The Defective Parking Space and Defective Kreweras Numbers
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arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Garcia, Rebecca E. Harris, Pamela E. Moon, Alex Ortiz, Aaron Quesada, Lauren J. Sánchez, Cynthia Marie Rivera Williams II, Dwight Anderson Wilson, Alexander N. |
| author_facet | Garcia, Rebecca E. Harris, Pamela E. Moon, Alex Ortiz, Aaron Quesada, Lauren J. Sánchez, Cynthia Marie Rivera Williams II, Dwight Anderson Wilson, Alexander N. |
| contents | A defective $(m,n)$-parking function with defect $d$ is a parking function with $m$ cars attempting to park on a street with $n$ parking spots in which exactly $d$ cars fail to park. We establish a way to compute the defect of a defective $(m,n)$-parking function and show that the defect of a parking function is invariant under the action of $\mathfrak{S}_m$, the symmetric group on $[m]=\{1,2,\ldots,m\}$. We introduce the defective parking space ${\sf DPark}_{m,n}$ spanned by defective parking functions and describe its Frobenius characteristic as an $\mathfrak{S}_m$ representation graded by defect via coefficients $\mathrm{Krew}_{d,n}(λ)$ called defective Kreweras numbers. We provide a conjectured formula for $\mathrm{Krew}_{d,n}(λ)$ for sufficiently large $n$. We also show that the set of nondecreasing defective $(m,n)$-parking functions with defect $d$ are in bijection with the set of standard Young tableaux of shape $(n + d, m - d)$. This implies that the number of $\mathfrak{S}_m$-orbits of defective $(m,n)$-parking functions with defect $d$ is given by $\frac{n-m+2d+1}{n+d+1}\binom{m+n}{n+d}$. We also give a multinomial formula for the size of an $\mathfrak{S}_m$-orbit of a nondecreasing $(m,n)$-parking function with defect $d$. We conclude by using these results to give a new formula for the number of defective parking functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Defective Parking Space and Defective Kreweras Numbers Garcia, Rebecca E. Harris, Pamela E. Moon, Alex Ortiz, Aaron Quesada, Lauren J. Sánchez, Cynthia Marie Rivera Williams II, Dwight Anderson Wilson, Alexander N. Combinatorics 05A19 (primary) 05A05, 20C30 (secondary) A defective $(m,n)$-parking function with defect $d$ is a parking function with $m$ cars attempting to park on a street with $n$ parking spots in which exactly $d$ cars fail to park. We establish a way to compute the defect of a defective $(m,n)$-parking function and show that the defect of a parking function is invariant under the action of $\mathfrak{S}_m$, the symmetric group on $[m]=\{1,2,\ldots,m\}$. We introduce the defective parking space ${\sf DPark}_{m,n}$ spanned by defective parking functions and describe its Frobenius characteristic as an $\mathfrak{S}_m$ representation graded by defect via coefficients $\mathrm{Krew}_{d,n}(λ)$ called defective Kreweras numbers. We provide a conjectured formula for $\mathrm{Krew}_{d,n}(λ)$ for sufficiently large $n$. We also show that the set of nondecreasing defective $(m,n)$-parking functions with defect $d$ are in bijection with the set of standard Young tableaux of shape $(n + d, m - d)$. This implies that the number of $\mathfrak{S}_m$-orbits of defective $(m,n)$-parking functions with defect $d$ is given by $\frac{n-m+2d+1}{n+d+1}\binom{m+n}{n+d}$. We also give a multinomial formula for the size of an $\mathfrak{S}_m$-orbit of a nondecreasing $(m,n)$-parking function with defect $d$. We conclude by using these results to give a new formula for the number of defective parking functions. |
| title | The Defective Parking Space and Defective Kreweras Numbers |
| topic | Combinatorics 05A19 (primary) 05A05, 20C30 (secondary) |
| url | https://arxiv.org/abs/2405.14635 |