88 research outputs found
Operators of equivalent sorting power and related Wilf-equivalences
We study sorting operators on permutations that are obtained
composing Knuth's stack sorting operator and the reversal operator
, as many times as desired. For any such operator , we
provide a size-preserving bijection between the set of permutations sorted by
and the set of those sorted by , proving that these sets are enumerated by the
same sequence, but also that many classical permutation statistics are
equidistributed across these two sets. The description of this family of
bijections is based on a bijection between the set of permutations avoiding the
pattern and the set of those avoiding which preserves many
permutation statistics. We also present other properties of this bijection, in
particular for finding pairs of Wilf-equivalent permutation classes.Comment: 18 pages, 8 figure
Longest Common Separable Pattern between Permutations
In this article, we study the problem of finding the longest common separable
pattern between several permutations. We give a polynomial-time algorithm when
the number of input permutations is fixed and show that the problem is NP-hard
for an arbitrary number of input permutations even if these permutations are
separable. On the other hand, we show that the NP-hard problem of finding the
longest common pattern between two permutations cannot be approximated better
than within a ratio of (where is the size of an optimal
solution) when taking common patterns belonging to pattern-avoiding classes of
permutations.Comment: 15 page
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
The Longest Common Pattern Problem for two Permutations
International audienceIn this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations
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
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