The Defective Parking Space and Defective Kreweras Numbers

TL;DR

This paper introduces defective parking functions and spaces, explores their symmetry properties, and connects them to Young tableaux and Kreweras numbers, providing formulas for counting and characterizing these structures.

Contribution

It defines defective parking functions and spaces, establishes their symmetry invariance, and links them to Young tableaux and Kreweras numbers, including conjectured formulas and counting methods.

Findings

Defective parking functions are invariant under symmetric group actions.

The set of nondecreasing defective parking functions corresponds to Young tableaux of specific shapes.

New formulas for counting defective parking functions and their orbits are derived.

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 $\mathrm{Krew}_{d,n}(\lambda)$ called defective Kreweras numbers. We provide a conjectured formula for $\mathrm{Krew}_{d,n}(\lambda)$ 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.