Posets and Permutations in the duplication-loss model

Mathilde Bouvel (1) , Elisa Pergola (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 DSI - Dipartimento di Sistemi e Informatica

In this paper, we are interested in the combinatorial analysis of the whole genome duplication - random loss model of genome rearrangement initiated in a paper of Chaudhuri, Chen, Mihaescu, and Rao in SODA 2006 and continued by Bouvel and Rossin in 2007. 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. 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 implicitly 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 a particular subset of B_d, namely the patterns in B_d of size 2d.

Mots clés: permutations; duplication-loss model; posets; Dyck paths; non-interval subsets; bijection

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Mathilde Bouvel, Elisa Pergola. Posets and Permutations in the duplication-loss model. GASCOM'08, Jun 2008, Bibbiena, Italy. ⟨hal-00322737⟩