Parking function symmetric functions

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#Parking function symmetric functions

A parking function of size $n$ is a list of positive integers $(a_1,a_2,\dotsc,a_n)$ with the property that if arranged in increasing order, then the $i$-th entry does not exceed $i.$ We therefore have a natural action of $\symS_n$ on the set of size $n$ parking functions. See [Yan15] for a background on parking functions.

Parking function symmetric functions first appeared in the study of diagonal invariants, see [Hai94]. The Parking function symmetric function $\parking_n(\xvec)$ is defined as the Frobenius characteristic of the above group action. This means (see [Ex. 7.48 (f), Sta01]) that \[\begin{aligned} \parking_n(\xvec) &= \sum_{\lambda \vdash n} (n+1)^{\ell(\lambda)-1} \frac{\powerSum_\lambda}{z_\lambda} \\ &=\sum_{\lambda \vdash n} \frac{1}{n+1} \schurS_\lambda(1^{n+1}) \schurS_\lambda \\ &=\sum_{\lambda \vdash n} \frac{n(n-1)\dotsm (n-\ell(\lambda)+2)}{m_1(\lambda)!\dotsm m_n(\lambda)!} \completeH_\lambda. \end{aligned}\] For the generalization to $(r,k)$-parking functions, see [SW18].

The parking function symmetric functions also occur as the path case of the Möbius symmetric function of a graph. R. S. G. D'León and M. L. Wachs prove \[(-1)^{n-1}M_{P_n}=\omega\parking_{n-1}\] in [DW26].

In [Sta24], R. Stanley introduces a shifted analog of $\parking_n(\xvec).$ These can be obtained from $\parking_n(\xvec)$ by applying the maps $\powerSum_{2i+1} \mapsto 2 \powerSum_{2i+1}$ and $\powerSum_{2i}\mapsto 0,$ when expressed in the power-sum basis.

A representation-theoretical model for the shifted parking functions is obtained in [HK25].

Bibliography

[DW26]Rafael S. González D’León and Michelle L. Wachs. Weighted bond posets and a new chromatic symmetric function. arXiv:2608.08692v1, 2026.

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@article{DLeonWachs2026x,
  author = {Rafael S. Gonz{\'a}lez D'Le{\'o}n and Michelle L. Wachs},
  title = {Weighted bond posets and a new chromatic symmetric function},
  year = {2026},
  eprint = {2608.08692v1},
  url = {https://arxiv.org/abs/2608.08692v1},
  journal = {arXiv e-prints}
}

[Hai94]Mark D. Haiman. Conjectures on the quotient ring by diagonal invariants. Journal of Algebraic Combinatorics, 3(1):17–76, 1994.

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@article{Haiman1994,
  doi = {10.1023/a:1022450120589},
  url2 = {https://doi.org/10.1023/a:1022450120589},
  title = {Conjectures on the Quotient Ring by Diagonal Invariants},
  year = {1994},
  publisher = {Springer Science and Business Media {LLC}},
  volume = {3},
  number = {1},
  pages = {17--76},
  author = {Mark D. Haiman},
  journal = {Journal of Algebraic Combinatorics}
}

[HK25]Zachary Hamaker and Jesse Kim. Odd shifted parking functions. arXiv:2505.10763, 2025.

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@article{HamakerKim2025x,
Author = {Zachary Hamaker and Jesse Kim},
Title = {Odd Shifted Parking Functions},
Year = {2025},
Eprint = {2505.10763},
  url = {https://arxiv.org/abs/2505.10763},
journal = {arXiv e-prints}
}

[Sta24]Richard P. Stanley. A shifted parking function symmetric function. arXiv:2405.02164, 2024.

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@article{Stanley2024x,
Author = {Richard P. Stanley},
Title = {A Shifted Parking Function Symmetric Function},
Year = {2024},
Eprint = {2405.02164},
  url = {https://arxiv.org/abs/2405.02164},
journal = {arXiv e-prints}
}

[Sta01]Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First, 2001.

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@book{StanleyEC2,
    author = {Richard P. Stanley},
    title = {Enumerative {C}ombinatorics: {V}olume 2},
    publisher = {Cambridge University Press},
    year = {2001},
    edition = {First},
    isbn = {0521789877},
    doi = {10.1017/CBO9780511609589}
}

[SW18]Richard P. Stanley and Yinghui Wang. Some aspects of $(r, k)$-parking functions. Journal of Combinatorial Theory, Series A, 159:54–78, October 2018.

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@article{StanleyWang2018,
  title = {Some aspects of $(r, k)$-parking functions},
  volume = {159},
  ISSN = {0097-3165},
  url = {http://dx.doi.org/10.1016/j.jcta.2018.05.003},
  DOI = {10.1016/j.jcta.2018.05.003},
  journal = {Journal of Combinatorial Theory,  Series A},
  publisher = {Elsevier BV},
  author = {Stanley,  Richard P. and Wang,  Yinghui},
  year = {2018},
  month = oct,
  pages = {54–78}
}

[Yan15]Catherine H. Yan. Parking functions. Handbook of enumerative combinatorics:835–893, March 2015.

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@incollection{Yan2015,
  doi = {10.1201/b18255-14},
  url2 = {https://doi.org/10.1201/b18255-14},
  year = {2015},
  month = mar,
  publisher = {Chapman and Hall/{CRC}},
  pages = {835--893},
  author = {Catherine H. Yan},
  title = {Parking Functions},
  booktitle = {Handbook of Enumerative Combinatorics}
}