Posets and permutations in the duplication–loss model: Minimal permutations with d descents

In this paper, we are interested in the combinatorial analysis of the whole genome duplication–random loss model of genome rearrangement initiated in Chaudhuri et al. (2006) [9], and Bouvel and Rossin (2009) [8]. In this model, genomes composed of n genes are modeled by permutations of the set of integers {1,2,…,n}, that can evolve through duplication–loss steps. It was previously shown that the class of permutations obtained in this model after a given number p of steps is a class of pattern-avoiding permutations of finite basis. The excluded patterns were described as the minimal permutations with d=2p descents, minimal being intended in the sense of the pattern-involvement relation on permutations. Here, we give a local and simpler characterization of the set Bd of minimal permutations with d descents. We also provide a more detailed analysis–characterization, bijection and enumeration–of two particular subsets of Bd, namely the patterns in Bd of size d+2 and 2d.