Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs

Genome rearrangements can be modeled as $k$-breaks, which break a genome at k positions and glue the resulting fragments in a new order. In particular, reversals, translocations, fusions, and fissions are modeled as $2$-breaks, and transpositions are modeled as $3$-breaks. While $k$-break rearrangements for $k>3$ have not been observed in evolution, they are used in cancer genomics to model chromothripsis, a catastrophic event of multiple breakages happening simultaneously in a genome. It is known that the $k$-break distance between two genomes (i.e., the minimum number of $k$-breaks required to transform one genome into the other) can be computed in terms of cycle lengths in the breakpoint graph of these genomes. In the current work, we address the combinatorial problem of enumerating genomes at a given $k$-break distance from a fixed unichromosomal genome. More generally, we enumerate genome pairs, whose breakpoint graph has a given distribution of cycle lengths. We further show how our enumeration can be used for uniform sampling of random genomes at a given $k$-break distance, and describe its connection to various combinatorial objects such as Bell polynomials.