Defective Parking Functions and Young Tableaux

Recall that 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 Sₘ the symmetric group on [m]=\1,2,…,m\. 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 Sₘ-orbits of defective $(m,n)$-parking functions with defect d is given by n-m+2d+1/n+d+1m+nn+d. We also give a multinomial formula for the size of an Sₘ-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.