Polyominoes Determined by Permutations: Enumeration via Bijections

Disanto, Filippo; Duchi, Enrica; Pinzani, Renzo; Rinaldi, Simone

doi:10.1007/s00026-011-0121-6

Cited by 3 publications

(6 citation statements)

References 6 publications

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“…Recently, a bijective proof of (1) was given in [8] by encoding convex permutominoes in terms of lattice paths.…”

Section: Permutominoes: Basic Definitions and Resultsmentioning

confidence: 99%

“…Without going further into formal definitions, a column-convex permutominide of size n is substantially a polyomino where the boundary of which is allowed to cross itself, while the columns remain connected, and with exactly one edge for every abscissa and exactly one edge for every ordinate between 1 and n + 1. These objects have been treated and enumerated, according to size, in [8].…”

Section: Permutominoes: Basic Definitions and Resultsmentioning

confidence: 99%

“…Figure 6. (a) the permutation π = (7,13,10,1,4,14,6,2,15,8,3,11,9,5,12), where two locally decomposable sub-permutations of π have been shaded; (b) a column-convex permutomino, with π as its first component, and evaluation v(π) = (0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1). The free points of π are encircled.…”

Section: A Characterization Of the First Components Of Column-convex mentioning

confidence: 99%

“…Finally, given the set F(π) it is not difficult to compute the number of evaluations satisfying it, then to determine the cardinality of [π]. For instance, there are 108 column-convex permutominoes associated with the permutation π = (7,13,10,1,4,14,6,2,15,8,3,11,9,5,12) in Figure 6.…”

Section: Proofmentioning

confidence: 99%

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Polygons Drawn from Permutations

Bilotta

Rinaldi

Socci

2013

Fundamenta Informaticae

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n this paper we consider the class of column-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations (π 1 , π 2 ). First, using a geometric construction, we prove that for every permutation π there is at least one column-convex permutomino P such that π 1 (P ) = π or π 2 (P ) = π. In the second part of the paper, we show how, for any given permutation π, it is possible to define a set of logical implications F(π) on the points of π, and prove that there exists a column-convex permutomino P such that π 1 (P ) = π if and only if F(π) is satisfiable. This property can be then used to give a characterization of the set of column-convex permutominoes P such that π 1 (P ) = π.

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“…Recently, a bijective proof of (1) was given in [8] by encoding convex permutominoes in terms of lattice paths.…”

Section: Permutominoes: Basic Definitions and Resultsmentioning

confidence: 99%

Section: Permutominoes: Basic Definitions and Resultsmentioning

confidence: 99%

Section: A Characterization Of the First Components Of Column-convex mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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Polygons Drawn from Permutations

Bilotta

Rinaldi

Socci

2013

Fundamenta Informaticae

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“…Permutominoes were introduced in [13] in some algebraic context and then they were considered by F. Incitti, in the study of the R-polynomials associated with a pair (π 1 , π 2 ) of permutations [12]. The most significant results on the enumeration of column-convex = (1,7,5,3,8,2,6,4) 2 = (7,5,8,1,6,3,4,2) π 1 π Figure 1: A permutomino of size 7 and the two associated permutations.…”

Section: Introductionmentioning

confidence: 99%

About Half Permutations

Rinaldi¹,

Socci²

2014

Electron. J. Combin.

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In 2011, Beaton et al. analytically proved that the number of directed column-convex permutominoes of size $n$ is given by $(n+1)!/2$. In this paper, we provide a different proof of this statement using a bijective method. More precisely, we present a bijective correspondence between the class $D_n$ of directed column-convex permutominoes of size $n$ and a set of permutations (called $dcc$-permutations) of length $n+1$, which we prove to be counted by $(n+1)!/2$. The class of $dcc$-permutations is a new class of permutations counted by half factorial numbers, and here we show some combinatorial characterizations of this class, using the concept of logical formulas determined by a permutation and the notion of mesh pattern.

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On the enumeration of column-convex permutominoes

Beaton¹,

Disanto²,

Guttmann³

et al. 2011

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis. Nous étudions l'énumeration des \emphpermutominos verticalement convexes, c.à.d. les polyominos verticalement convexes définis par un couple de permutations. Nous donnons une construction recursive directe pour ces permutominos de taille fixée, basée sur une application de la méthode ECO et les arbres de génération, qui nous amène à une équat ion fonctionelle. Ensuite nous obtenons des bornes superieures et inférieures pour le nombre de ces permutominos convexes et nous conjecturons leur comportement asymptotique à l'aide d'analyses numériques.

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Polyominoes Determined by Permutations: Enumeration via Bijections

Cited by 3 publications

References 6 publications

Polygons Drawn from Permutations

Polygons Drawn from Permutations

About Half Permutations

On the enumeration of column-convex permutominoes

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