2023/11/27 by Chen, Douglas M. · 1 citation
#05A15 #05A19 #05D05 #05E18 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.15699
We continue the study of parking assortments, a generalization of parking functions introduced by Chen, Harris, Martínez, Pabón-Cancel, and Sargent. Given n cars of lengths y=(y1,y2,…,yn) ∈ ℕn, we focus on the sets PAinvn(y) and PAinv,\uparrown(y) of permutation invariant (resp. nondecreasing) parking assortments for y. For x ∈ PAinvn(y), we introduce the degree of x, the number of non-1 entries of x, and the characteristic χ(y) of y, the greatest degree of z ∈ PAinvn(y). We establish direct necessary conditions for y with χ(y)=0 and a characterization for y with χ(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 W(y)=\ w ∈ ℕ:(1n-1,w) ∈ PAinvn(y) \. We obtain tight upper bounds of this set and prove that its size is at most 2n-1, providing constraints on the subsequence sums of y for equality to hold. Finally, we show that if x ∈ PAinv,\uparrown(y), then x ∈ \ 1 \n-χ(y) × W(y)χ(y), which implies a new upper bound on |PAinv,\uparrown(y)|. Our results generalize several theorems by Chen et al.