Convex Polyominoes Research Articles

Bijections between directed-column convex polyominoes and restricted compositions

A bijection is given between the set of directed column-convex polyominoes on triangular and honeycomb lattices of area n and some families of restricted compositions. This is an analogous result to one given by Deutsch and Prodinger for polyominoes over square lattices. As a byproduct, we deduce new close forms for the number of hexagonal and triangular directed column-convex polyominoes of area n with k columns.

Asymptotics of Z-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we show that one can compute in polynomial time the number of polyominoes of area n and degree of convexity at most 2 (the so-called Z-convex polyominoes). The integer sequence that we have computed allows us to conjecture the asymptotic number an of Z-convex polyominoes of area n, ɑn ∼ C·exp(π)√11n/4⁄n3/2.

The $h$-Polynomial and the Rook Polynomial of some Polyominoes

Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, J. Algebra, 2012. Write the Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots )/(1-t)^{\dim(\Bbbk[X])}$. For $k \in \mathbb{N}$, let $r_k$ be the number of configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2 < r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured characterization of thin polyominoes by Rinaldo and Romeo, J. Algebraic Combin., 2021.

On counting Z-convex polyominoes

Abstract We show a decomposition that allows to compute the number of convex polyominoes of area n and degree of convexity at most 2 (the so-called Z-convex polyominoes) in polynomial time.

Large deviations of convex polyominoes

Many open problems in machine learning, pattern recognition, and geometric analysis require enumeration of different types of lattice polygons, and in particular convex polyominoes. In this work, we develop a large deviation principle for convex polyominoes under different restrictions, such as fixed area and/or perimeter.

Concatenation arguments and their applications to polyominoes and polycubes

In this paper, we develop a method for setting lower and upper bounds on growth constants of polyominoes and polycubes whose enumerating sequences are so-called quasi sub- or super-multiplicative. The method is based on concatenation arguments, applied directly or recursively. Inter alia, we demonstrate the method on general polycubes, tree polyominoes and polycubes, and convex polyominoes.

Further steps on the reconstruction of convex polyominoes from orthogonal projections

A remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

Regularity and the Gorenstein property of $L$-convex Polyominoes

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen–Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it. Though the main results are of algebraic nature, all proofs are combinatorial.

Algebraic Properties of the Coordinate Ring of a Convex Polyomino

We classify all convex polyominoes whose coordinate rings are Gorenstein. We also give an upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of any convex polyomino in terms of the smallest interval which contains its vertices. We give a recursive formula for computing the multiplicity of a stack polyomino.

The perimeter generating function for nondirected diagonally convex polyominoes

A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes, researchers also gave a look at diagonally convex polyominoes (DCPs), but noticed an awkward feature: when the last diagonal of a DCP is deleted, the remaining object is not always a polyomino. So researchers focused their attention on directed DCPs. (A directed DCP is such a DCP that remains a polyomino when, for any i, its last i diagonals are deleted.) Directed DCPs gradually became well understood, whereas general DCPs have remained unexplored up to now.In this paper, we finally face general DCPs. Modulo a little trick, which saves us from dealing with non-polyominoes, we use the layered approach (described in chapter 3 of the book “Polygons, Polyominoes and Polycubes”, edited by Anthony Guttmann). The computations are of remarkable bulk. Our main result is the perimeter generating function for DCPs; we denote it D(d,x). The function D(d,x) is algebraic and satisfies an equation of degree eight. The formula for D(d,x) is about three pages long. That formula involves nine polynomials in d and x, and each of those polynomials is of degree 58 or more in x.

Convex polyominoes revisited: enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees

The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that (A) the average number of interior vertices over all convex polyominoes of perimeter is asymptotic to (B) the average number of boundary vertices with degree two over all convex polyominoes of perimeter is asymptotic to Additionally, we obtain an explicit generating function counting the number of convex polyominoes with n boundary vertices of degrees at most three and show that this number is asymptotic to Moreover, we show that the expected number of the boundary vertices of degree four over all convex polyominoes with n vertices of degrees at most three is asymptotically (C) the number of convex polyominoes with the outer-site perimeter n is asymptotic to and show the expected number of the outer-site perimeter over all convex polyominoes with perimeter is asymptotic to Lastly, we prove that the expected perimeter over all convex polyominoes with the outer-site perimeter n is asymptotic to .

The number of directed [formula omitted]-convex polyominoes

We present a new general approach for the enumeration of directed convex polyominoes, which lets us easily control several statistics. This method relies on a bijection between directed convex polyominoes with semi-perimeter equal to n+2 and triplets (Fe,Fs,λ), where Fe and Fs are forests with a total number of nodes equal to n, and λ is a lattice path of length n+2. Based on this bijection, we develop a new method for the enumeration of directed convex polyominoes, according to several different parameters, including: the semi-perimeter, the degree of convexity, the width, the height, the size of the last row/column and the number of corners. We point out that most of these statistics have already been computed in the literature, but applying our method every statistic which can be read on the two forests Fe and Fs, can be in principle computed. The most important and original result of the paper consists of applying our method to the enumeration of directed convex polyominoes which are also k-convex. Let us recall that a convex polyomino is k-convex if every pair of its cells can be connected by a monotone path with at most k changes of direction. The application of our method eventually lets us prove that the generating function of directed k-convex polyominoes, for any k≥1, is a rational function, which can be suitably expressed in terms of the known Fibonacci polynomials Fk: z2Fk+2F2k+32F2k+2.

Colourability and word-representability of near-triangulations

Abstract A graph G = (V;E) is word-representable if there is a word w over the alphabet V such that x and y alternate in w if and only if the edge (x; y) is in G. It is known [6] that all 3-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not. There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al. [1] showed that a triangulation of a convex polyomino is word-representable if and only if it is 3-colourable; and Glen and Kitaev [5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed. It was shown in [4] that a near-triangulation is 3-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5].

Efficient Parallel Self-Assembly Under Uniform Control Inputs

We prove that by successively combining subassemblies, we can achieve sublinear construction times for “staged” assembly of microscale objects from a large number of tiny particles, for vast classes of shapes; this is a significant advance in the context of programmable matter and self-assembly for building high-yield microfactories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle; particles bond when forced together with a compatible particle. Previous work considered sequential composition of objects, resulting in construction time that is linear in the number $N$ of particles, which is inefficient for large $N$ . Our progress implies critical speedup for constructible shapes; for convex polyominoes, even a constant construction time is possible. We also show that our construction process can be used for pipelining, resulting in an amortized constant production time.

Toric representations of algebras defined by certain nonsimple polyominoes

In this paper, we give a toric representation of the associated ring of a polyomino which is obtained by removing a convex polyomino from its ambient rectangle.

Statistics on bargraphs viewed as cornerless Motzkin paths

A bargraph is a self-avoiding lattice path with steps U=(0,1), H=(1,0) and D=(0,−1) that starts at the origin and ends on the x-axis, and stays strictly above the x-axis everywhere except at the endpoints. Bargraphs have been studied as a special class of convex polyominoes, and enumerated using the so-called wasp-waist decomposition of Bousquet-Mélou and Rechnitzer. In this paper we note that there is a trivial bijection between bargraphs and Motzkin paths without peaks or valleys. This allows us to use the recursive structure of Motzkin paths to enumerate bargraphs with respect to several statistics, finding simpler derivations of known results and obtaining many new ones. We also count symmetric bargraphs and alternating bargraphs. In some cases we construct statistic-preserving bijections between different combinatorial objects, proving some identities that we encounter along the way.

On the exhaustive generation of k-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates, given an integer k, all k-convex polyominoes of area n in constant amortized time, using space O(n). Furthermore, by applying few changes, we are able to generate all convex polyominoes whose degree of convexity is exactly k.

Reconstruction of convex polyominoes with a blocking component

Several papers in the available literature tackled problems concerning convex polyominoes in discrete tomography. An interesting subclass consists of L-convex polyominoes, since the related reconstruction problem can have only a unique solution. On the other hand, recent studies have modeled an approach to reconstruct objects even in the case that some of the projections are unavailable, due to a particularly dense part of the scanned object, that we refer to as a blocking component. In this work we merge the two problems in order to obtain efficient reconstruction algorithms for convex and L-convex polyominoes, in case a blocking component is included.

On Computing the Degree of Convexity of Polyominoes

In this paper we present an algorithm which has as input a convex polyomino $P$ and computes its degree of convexity, defined as the smallest integer $k$ such that any two cells of $P$ can be joined by a monotone path inside $P$ with at most $k$ changes of direction. The algorithm uses space $O(m + n)$ to represent a polyomino $P$ with $n$ rows and $m$ columns, and has a running time $O(min(m; r k))$, where $r$ is the number of corners of $P$. Moreover, the algorithm leads naturally to a decomposition of $P$ into simpler polyominoes.

The number of directed $k$-convex polyominoes

We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed $k$-convex polyominoes.We show it is a rational function and we study its asymptotic behavior. Nous présentons une nouvelle méthode générique pour obtenir facilement et rapidement les fonctions génératrices des polyominos dirigés convexes avec différentes combinaisons de statistiques : hauteur, largeur, longueur de la dernière ligne/colonne et nombre de coins. La méthode peut être utilisée pour énumérer différentes familles de polyominos dirigés convexes: les polyominos symétriques, les polyominos parallélogrammes. De cette façon, nouscalculons la fonction génératrice des polyominos dirigés $k$-convexes, nous montrons qu’elle est rationnelle et nous étudions son comportement asymptotique.