The symmetric functions catalog

An overview of symmetric functions and related topics

2024-05-06

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 that (see [Ex. 7.48 (f), Sta01]) that

\begin{align*} \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{align*}

For the generalization to $(r,k)$-parking functions, see [SW18b].

In [Sta24], Richard 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.

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