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

TL;DR

This paper analyzes the combinatorial structure of permutations arising from a genome duplication-loss model, providing simplified characterizations and enumerations of minimal permutations with a given number of descents.

Contribution

It offers a new, simpler characterization of minimal permutations with d descents and detailed analysis of specific subclasses within this set.

Findings

Characterization of minimal permutations with d descents

Bijection and enumeration of permutations of size d+2 and 2d

Simplified description of pattern-avoiding permutations in the model

Abstract

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 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.