Vigemers: on the number of $k$-mers sharing the same XOR-based minimizer

In bioinformatics, minimizers have become an inescapable method for handling $k$-mers (words of fixed size $k$) extracted from DNA or RNA sequencing, whether for sampling, storage, querying or partitioning. According to some fixed order on $m$-mers ($m<k$), the minimizer of a $k$-mer is defined as its smallest $m$-mer -- and acts as its fingerprint. Although minimizers are widely used for partitioning purposes, there is almost no theoretical work on the quality of the resulting partitions. For instance, it has been known for decades that the lexicographic order empirically leads to highly unbalanced partitions that are unusable in practice, but it was not until very recently that this observation was theoretically substantiated. The rejection of the lexicographic order has led the community to resort to (pseudo-)random orders using hash functions. In this work, we extend the theoretical results relating to the partitions obtained by the lexicographical order, departing from it to a (exponentially) large family of hash functions, namely where the $m$-mers are XORed against a fixed key. More precisely, provided a key $\gamma$ and a $m$-mer $w$, we investigate the function that counts how many $k$-mers admit $w$ as their minimizer (i.e. where $w\oplus\gamma$ is minimal among all $m$-mers of said $k$-mers). This number, denoted by $\pi_k^{\gamma}(w)$, represents the maximum size of the bucket associated with $w$, if all possible $k$-mers were to be seen and partitioned. We adapt the (lexicographical order) method of the literature to our framework and propose combinatorial equations that allow to compute, using dynamic programming, $\pi_k^{\gamma}(w)$ in $O(km^2)$ time and $O(km)$ space.