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 $[m]=\{1,2,\dots ,m\}$. We introduce the defective parking space ${\mathsf{DPark}}_{m,n}$ spanned by defective parking functions and describe its Frobenius characteristic as an ${\mathfrak{S}}_m$ representation graded by defect via coefficients ${\mathrm{Krew}}_{d,n}(\hat{I}»)$ called defective Kreweras numbers. We provide a conjectured formula for ${\mathrm{Krew}}_{d,n}(\hat{I}»)$ 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 $\frac{n-m+2d+1}{n+d+1}\left(\genfrac{}{}{0ex}{}{m+n}{n+d}\right)$. 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.

34 pages, 6 figures, 4 tables