  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.
