Posets and Permutations in the duplication-loss model

Combinatorial analysis of the whole genome duplication - random loss model of genome rearrangement . In this model, genomes composed of n genes are modeled by permutations of the set of integers [1..n], that can evolve through duplication-loss steps. 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 implicitly described as the minimal permutations with d = 2p descents, minimal being intended in the sense of the pattern-involvement relation on permutations. A local and simpler characterization of the set Bd of minimal permutations with d descents is studied. A more detailed analysis characterization, bijection and enumeration - of a particular subset of Bd, namely the patterns in Bd of size 2d is provided.