Symmetry in Tree Parking Distributions

Amanuel T. Getachew

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arXiv · 2509.00436

Symmetry in Tree Parking Distributions

Amanuel T. Getachew

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Abstract

In this paper, we explore parking distributions on caterpillar trees, focusing on two primary statistics: the number of lucky cars and the frequency with which cars prefer specific parking spaces. We use first-return decomposition to reveal a symmetry in their joint distribution and develop a $q, t$-analog of the Fuss-Catalan generating function. We prove that this generating function exhibits specific symmetry and satisfies a functional equation. Additionally, we extend our findings to any m statistics that satisfy certain criteria, presenting a concrete example of such $m$ statistics to illustrate the broader applicability of our results.

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BibTeX RIS

Amanuel T. Getachew. 2025-08-30. Symmetry in Tree Parking Distributions. https://arxiv.org/abs/2509.00436

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