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.
Categories: math.CO