Parking Garage Functions

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College. This project is about a generalization of parking functions called parking garage functions. Parking functions have been well studied, but the concept of parking garage functions is new and introduced in the project. Parking garage functions are sequences that represent the parking garage level preferences of cars which lead to all cars parking on a level after a systematic placement. We found a recursive formula for the number of sequences that are a parking garage function. We also found a closed formula for a subset of parking garage functions, descending parking garage functions, via a bijection between descending parking garage functions and Dyck paths which are paths on a rectangular grid which only take right and upward steps starting at the origin and remain under a positively sloped diagonal that goes through the origin

A family of bijections between G-parking functions and spanning trees

For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j edges going from j to G-U. We construct a family of bijective maps between the set P_G of G-parking functions and the set T_G of spanning trees of G rooted at 0, thus providing a combinatorial proof of |P_G| = |T_G|.Comment: 11 pages, 4 figures; a family of bijections containing the two original bijections is presented; submitted to J. Combinatorial Theory, Series

Connecting kk-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.Comment: 14 pages; corrected a mistake in the last part of v1 regarding the factor of 1/2 in the upper bounds obtaine

New combinatorial perspectives on MVP parking functions and their outcome map

In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars have the same preferred spot. We study a model introduced by Harris, Kamau, Mori, and Tian in recent work, called the MVP parking problem. In this model, priority is given to the cars arriving later in the sequence. When a car finds its preferred spot occupied by a previous car, it "bumps" that car out of the spot and parks there. The earlier car then has to drive on, and parks in the first available spot it can find. If all cars manage to park through this procedure, we say that the list of preferences is an MVP parking function. We study the outcome map of MVP parking functions, which describes in what order the cars end up. In particular, we link the fibres of the outcome map to certain subgraphs of the inversion graph of the outcome permutation. This allows us to reinterpret and improve bounds from Harris et al. on the fibre sizes. We then focus on a subset of parking functions, called Motzkin parking functions, where every spot is preferred by at most two cars. We generalise results from Harris et al., and exhibit rich connections to Motzkin paths. We also give a closed enumerative formula for the number of MVP parking functions whose outcome is the complete bipartite permutation. Finally, we give a new interpretation of the MVP outcome map in terms of an algorithmic process on recurrent configurations of the Abelian sandpile model.Comment: 33 pages, 25 figures, 6 table

Rational Parking Functions and LLT Polynomials

We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.Comment: 14 pages, 8 figure