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Parking with Frustrated Drivers — arXiv:2609.35638

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.35638
License (as archived): CC0-1.0
Archived: 2026-10-07T23:29:19.157Z
SHA-256 of the archived text: 1a71e9e41bbeced98d6bf93b16629faaea740ca9637cd78bb784f0fff2b49a44
Stable URL of this page: https://metraipolis.com/documento/dc_08cb323cd327

How to cite

Parking with Frustrated Drivers — arXiv:2609.35638 — arXiv (en), https://arxiv.org/abs/2609.35638 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_08cb323cd327 (Metraipolis archive dc_08cb323cd327, sha256 1a71e9e41bbeced98d6bf93b16629faaea740ca9637cd78bb784f0fff2b49a44)

BibTeX

@misc{dc08cb323cd327,
  title = {Parking with Frustrated Drivers — arXiv:2609.35638},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.35638},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_08cb323cd327; sha256 1a71e9e41bbeced98d6bf93b16629faaea740ca9637cd78bb784f0fff2b49a44},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.35638. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Imagine there are $n$ cars lined up along a one-way street containing $n$ spots. Each car contains a group of friends, including a reluctant driver. Each car has a preferred spot and cars enter one by one. The cars drive to their preferred spot and if it is empty park there. If it is not empty, a friend in the back yells out ``Hey! You should have driven faster!". Frustrated by this, the driver continues down the road until they find the last unoccupied spot (if one exists) and parks there. We say a sequence $(a_1,a_2,\dots, a_n)$ of preferred spots is a frustrated parking function if all cars can park under this rule. In this paper, we study the enumerative properties of frustrated parking functions. In particular, we show that the number of frustrated parking functions of length $n$ is $(2n-1)!!$. This is done by associating frustrated parking functions with height labeled Dyck paths. Using this association, we are then able to better understand the sets of lucky cars and lucky spots for frustrated parking functions. We show that the frustrated parking functions of length $n$ where the first $k$ cars (or first $k$ spots) are lucky is given by $k!S(n,k)$ where $S(n,k)$ is the Stirling number of the second kind. This in turn implies that the number of frustrated parking functions where once a car (or spot) is unlucky, the remaining cars (or spots) are unlucky is counted by the $n^{th}$ Fubini number. We also show that the number of frustrated parking functions with $k$ lucky cars (or spots) is given by the second order Eulerian number.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.35638 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T23:08:10.869Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).