrence relation $ f_n=2nf_{n-1}-n(n-1)f_{n-2}+2(n-1)! $ with $f_0=f_1=0$.

math.CO↗
arXiv2023

On the Outcome Map of MVP Parking Functions: Permutations Avoiding 321 and 3412, and Motzkin Paths

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, they park there and the car parked in that spot is bumped to the next unoccupied 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 $\mathfrak{S}_n$, which does not avoid the pattern 321) is given by the $n$th 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.

math.CO↗
arXiv2022

On Parking Functions and The Tower of Hanoi

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.

math.CO↗
arXiv2021

On the Request-Trip-Vehicle Assignment Problem

The request-trip-vehicle assignment problem is at the heart of a popular decomposition strategy for online vehicle routing. In this framework, assignments are done in batches in order to exploit any shareability among vehicles and incoming travel requests. We study a natural ILP formulation and its LP relaxation. Our main result is an LP-based randomized rounding algorithm that, whenever the instance is feasible, leverages mild assumptions to return an assignment whose: i) expected cost is at most that of an optimal solution, and ii) expected fraction of unassigned requests is at most $1/e$. If trip-vehicle assignment costs are $α$-approximate, we pay an additional factor of $α$ in the expected cost. We can relax the feasibility requirement by considering the penalty version of the problem, in which a penalty is paid for each unassigned request. We find that, whenever a request is repeatedly unassigned after a number of rounds, with high probability it is so in accordance with the sequence of LP solutions and not because of a rounding error. We additionally introduce a deterministic rounding heuristic inspired by our randomized technique. Our computational experiments show that our rounding algorithms achieve a performance similar to that of the ILP at a reduced computation time, far improving on our theoretical guarantee. The reason for this is that, although the assignment problem is hard in theory, the natural LP relaxation tends to be very tight in practice.

cs.DS↗