Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.03549

On the combinatorics of $k$-Naples parking functions and parking strategies

Francesco Verciani
View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose a characterization of $k$-Naples parking functions in terms of subsequences with the structure of a complete $k$-Naples parking function. We define complete parking preferences by requiring that for all $j=2,\dots,n$, the number of cars having preference at least $j$ is strictly greater than the number of spots in $[j,n]$. We also provide a characterization of permutation invariant $k$-Naples parking functions. Finally, we introduce a generalization of the parking problem where each car is given its own parking rule, by defining parking strategies as vectors of rules that allow all cars to park. Given a parking preference, we also investigate ways to find parking strategies that minimize certain natural parameters, such as the total number of backward steps, or the number of cars that need to drive backwards.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Francesco Verciani. 2023-11-06. On the combinatorics of $k$-Naples parking functions and parking strategies. https://arxiv.org/abs/2311.03549

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗