Minimal permutations with [formula omitted] descents

Abstract

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 d n , as n → ∞ .