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In this paper, we are interested in the combinatorial analysis of the wholegenome duplication - random loss model of genome rearrangement initiated in apaper of Chaudhuri, Chen, Mihaescu, and Rao in SODA 2006 and continued byBouvel and Rossin in 2007. In this model, genomes composed of n genes aremodeled by permutations of the set of integers [1..n], that can evolve throughduplication-loss steps. It was previously shown that the class of permutationsobtained in this model after a given number p of steps is a class ofpattern-avoiding permutations of finite basis. The excluded patterns weredescribed as the minimal permutations with d=2^p descents, minimal beingintended in the sense of the pattern-involvement relation on permutations.Here, we give a local and simpler characterization of the set B_d of minimalpermutations 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.

Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents