Minimal permutations with d descents

Recently, Bouvel and Pergola initiated the study of a special class of permutations, minimal permutations with a given number of descents, which arise from the whole genome duplication-random loss model of genome rearrangement. In this paper, we show that the number of minimal permutations of length 2d− 1 with d descents is given by 2d−3(d− 1)cd, where cd is the d-th Catalan number. We also derive a recurrence relation on the generating functions for the number of minimal permutations π of length n with respect to the number of descents, and the values of the first and second elements of π. Furthermore, we show that given d ≥ 1 there exists a constant ad such that the number of minimal permutations of length n with n− d descents is asymptotically equivalent to add , as n → ∞.