10 research outputs found
Posets and Permutations in the duplication-loss model
Version courte de "Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents.", présentée à GASCom'08In 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
Some equinumerous pattern-avoiding classes of permutations
Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X(p,q,r,s) of permutations that contain no pattern of the form α β γ where |α |=r, |γ |=s and β is any arrangement of \1,2,\ldots,p\∪ \m-q+1, m-q+2, \ldots,m\ is considered. A recurrence relation to enumerate the permutations of X(p,q,r,s) is established. The method of proof also shows that X(p,q,r,s)=X(p,q,1,0)X(1,0,r,s) in the sense of permutational composition.\par 2000 MATHEMATICS SUBJECT CLASSIFICATION: 05A0
A variant of the tandem duplication - random loss model of genome rearrangement
In Soda'06, Chaudhuri, Chen, Mihaescu and Rao study algorithmic properties of
the tandem duplication - random loss model of genome rearrangement, well-known
in evolutionary biology. In their model, the cost of one step of
duplication-loss of width k is for or . In
this paper, we study a variant of this model, where the cost of one step of
width is 1 if , for any value of the
parameter . We first show that permutations obtained after steps of
width define classes of pattern-avoiding permutations. We also compute the
numbers of duplication-loss steps of width necessary and sufficient to
obtain any permutation of , in the worst case and on average. In this
second part, we may also consider the case , a function of the size
of the permutation on which the duplication-loss operations are performed
Compositions of pattern restricted sets of permutations
The composition of two pattern restricted classes X,Y is the set of all permutation products ## where # X,# Y . This set is also defined by pattern restrictions. Examples are given where this set of restrictions is finite and where it is infinite. The composition operation is studied in terms of machines that sort and generate permutations. The theory is then applied to a multistage sorting network where each stage can exchange any number of adjacent disjoint pairs