Information Processing Letters 110 (2010) 474–480 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Whole mirror duplication-random loss model and pattern avoiding permutations Jean-Luc Baril ∗ , Rémi Vernay LE2I UMR-CNRS 5158, Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France article info abstract Article history: Received 29 June 2009 Received in revised form 10 February 2010 Accepted 20 April 2010 Available online 21 April 2010 Communicated by A.A. Bertossi Keywords: Algorithms Combinatorial problems Pattern avoiding permutation Whole duplication-random loss model Genome Generating algorithm Binary reflected Gray code In this paper we study the problem of the whole mirror duplication-random loss model in terms of pattern avoiding permutations. We prove that the class of permutations obtained with this model after a given number p of duplications of the identity is the class of permutations avoiding the alternating permutations of length 2 p + 1. We also compute the number of duplications necessary and sufficient to obtain any permutation of length n. We provide two efficient algorithms to reconstitute a possible scenario of whole mirror duplications from identity to any permutation of length n. One of them uses the well-known binary reflected Gray code (Gray, 1953) [10]. Other relative models are also considered.  2010 Elsevier B.V. All rights reserved. 1. Introduction and notation The well-known genome duplication consists in copy- ing a part of the original genome inserted into itself, fol- lowed by the loss of one copy of each of the duplicated genes (see [2,9,11,14,17] for an explanation of different methods of duplication). From a formal point of view, a genome of n genes is represented by a permutation of length n. In a previous article, Chaudhuri et al. [7] investi- gated a variant called the tandem duplication-random loss model: the duplicated part (of size K ) of the genome is in- serted immediately after the original portion, followed by the loss procedure. This model comes from evolutionary biology where it has been applied to the vertebrate mito- chondrial genomes. Chaudhuri et al. introduce a notion of distance between two genomes and they provide an algo- rithm to compute it efficiently for certain regions of the parameter space. Bouvel and Rossin [5] have also studied * Corresponding author. E-mail addresses: barjl@u-bourgogne.fr (J.-L. Baril), remi.vernay@u-bourgogne.fr (R. Vernay). this model. They proved that the class of permutations ob- tained from the identity after p steps (of width K ) is also a class of pattern avoiding permutations. More particularly, they investigate the restricted case of a whole duplication (W-duplication for short): the whole duplication consists in copying entirely the permutation on its right and the loss procedure consists to delete one of the two copies of each gene. Here, we give an example of the process of a W-duplication followed by the loss procedure on the per- mutation 123456: 123456  123456123456    duplication  12  34  56  1  23  45  6    loss procedure  124635. So, they prove that the permutations obtained after p W- duplications is the class of permutations avoiding all min- imal permutations with 2 p descents, minimal in the sense of pattern-involvement relation on permutations. More- over, they computed the number of duplication-loss steps 0020-0190/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2010.04.016