{"title":"具有固定周长缺陷的多面体公式的自动生成","authors":"Gill Barequet, Bar Magal","doi":"10.1016/j.comgeo.2022.101919","DOIUrl":null,"url":null,"abstract":"<div><p>A <em>polyomino</em><span> is a shape best described as a connected set of cells in the square lattice. As part of recreational mathematics, polyominoes have seen active research since the 1950s. Simultaneously, polyominoes have been investigated in statistical physics under the name “lattice animals,” mainly in regards to percolation problems. One of the main points of interest is to solve the yet unanswered question of how many different polyominoes exist. Most of the focus, so far, went to estimating the number of different polyominoes that can be made with a given fixed number of cells. Recently, there are increased efforts to discover the number of polyominoes with not only a given area, but with a given perimeter size or perimeter defect as well.</span></p><p>Roughly speaking, the perimeter defect is a number that measures how many twists a polyomino has. Formally, the <em>defect</em> of a polyomino <em>P</em> is defined as the deviation of the perimeter size of <em>P</em> from the maximum possible perimeter size taken over all polyominoes of the same area as <em>P</em>. Interestingly, a polyomino might contain a column or row which can be “cut” out of the polyomino, then the remaining parts of the polyomino are “glued” back together along the cut, and result in a smaller polyomino with equal perimeter defect to the original. By repeating such “cut-and-glue” operations on a polyomino until all matching columns and rows have been removed, one obtains a so-called “reduced polyomino.”</p><p><span>We expand on the efforts regarding perimeter defect and reduced polyominoes, in two directions. First, given a fixed perimeter defect, we demonstrate and prove upper and lower bounds on the width and height, as well as an upper bound on the area, of any reduced polyomino with a given perimeter defect. Second, we present an algorithm for enumerating all reduced polyominoes with a given perimeter defect, and calculate their combined generating function. From the generating function, we can extract a formula </span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for the number of polyominoes that have the fixed perimeter defect <em>k</em> and area <em>n</em>, for any <em>n</em>. Using our new algorithm, and in the case of <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span> some additional manual calculations, we provide closed formulae of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, for up to <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span>, as well as the generating functions for up to <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span>. This is an improvement over the previously known formulae, which were known only up to <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Automatic generation of formulae for polyominoes with a fixed perimeter defect\",\"authors\":\"Gill Barequet, Bar Magal\",\"doi\":\"10.1016/j.comgeo.2022.101919\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A <em>polyomino</em><span> is a shape best described as a connected set of cells in the square lattice. As part of recreational mathematics, polyominoes have seen active research since the 1950s. Simultaneously, polyominoes have been investigated in statistical physics under the name “lattice animals,” mainly in regards to percolation problems. One of the main points of interest is to solve the yet unanswered question of how many different polyominoes exist. Most of the focus, so far, went to estimating the number of different polyominoes that can be made with a given fixed number of cells. Recently, there are increased efforts to discover the number of polyominoes with not only a given area, but with a given perimeter size or perimeter defect as well.</span></p><p>Roughly speaking, the perimeter defect is a number that measures how many twists a polyomino has. Formally, the <em>defect</em> of a polyomino <em>P</em> is defined as the deviation of the perimeter size of <em>P</em> from the maximum possible perimeter size taken over all polyominoes of the same area as <em>P</em>. Interestingly, a polyomino might contain a column or row which can be “cut” out of the polyomino, then the remaining parts of the polyomino are “glued” back together along the cut, and result in a smaller polyomino with equal perimeter defect to the original. By repeating such “cut-and-glue” operations on a polyomino until all matching columns and rows have been removed, one obtains a so-called “reduced polyomino.”</p><p><span>We expand on the efforts regarding perimeter defect and reduced polyominoes, in two directions. First, given a fixed perimeter defect, we demonstrate and prove upper and lower bounds on the width and height, as well as an upper bound on the area, of any reduced polyomino with a given perimeter defect. Second, we present an algorithm for enumerating all reduced polyominoes with a given perimeter defect, and calculate their combined generating function. From the generating function, we can extract a formula </span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for the number of polyominoes that have the fixed perimeter defect <em>k</em> and area <em>n</em>, for any <em>n</em>. Using our new algorithm, and in the case of <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span> some additional manual calculations, we provide closed formulae of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, for up to <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span>, as well as the generating functions for up to <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span>. This is an improvement over the previously known formulae, which were known only up to <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span>.</p></div>\",\"PeriodicalId\":51001,\"journal\":{\"name\":\"Computational Geometry-Theory and Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Geometry-Theory and Applications\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0925772122000621\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Geometry-Theory and Applications","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0925772122000621","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Automatic generation of formulae for polyominoes with a fixed perimeter defect
A polyomino is a shape best described as a connected set of cells in the square lattice. As part of recreational mathematics, polyominoes have seen active research since the 1950s. Simultaneously, polyominoes have been investigated in statistical physics under the name “lattice animals,” mainly in regards to percolation problems. One of the main points of interest is to solve the yet unanswered question of how many different polyominoes exist. Most of the focus, so far, went to estimating the number of different polyominoes that can be made with a given fixed number of cells. Recently, there are increased efforts to discover the number of polyominoes with not only a given area, but with a given perimeter size or perimeter defect as well.
Roughly speaking, the perimeter defect is a number that measures how many twists a polyomino has. Formally, the defect of a polyomino P is defined as the deviation of the perimeter size of P from the maximum possible perimeter size taken over all polyominoes of the same area as P. Interestingly, a polyomino might contain a column or row which can be “cut” out of the polyomino, then the remaining parts of the polyomino are “glued” back together along the cut, and result in a smaller polyomino with equal perimeter defect to the original. By repeating such “cut-and-glue” operations on a polyomino until all matching columns and rows have been removed, one obtains a so-called “reduced polyomino.”
We expand on the efforts regarding perimeter defect and reduced polyominoes, in two directions. First, given a fixed perimeter defect, we demonstrate and prove upper and lower bounds on the width and height, as well as an upper bound on the area, of any reduced polyomino with a given perimeter defect. Second, we present an algorithm for enumerating all reduced polyominoes with a given perimeter defect, and calculate their combined generating function. From the generating function, we can extract a formula for the number of polyominoes that have the fixed perimeter defect k and area n, for any n. Using our new algorithm, and in the case of some additional manual calculations, we provide closed formulae of , for up to , as well as the generating functions for up to . This is an improvement over the previously known formulae, which were known only up to .
期刊介绍:
Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems.
Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.