arXiv subjects

Roger Tian

Publications and source records attributed to Roger Tian.

9 recordsLinked to original sources

arXiv2026

Characterizing $k$-Naples parking functions and refining bounds via obstructed parking functions

The Naples parking functions are a major generalization of the classical parking functions, where cars have more options to park because they are allowed to check (and possibly park) backward up to a fixed number of spots before driving forward as in the classical case. In a previous work, using involutions we established an injection from the Naples parking functions to a special set of obstructed parking functions, which we referred to as \textit{left obstructed}, thereby immediately obtaining a crude upper bound on the former set. We continue that work here by identifying the image of this injection, providing a characterization of Naples parking functions as a certain (smaller) set of obstructed parking functions, which we will refer to as \textit{long left obstructed}. We also derive a formula for the number of left obstructed parking functions. These combined allow us to improve the upper bound on the Naples parking functions. In addition, we demonstrate how lower bounds for this set can be obtained. As will become clear, the long left obstructed parking functions have certain combinatorial features that are useful in estimating the cardinality of the set, which are not obvious from solely looking at Naples parking functions.

math.CO↗

arXiv2025

Counting occurrences of a pattern in a binary word

Enumerating the number of times one word occurs in another is a much-studied combinatorial subject. By utilizing a method that we call ``lexicographic extreme referencing'', we provide a formula for computing occurrences of one binary word in another. We then study $B_{n,p}(k)$, the number of binary words of length $n$ containing a given word $p$ exactly $k$ times. For this purpose, we first use lexicographic extreme referencing to provide an algorithm for constructing all words $w$ that contain a given word $p$. Afterward, we give a modified version of this algorithm for constructing the subset of binary words that are ``primitive'' with respect to $p$, and we discuss approaches for finding $B_{n,p}(k)$ via primitive words.

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↗

arXiv2021

Connecting $k$-Naples parking functions and obstructed parking functions via involutions

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.

math.CO↗

arXiv2020

Parking functions on directed stars and orientation reversal with an extension to general directed trees

Parking functions, classically defined in terms of cars with preferred parking spots on a directed path attempting to park there, arise in many combinatorial situations and have seen various generalizations. In particular, parking functions have been defined for general digraphs, which yields many more enumeration problems. For example, in a directed tree whose edges are orientated away from the root, it is unknown in general how the number of parking functions on it changes once the orientation is reversed, even in the case when the tree is a star. We show that this orientation reversal results in more parking functions on the directed star in most cases, after which we extend these methods to show that this also results in more parking functions on the general directed tree if, in some sense, the number of vertices greatly exceeds the number of cars.

math.CO↗

arXiv2016

Characterization of ${\cal B}(\infty)$ using marginally large tableaux and rigged configurations in the $A_n$ case via integer sequences

Rigged configurations are combinatorial objects prominent in the study of solvable lattice models. Marginally large tableaux are semi-standard Young tableaux of special form that give a realization of the crystals ${\cal B}(\infty)$. We introduce cascading sequences to characterize marginally large tableaux. Then we use cascading sequences and a non-explicit crystal isomorphism between marginally large tableaux and rigged configurations to give a characterization of the latter set, and to give an explicit bijection between the two sets.

math.CO↗

arXiv2014

Generalizations of an Expansion Formula for Top to Random Shuffles

In the top to random shuffle, the first a cards are removed from a deck of n cards 12 \cdots n and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element B_a, which we define formally in Section 2, of the algebra Q[S_n]. For a = 1, Adriano Garsia in "On the Powers of Top to Random Shuffling" (2002) derived an expansion formula for B_1^k for k \leq n, though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product B_{a_1}B_{a_2} \cdots B_{a_k} where a_1, \ldots, a_k are positive integers, from which an improved version of Garsia's aforementioned formula follows. We show some applications of this formula for B_{a_1}B_{a_2} \cdots B_{a_k}, which include enumeration and calculating probabilities. Then for an arbitrary group G we define the group of G-permutations S_n^G := G \wr S_n and further generalize the aforementioned expansion formula to the algebra Q[S_n^G] for the case of finite G, and we show how other similar expansion formulae in Q[S_n] can be generalized to Q[S_n^G].

math.CO↗

arXiv2009

Identities of the Function f(x,y) = x^2 + y^3

Harvey Friedman asked in 1986 whether the function f(x,y) = x^2 + y^3 on the real plane R^2 satisfies any identities; examples of identities are commutativity and associativity. To solve this problem of Friedman, we must either find a nontrivial identity involving expressions formed by recursively applying f to a set of variables {x_1,x_2, ..., x_n} that holds in the real numbers or to prove that no such identities hold. In this paper, we will solve certain special cases of Friedman's problem and explore the connection between this problem and certain Diophantine equations.

math.GM↗

arXiv2009

On the Diophantine Equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h

This paper is a continuation of [1], in which I studied Harvey Friedman's problem of whether the function f(x,y) = x^2 + y^3 satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the exponential Diophantine equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h into subcases that are easier to analyze. Then we will solve an equation obtained by imposing a restriction on one of these subcases, after which we will solve a generalization of this equation.

math.GM↗