A Generalized Hamming Distance of Sequence Patterns

We define sequence patterns of length $n$ and level $\ell$ to be equivalence classes of sequences that have $n$ elements from the set of $\ell$ integer symbols $\{1,2,\ldots,\ell\}$ with no restriction on repetition, where the equivalence relation is induced by symbol relabeling without swapping positions of symbols. We define a distance for a set of $k$ sequence patterns of length $n$ and level $\ell$ by generalizing the Hamming distance between sequences. We compute the maximal distance for $k$ sequence patterns of length $n$ and level $\ell$ and demonstrate how to calculate the exact distance between a pair of length-$n$ level-$\ell$ sequence patterns.