Permutation Diagrams, Fixed Points and Kazhdan-Lusztig R-Polynomials

Incitti, Federico

doi:10.1007/s00026-006-0294-6

Cited by 12 publications

(29 citation statements)

References 11 publications

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“…Permutominoes were introduced in [15], and then considered by F. Incitti while studying the problem of determining the R-polynomials associated with a pair (π 1 , π 2 ) of permutations [14]. In recent years, a particular class of permutominoes, namely convex permutominoes, and their associated permutations, have been widely studied [3,9].…”

Section: Introductionmentioning

confidence: 99%

On the shape of permutomino tiles

Massé¹,

Frosini²,

Rinaldi³

et al. 2013

Discrete Applied Mathematics

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In this paper we explore the connections between two classes of polyominoes, namely the permutominoes and the pseudo-square polyominoes. A permutomino is a polyomino uniquely determined by a pair of permutations. Permutominoes, and in particular convex permutominoes, have been considered in various kinds of problems such as: enumeration, tomographical reconstruction, and algebraic characterization.On the other hand, pseudo-square polyominoes are a class of polyominoes tiling the the plane by translation. The characterization of such objects has been given by Beauquier and Nivat, who proved that a polyomino tiles the plane by translation if and only if it is a pseudo-square or a pseudohexagon. In particular, a polyomino is pseudo-square if its boundary word may be factorized as XY X Y , where X denotes the path X traveled in the opposite direction.In this paper we relate the two concepts by considering the pseudo-square polyominoes which are also convex permutominoes. By using the Beauquier-Nivat characterization we provide some geometrical and combinatorial properties of such objects, and we show for any fixed X, each word Y such that XY X Y is pseudo-square is prefix of an infinite word Y ∞ with period 4 |X| N |X| E . Also, we show that XY X Y are centrosymmetric, i.e. they are fixed by rotation of angle π. The proof of this fact is based on the concept of pseudoperiods, a natural generalization of periods.

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Section: Introductionmentioning

confidence: 99%

On the shape of permutomino tiles

Massé¹,

Frosini²,

Rinaldi³

et al. 2013

Discrete Applied Mathematics

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“…It naturally generalizes the concept of permutomino, recently investigated by several authors and from different points of view [1,2,4,6,7]. In this paper, using bijective methods, we determine the enumeration of various classes of convex permutominides, including, parallelogram, directed convex, convex, and row convex permutominides.…”

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confidence: 99%

Polyominoes Determined by Permutations: Enumeration via Bijections

et al. 2011

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A permutominide is a set of cells in the plane satisfying special connectivity constraints and uniquely defined by a pair of permutations. It naturally generalizes the concept of permutomino, recently investigated by several authors and from different points of view. In this paper, using bijective methods, we determine the enumeration of various classes of convex permutominides, including, parallelogram, directed convex, convex, and row convex permutominides. As a corollary we have a bijective proof for the number of convex permutominoes, which was still an open problem

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“…We give an algorithm for direct enumeration of permutominoes [12] by size, or, equivalently, for the enumeration of grid orthogonal polygons [19].We show how the construction technique allows us to derive a simple characterization of the class of convex permutominoes, which has been extensively investigated recently [4]. The approach extends to some of its subclasses, namely to the row convex and the directed convex permutominoes.…”

mentioning

confidence: 99%

“…A grid ogon is an orthogonal polygon without collinear edges, embedded in a regular square grid, and having exactly one edge in each line of its minimal bounding square. They were addressed more recently under the name of permutominoes [4,8,12], because they correspond to polyominoes that are determined by a suitable pair of permutations having the same size. All polyominoes we will consider are simply-connected and, similarly, all polygons are simple and without holes.…”

Section: Introductionmentioning

confidence: 99%

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On the Enumeration of Permutominoes

Tomás

2015

Fundamentals of Computation Theory

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Although the exact counting and enumeration of polyominoes remain challenging open problems, several positive results were achieved for special classes of polyominoes. We give an algorithm for direct enumeration of permutominoes [12] by size, or, equivalently, for the enumeration of grid orthogonal polygons [19].We show how the construction technique allows us to derive a simple characterization of the class of convex permutominoes, which has been extensively investigated recently [4]. The approach extends to some of its subclasses, namely to the row convex and the directed convex permutominoes.

show abstract

Permutation Diagrams, Fixed Points and Kazhdan-Lusztig R-Polynomials

Cited by 12 publications

References 11 publications

On the shape of permutomino tiles

On the shape of permutomino tiles

Polyominoes Determined by Permutations: Enumeration via Bijections

On the Enumeration of Permutominoes

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