Generalised Random Parking for Trapeziums on a Strip
Abstract
In this article, we study a generalisation of Rényi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.
Related
- Distribution of the First Hitting Point on a Boundary Set for a Random Walk in a Lattice Strip
- Pattern restricted parking functions with limited preferences
- Characterizing k-Naples parking functions and refining bounds via obstructed parking functions
- A new generalization of parking functions: ρ-Naples parking functions
- A parking function analog of the Schröder numbers
BibTeX
@misc{kramerbang2026generalised,
title = {{Generalised Random Parking for Trapeziums on a Strip}},
author = {David Kramer-Bang and Stjepan Šebek},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02491},
eprint = {2609.02491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02491},
}
RIS
TY - UNPB
TI - Generalised Random Parking for Trapeziums on a Strip
AU - David Kramer-Bang
AU - Stjepan Šebek
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02491
UR - https://arxiv.org/abs/2609.02491
AB - In this article, we study a generalisation of Rényi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.
ER -