Lucky Cars in Fubini Rankings and Unit Fubini Rankings
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arXiv
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915588728684544 |
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| author | Barreto, Camilo Beerbower, Melissa Elder, Jennifer Harris, Pamela E. Martinez, Lucy Ramírez, José L. Ramírez, Samuel Shirley, Grant Vásquez, Julio C. |
| author_facet | Barreto, Camilo Beerbower, Melissa Elder, Jennifer Harris, Pamela E. Martinez, Lucy Ramírez, José L. Ramírez, Samuel Shirley, Grant Vásquez, Julio C. |
| contents | We study lucky cars in subsets of parking functions, called Fubini rankings and unit Fubini rankings. A Fubini ranking is a sequence of nonnegative integers that encodes a valid ranking of competitors, where ties are allowed. A car (or competitor) is said to be lucky if it is the first instance of that rank appearing in the sequence. We present combinatorial characterizations and enumeration formulas for lucky cars in both Fubini rankings and unit Fubini rankings, and establish connections between these objects and ordered set partitions, as well as integer compositions. To obtain our results, we use several techniques to enumerate statistics over these families of objects.
In particular, we employ generating functions, bijective and combinatorial arguments, recurrence relations, and Zeilberger's creative telescoping method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lucky Cars in Fubini Rankings and Unit Fubini Rankings Barreto, Camilo Beerbower, Melissa Elder, Jennifer Harris, Pamela E. Martinez, Lucy Ramírez, José L. Ramírez, Samuel Shirley, Grant Vásquez, Julio C. Combinatorics 05A05, 05A15, 05A18 We study lucky cars in subsets of parking functions, called Fubini rankings and unit Fubini rankings. A Fubini ranking is a sequence of nonnegative integers that encodes a valid ranking of competitors, where ties are allowed. A car (or competitor) is said to be lucky if it is the first instance of that rank appearing in the sequence. We present combinatorial characterizations and enumeration formulas for lucky cars in both Fubini rankings and unit Fubini rankings, and establish connections between these objects and ordered set partitions, as well as integer compositions. To obtain our results, we use several techniques to enumerate statistics over these families of objects. In particular, we employ generating functions, bijective and combinatorial arguments, recurrence relations, and Zeilberger's creative telescoping method. |
| title | Lucky Cars in Fubini Rankings and Unit Fubini Rankings |
| topic | Combinatorics 05A05, 05A15, 05A18 |
| url | https://arxiv.org/abs/2510.27574 |