Relevant bibliographies by topics / Polyomino Ideal

Academic literature on the topic 'Polyomino Ideal'

Author: Grafiati
Published: 11 March 2023

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Journal articles on the topic "Polyomino Ideal"

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Yonatan Hamonangan, Yoshua, and Intan Muchtadi-Alamsyah. "On Radical Property of Cross Polyomino Ideal." Journal of Physics: Conference Series 1306 (August 2019): 012023. http://dx.doi.org/10.1088/1742-6596/1306/1/012023.

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Muzika-Dizdarevic, Manuela, and Rade Zivaljevic. "Symmetric polyomino tilings, tribones, ideals, and Gröbner bases." Publications de l'Institut Math?matique (Belgrade) 98, no. 112 (2015): 1–23. http://dx.doi.org/10.2298/pim1512001m.

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We apply the theory of Grobner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions TN = T3k?1 and TN = T3k in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120? rotation of the triangle if and only if either N = 27r ? 1 or N = 27r for some integer r > 0. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.
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3

Mascia, C., G. Rinaldo, and F. Romeo. "Primality of polyomino ideals by quadratic Gröbner basis." Mathematische Nachrichten 295, no. 3 (February 6, 2022): 593–606. http://dx.doi.org/10.1002/mana.202000252.

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Hibi, Takayuki, and Ayesha Asloob Qureshi. "Nonsimple polyominoes and prime ideals." Illinois Journal of Mathematics 59, no. 2 (2015): 391–98. http://dx.doi.org/10.1215/ijm/1462450707.

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Asloob Qureshi, Ayesha. "Ideals generated by 2-minors, collections of cells and stack polyominoes." Journal of Algebra 357 (May 2012): 279–303. http://dx.doi.org/10.1016/j.jalgebra.2012.01.032.

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Cisto, Carmelo, and Francesco Navarra. "Primality of closed path polyominoes." Journal of Algebra and Its Applications, December 2, 2021. http://dx.doi.org/10.1142/s021949882350055x.

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In this paper, we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture, we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.
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Cisto, Carmelo, Francesco Navarra, and Rosanna Utano. "On Gröbner Basis and Cohen-Macaulay Property of Closed Path Polyominoes." Electronic Journal of Combinatorics 29, no. 3 (September 9, 2022). http://dx.doi.org/10.37236/11122.

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In this paper we introduce some monomial orders for the class of closed path polyominoes and we prove that the set of the generators of the polyomino ideal attached to a closed path forms the reduced Gröbner basis with respect to these monomial orders. It is known that the polyomino ideal attached to a closed path containing an $L$-configuration or a ladder of at least three steps, equivalently having no zig-zag walks, is prime. As a consequence, we obtain that the coordinate ring of a closed path having no zig-zag walks is a normal Cohen-Macaulay domain.
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Del Lungo, A., E. Duchi, A. Frosini, and S. Rinaldi. "On the Generation and Enumeration of some Classes of Convex Polyominoes." Electronic Journal of Combinatorics 11, no. 1 (September 13, 2004). http://dx.doi.org/10.37236/1813.

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ECO is a method for the recursive generation, and thereby also the enumeration of classes of combinatorial objects. It has already found successful application in recent literature both to the exhaustive generation and to the uniform random generation of various objects classified according to several parameters of interest, as well as to their enumeration. In this paper we extend this approach to the generation and enumeration of some classes of convex polyominoes. We begin with a review of the ECO method and of the closely related notion of a succession rule. From this background, we develop the following principal findings: i) ECO constructions for both column-convex and convex polyominoes; ii) translations of these constructions into succession rules; iii) the consequent deduction of the generating functions of column-convex and of convex polyominoes according to their semi-perimeter, first of all analytically by means of the so-called kernel method, and then in a more novel manner by drawing on some ideas of Fedou and Garcia; iv) algorithms for the exhaustive generation of column convex and of convex polyominoes which are based on the ECO constructions of these object and which are shown to run in constant amortized time.
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Dissertations / Theses on the topic "Polyomino Ideal"

1

Romeo, Francesco. "Algebraic Properties and Invariants of Polyominoes." Doctoral thesis, Università degli studi di Trento, 2022. http://hdl.handle.net/11572/346499.

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Polyominoes are two-dimensional objects obtained by joining edge by edge squares of same size. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. Recently Qureshi introduced a binomial ideal induced by the geometry of a given polyomino, called polyomino ideal, and its related algebra. From that moment different authors studied algebraic properties and invariants related to this ideal, such as primality, Gröbner bases, Gorensteinnes and Castelnuovo-Mumford regularity. In this thesis, we provide an overview on the results that we obtained about polyomino ideals and its related algebra. In the first part of the thesis, we discuss questions about the primality and the Gröbner bases of the polyomino ideal. In the second part of the thesis, we talk over the Castelnuovo-Mumford regularity, Hilbert series, and Gorensteinnes of the polyomino ideal and its coordinate ring.
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2

Mascia, Carla. "Ideals generated by 2-minors: binomial edge ideals and polyomino ideals." Doctoral thesis, Università degli studi di Trento, 2020. http://hdl.handle.net/11572/252052.

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Since the early 1990s, a classical object in commutative algebra has been the study of binomial ideals. A widely-investigated class of binomial ideals is the one containing those generated by a subset of 2-minors of an (m x n)-matrix of indeterminates. This thesis is devoted to illustrate some algebraic and homological properties of two classes of ideals of 2-minors: binomial edge ideals and polyomino ideals. Binomial edge ideals arise from finite graphs and their appeal results from the fact that their homological properties reflect nicely the combinatorics of the underlying graph. First, we focus on the binomial edge ideals of block graphs. We give a lower bound for their Castelnuovo-Mumford regularity by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs. Secondly, we consider some classes of Cohen-Macaulay binomial edge ideals. We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs. Polyomino ideals arise from polyominoes, plane figures formed by joining one or more equal squares edge to edge. It is known that the polyomino ideal of simple polyominoes is prime. We consider multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.
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3

Mascia, Carla. "Ideals generated by 2-minors: binomial edge ideals and polyomino ideals." Doctoral thesis, Università degli studi di Trento, 2020. http://hdl.handle.net/11572/252052.

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Abstract:
Since the early 1990s, a classical object in commutative algebra has been the study of binomial ideals. A widely-investigated class of binomial ideals is the one containing those generated by a subset of 2-minors of an (m x n)-matrix of indeterminates. This thesis is devoted to illustrate some algebraic and homological properties of two classes of ideals of 2-minors: binomial edge ideals and polyomino ideals. Binomial edge ideals arise from finite graphs and their appeal results from the fact that their homological properties reflect nicely the combinatorics of the underlying graph. First, we focus on the binomial edge ideals of block graphs. We give a lower bound for their Castelnuovo-Mumford regularity by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs. Secondly, we consider some classes of Cohen-Macaulay binomial edge ideals. We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs. Polyomino ideals arise from polyominoes, plane figures formed by joining one or more equal squares edge to edge. It is known that the polyomino ideal of simple polyominoes is prime. We consider multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.
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