Expected number of breakpoints after t random reversals in genomes with duplicate genes

In comparative genomics, one wishes to deduce the evolutionary distance between different species by studying their genomes. Using gene order information, we seek the number of times the gene order has changed between two species. One approach is to compute the method of moments estimate of this edit distance from a measure of dissimilarity called the breakpoint measure. In this paper, we extend formulae and bounds of this estimate on gene permutations to genomes with duplicate genes