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 2 d - 1 with d descents is given by 2^{d - 3} (d - 1) c_d, where c_d is the d-th Catalan number. For fixed n, we also derive a recurrence relation on the multivariate generating function for the number of minimal permutations of length n counted by the number of descents, and the values of the first and second elements of the permutation. For fixed d, on the basis of this recurrence relation, we obtain a recurrence relation on the multivariate generating function for the number of minimal permutations of length n with n - d descents, counted by the length, and the values of the first and second elements of the permutation. As a consequence, the explicit generating functions for the numbers of minimal permutations of length n with n - d descents are obtained for d ≤ 5. Furthermore, we show that for fixed d ≥ 1, there exists a constant a_d such that the number of minimal permutations of length n with n - d descents is asymptotically equivalent to a_d dⁿ, as n → ∞.