On the Structure of Permutation Invariant Parking

We continue the study of parking assortments, a generalization of parking functions introduced by Chen, Harris, Mart\'{i}nez, Pab\'{o}n-Cancel, and Sargent. Given $n$ cars of lengths $\mathbf{y}=(y_1,y_2,\dots,y_n) \in \mathbb{N}^n$, we focus on the sets $\mathsf{PA}^{\mathrm{inv}}_n(\mathbf{y})$ and $\mathsf{PA}^{\mathrm{inv},\uparrow}_n(\mathbf{y})$ of permutation invariant (resp. nondecreasing) parking assortments for $\mathbf{y}$. For $\mathbf{x} \in \mathsf{PA}^{\mathrm{inv}}_n(\mathbf{y})$, we introduce the degree of $\mathbf{x}$, the number of non-$1$ entries of $\mathbf{x}$, and the characteristic $\chi(\mathbf{y})$ of $\mathbf{y}$, the greatest degree of $\mathbf{z} \in \mathsf{PA}^{\mathrm{inv}}_n(\mathbf{y})$. We establish direct necessary conditions for $\mathbf{y}$ with $\chi(\mathbf{y})=0$ and a characterization for $\mathbf{y}$ with $\chi(\mathbf{y})=n-1$. For the latter, we derive a closed form for its invariant parking set and enumerate its size using properties of the Pitman-Stanley polytope. Next, we prove closure and embedding properties of the invariant parking set. We apply these results to study the degree as a function and the characteristic under sequences of successive prefix length vectors. We then examine the invariant solution set $\mathcal{W}(\mathbf{y})=\{ w \in \mathbb{N}:(1^{n-1},w) \in \mathsf{PA}^{\mathrm{inv}}_n(\mathbf{y}) \}$. We obtain tight upper bounds of this set and prove that its size is at most $2^{n-1}$, providing constraints on the subsequence sums of $\mathbf{y}$ for equality to hold. Finally, we show that if $\mathbf{x} \in \mathsf{PA}^{\mathrm{inv},\uparrow}_n(\mathbf{y})$, then $\mathbf{x} \in \{ 1 \}^{n-\chi(\mathbf{y})} \times \mathcal{W}(\mathbf{y})^{\chi(\mathbf{y})}$, which implies a new upper bound on $|\mathsf{PA}^{\mathrm{inv},\uparrow}_n(\mathbf{y})|$. Our results generalize several theorems by Chen et al.