Posets and permutations in the duplication-loss model: minimal permutations with d descets

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) and Bouvel and Rossin (2009). 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 = 2^p 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 B_d of minimal permutations with d descents. We also provide a more detailed analysis - characterization, bijection and enumeration - of two particular subsets of B_d, namely the patterns in B_d of size d + 2 and 2d.