### Patterns in Permutations and Words (Monographs in Theoretical Computer Science. An EATCS Series)

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In this way, a separating tree is equivalent to a construction of the permutation by direct and skew sums, starting from the trivial permutation. The separable permutations also have a characterization from algebraic geometry : if a collection of distinct real polynomials all have equal values at some number x , then the permutation that describes how the numerical ordering of the polynomials changes at x is separable, and every separable permutation can be realized in this way.

That is, there is one separable permutation of length one, two of length two, and in general the number of separable permutations of a given length starting with length one is. Another proof applying more directly to separable permutations themselves, was given by West The problem of finding the longest separable pattern that is common to a set of input permutations may be solved in polynomial time for a fixed number of input permutations, but is NP-hard when the number of input permutations may be variable, and remains NP-hard even when the inputs are all themselves separable.

As they show, the class of permutations that are transformed by this process into the all-one matrix is exactly the class of separable permutations.

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Every permutation can be used to define a permutation graph , a graph whose vertices are the elements of the permutation and whose edges are the inversions of the permutation. In the case of a separable permutation, the structure of this graph can be read off from the separation tree of the permutation: two vertices of the graph are adjacent if and only if their lowest common ancestor in the separation tree is negative.

The graphs that can be formed from trees in this way are called cographs short for complement-reducible graphs and the trees from which they are formed are called cotrees.

## [PDF] Quadrant marked mesh patterns in avoiding permutations II - Semantic Scholar

Banderier and F. Luca , On the period mod m of polynomially-recursive sequences: a case study , preparation , Banderier, M. Denise, and P. Generating functions for generating trees , Discrete Math , vol.

## Patterns in Permutations and Words

Barcucci, A. Lungo, E. Pergola, and R. Pinzani , ECO: a methodology for the enumeration of combinatorial objects , J.

Equations Appl , vol. Baril , Gray code for permutations with a fixed number of left-to-right minima , Ars Combin , vol. Baril and R.

Vernay , Whole mirror duplication-random loss model and pattern avoiding permutations , Inform. Lett , vol.

DOI : Discrete Mathematics and its Applications , Bouvel and L. Ferrari , On the enumeration of d-minimal permutations , Discrete Math.

## Separable permutation

Sci , vol. Bouvel and E. Patterns in Permutations and Words Sergey Kitaev There has been considerable interest recently in the subject of patterns in permutations and words, a new branch of combinatorics with its roots in the works of Rotem, Rogers, and Knuth in the s. Consideration of the patterns in question has been extremely interesting from the combinatorial point of view, and it has proved to be a useful language in a variety of seemingly unrelated problems, including the theory of Kazhdan-Lusztig polynomials, singularities of Schubert varieties, interval orders, Chebyshev polynomials, models in statistical mechanics, and various sorting algorithms, including sorting stacks and sortable permutations.

Les mer. Om boka There has been considerable interest recently in the subject of patterns in permutations and words, a new branch of combinatorics with its roots in the works of Rotem, Rogers, and Knuth in the s.