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

Minimal Permutations and 2-Regular Skew Tableaux

Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that such structures are in one-to-one correspondence with a class of skew Young tableaux, which we call $2$-regular skew tableaux. Using the determinantal formula for the number of skew Young tableaux of a given shape, we find an explicit formula for $f_{n-3}(n)$. Furthermore, by using the Knuth equivalence, we give a combinatorial interpretation of a formula for a refinement of the number $f_{n+1}(2n+1)$.Comment: 19 page

On the enumeration of d-minimal permutations

International audienceWe suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers

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 $\alpha^k$ for $\alpha =1$ or $\alpha >=2$. In this paper, we study a variant of this model, where the cost of one step of width $k$ is 1 if $k K$, for any value of the parameter $K in N$. We first show that permutations obtained after $p$ steps of width $K$ define classes of pattern-avoiding permutations. We also compute the numbers of duplication-loss steps of width $K$ necessary and sufficient to obtain any permutation of $S_n$, in the worst case and on average. In this second part, we may also consider the case $K=K(n)$, a function of the size $n$ of the permutation on which the duplication-loss operations are performed