The Defective Parking Space and Defective Kreweras Numbers

Abstract

A defective $(m,n)$-parking function with defect $d$ is a parking function with $m$ cars attempting to park on a street with $n$ parking spots in which exactly $d$ cars fail to park. We establish a way to compute the defect of a defective $(m,n)$-parking function and show that the defect of a parking function is invariant under the action of $\mathfrak{S}_m$, the symmetric group on $[m]=\{1,2,\ldots,m\}$. We introduce the defective parking space ${\sf DPark}_{m,n}$ spanned by defective parking functions and describe its Frobenius characteristic as an $\mathfrak{S}_m$ representation graded by defect via coefficients $\mathrm{Krew}_{d,n}(λ)$ called defective Kreweras numbers. We provide a conjectured formula for $\mathrm{Krew}_{d,n}(λ)$ for sufficiently large $n$. We also show that the set of nondecreasing defective $(m,n)$-parking functions with defect $d$ are in bijection with the set of standard Young tableaux of shape $(n + d, m - d)$. This implies that the number of $\mathfrak{S}_m$-orbits of defective $(m,n)$-parking functions with defect $d$ is given by $\frac{n-m+2d+1}{n+d+1}\binom{m+n}{n+d}$. We also give a multinomial formula for the size of an $\mathfrak{S}_m$-orbit of a nondecreasing $(m,n)$-parking function with defect $d$. We conclude by using these results to give a new formula for the number of defective parking functions.

Figures (4)

• Figure 1: Parking position of cars with preference list $(3,5,5,6,9,9,10)\in\mathrm{DPF}^\uparrow_{7,9,2}$.
• Figure 2: A $(7,5)$-lattice path with $\mathrm{dip}(\mathbf{w})=2$ and $\mathrm{runs}(\mathbf{w})=(2,1,1,1)$.
• Figure 3: Illustration depicting how we use the bijections defined in this section to inductively establish \ref{['thm:main-bij']}. The maps $\pi$ denote projection to the first coordinate.
• Figure 4: The lattice path $\mathbf{w}$ corresponding to $\mathbf{x}=(1,1,2,3,5,5,6)$ as well as its conjugate lattice path $\bar{\mathbf{w}}$ corresponding to $\bar{\mathbf{x}} = (2,4,4,5,6)$. This figure also illustrates how the line $y=x+2$ is taken to the line $y=x$ by conjugation.

Theorems & Definitions (70)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Definition 2.2
• Definition 2.3
• Example 2.4