Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation sphere sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing tight upper and lower bounds on multipermutation Ulam sphere sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric.