Tiling the Plane with Permutations
Identifieur interne : 000E18 ( Istex/Corpus ); précédent : 000E17; suivant : 000E19Tiling the Plane with Permutations
Auteurs : Alexandre Blondin Massé ; Andrea Frosini ; Simone Rinaldi ; Laurent VuillonSource :
- Lecture Notes in Computer Science [ 0302-9743 ]
Abstract
Abstract: A permutomino is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as $XY\overline{X} \,\overline{Y}$ . In this paper we consider 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 $XY\overline{X} \,\overline{Y}$ is pseudo-square is prefix of an infinite word Y ∞ with period 4 |X| N |X| E . Some conjectures obtained through exhaustive search are also presented and discussed in the final section.
Url:
DOI: 10.1007/978-3-642-19867-0_32
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Tiling the Plane with Permutations</title><author><name sortKey="Blondin Masse, Alexandre" sort="Blondin Masse, Alexandre" uniqKey="Blondin Masse A" first="Alexandre" last="Blondin Massé">Alexandre Blondin Massé</name><affiliation><mods:affiliation>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</mods:affiliation></affiliation></author><author><name sortKey="Frosini, Andrea" sort="Frosini, Andrea" uniqKey="Frosini A" first="Andrea" last="Frosini">Andrea Frosini</name><affiliation><mods:affiliation>Dipartimento di Sistemi e Informatica, Università di Firenze, viale Morgagni 65, 50134, Firenze</mods:affiliation></affiliation></author><author><name sortKey="Rinaldi, Simone" sort="Rinaldi, Simone" uniqKey="Rinaldi S" first="Simone" last="Rinaldi">Simone Rinaldi</name><affiliation><mods:affiliation>Dipartimento di Matematica e Informatica, Università di Siena, Pian dei Mantellini 44, 53100, Siena</mods:affiliation></affiliation></author><author><name sortKey="Vuillon, Laurent" sort="Vuillon, Laurent" uniqKey="Vuillon L" first="Laurent" last="Vuillon">Laurent Vuillon</name><affiliation><mods:affiliation>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:3C7B117A1CD87393EE2957754EE0B53E2C5BDB8F</idno><date when="2011" year="2011">2011</date><idno type="doi">10.1007/978-3-642-19867-0_32</idno><idno type="url">https://api.istex.fr/ark:/67375/HCB-ZSQV3PRX-N/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">000E18</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000E18</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Tiling the Plane with Permutations</title><author><name sortKey="Blondin Masse, Alexandre" sort="Blondin Masse, Alexandre" uniqKey="Blondin Masse A" first="Alexandre" last="Blondin Massé">Alexandre Blondin Massé</name><affiliation><mods:affiliation>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</mods:affiliation></affiliation></author><author><name sortKey="Frosini, Andrea" sort="Frosini, Andrea" uniqKey="Frosini A" first="Andrea" last="Frosini">Andrea Frosini</name><affiliation><mods:affiliation>Dipartimento di Sistemi e Informatica, Università di Firenze, viale Morgagni 65, 50134, Firenze</mods:affiliation></affiliation></author><author><name sortKey="Rinaldi, Simone" sort="Rinaldi, Simone" uniqKey="Rinaldi S" first="Simone" last="Rinaldi">Simone Rinaldi</name><affiliation><mods:affiliation>Dipartimento di Matematica e Informatica, Università di Siena, Pian dei Mantellini 44, 53100, Siena</mods:affiliation></affiliation></author><author><name sortKey="Vuillon, Laurent" sort="Vuillon, Laurent" uniqKey="Vuillon L" first="Laurent" last="Vuillon">Laurent Vuillon</name><affiliation><mods:affiliation>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s" type="main" xml:lang="en">Lecture Notes in Computer Science</title><idno type="ISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="ISSN">0302-9743</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0302-9743</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: A permutomino is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as $XY\overline{X} \,\overline{Y}$ . In this paper we consider 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 $XY\overline{X} \,\overline{Y}$ is pseudo-square is prefix of an infinite word Y ∞ with period 4 |X| N |X| E . Some conjectures obtained through exhaustive search are also presented and discussed in the final section.</div></front></TEI><istex><corpusName>springer-ebooks</corpusName><author><json:item><name>Alexandre Blondin Massé</name><affiliations><json:string>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</json:string></affiliations></json:item><json:item><name>Andrea Frosini</name><affiliations><json:string>Dipartimento di Sistemi e Informatica, Università di Firenze, viale Morgagni 65, 50134, Firenze</json:string></affiliations></json:item><json:item><name>Simone Rinaldi</name><affiliations><json:string>Dipartimento di Matematica e Informatica, Università di Siena, Pian dei Mantellini 44, 53100, Siena</json:string></affiliations></json:item><json:item><name>Laurent Vuillon</name><affiliations><json:string>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</json:string></affiliations></json:item></author><arkIstex>ark:/67375/HCB-ZSQV3PRX-N</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: A permutomino is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as $XY\overline{X} \,\overline{Y}$ . In this paper we consider 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 $XY\overline{X} \,\overline{Y}$ is pseudo-square is prefix of an infinite word Y ∞ with period 4 |X| N |X| E . Some conjectures obtained through exhaustive search are also presented and discussed in the final section.</abstract><qualityIndicators><refBibsNative>false</refBibsNative><abstractWordCount>161</abstractWordCount><abstractCharCount>1117</abstractCharCount><keywordCount>0</keywordCount><score>8.914</score><pdfWordCount>4982</pdfWordCount><pdfCharCount>23498</pdfCharCount><pdfVersion>1.6</pdfVersion><pdfPageCount>13</pdfPageCount><pdfPageSize>429.442 x 659.895 pts</pdfPageSize></qualityIndicators><title>Tiling the Plane with Permutations</title><chapterId><json:string>32</json:string><json:string>Chap32</json:string></chapterId><genre><json:string>conference</json:string></genre><serie><title>Lecture Notes in Computer Science</title><language><json:string>unknown</json:string></language><copyrightDate>2011</copyrightDate><issn><json:string>0302-9743</json:string></issn><eissn><json:string>1611-3349</json:string></eissn><editor><json:item><name>David Hutchison</name><affiliations><json:string>Lancaster University, Lancaster, UK</json:string></affiliations></json:item><json:item><name>Takeo Kanade</name><affiliations><json:string>Carnegie Mellon University, Pittsburgh, PA, USA</json:string></affiliations></json:item><json:item><name>Josef Kittler</name><affiliations><json:string>University of Surrey, Guildford, UK</json:string></affiliations></json:item><json:item><name>Jon M. 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type="department">Dipartimento di Sistemi e Informatica</orgName><orgName type="institution">Università di Firenze</orgName><address><street>viale Morgagni 65</street><postCode>50134</postCode><settlement>Firenze</settlement></address></affiliation></author><author><persName><forename type="first">Simone</forename><surname>Rinaldi</surname></persName><affiliation><orgName type="department">Dipartimento di Matematica e Informatica</orgName><orgName type="institution">Università di Siena</orgName><address><street>Pian dei Mantellini 44</street><postCode>53100</postCode><settlement>Siena</settlement></address></affiliation></author><author><persName><forename type="first">Laurent</forename><surname>Vuillon</surname></persName><affiliation><orgName type="department">Laboratoire de Mathématiques</orgName><orgName type="institution">Université de Savoie, CNRS UMR 5127</orgName><address><postCode>73376</postCode><settlement>Le Bourget du Lac</settlement></address></affiliation></author><idno 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Proceedings</title><title level="m" type="part">Discrete Tomography</title><idno type="DOI">10.1007/978-3-642-19867-0</idno><idno type="book-id">978-3-642-19867-0</idno><idno type="ISBN">978-3-642-19866-3</idno><idno type="eISBN">978-3-642-19867-0</idno><idno type="chapter-id">Chap32</idno><idno type="part-id">Part6</idno><editor><persName><forename type="first">Isabelle</forename><surname>Debled-Rennesson</surname></persName><email>isabelle.debled-rennesson@loria.fr</email><affiliation><orgName type="institution">LORIA, Equipe ADAGIo</orgName><address><street>Campus Scientifique, BP 239</street><postCode>54506</postCode><settlement>Vandœuvre-lès-Nancy Cedex</settlement><country key="FR">FRANCE</country></address></affiliation></editor><editor><persName><forename type="first">Eric</forename><surname>Domenjoud</surname></persName><email>eric.domenjoud@loria.fr</email><affiliation><orgName type="institution">LORIA, Equipe ADAGIo</orgName><address><street>Campus Scientifique, BP 239</street><postCode>54506</postCode><settlement>Vandœuvre-lès-Nancy Cedex</settlement><country key="FR">FRANCE</country></address></affiliation></editor><editor><persName><forename type="first">Bertrand</forename><surname>Kerautret</surname></persName><email>bertrand.kerautret@loria.fr</email><affiliation><orgName type="institution">LORIA, Equipe ADAGIo</orgName><address><street>Campus Scientifique, BP 239</street><postCode>54506</postCode><settlement>Vandœuvre-lès-Nancy Cedex</settlement><country key="FR">FRANCE</country></address></affiliation></editor><editor><persName><forename type="first">Philippe</forename><surname>Even</surname></persName><email>philippe.even@loria.fr</email><affiliation><orgName type="institution">LORIA, Equipe ADAGIo</orgName><address><street>Campus Scientifique, BP 239</street><postCode>54506</postCode><settlement>Vandœuvre-lès-Nancy Cedex</settlement><country key="FR">FRANCE</country></address></affiliation></editor><imprint><biblScope unit="vol">6607</biblScope><biblScope unit="page" from="381">381</biblScope><biblScope unit="page" to="393">393</biblScope><biblScope unit="chapter-count">43</biblScope><biblScope unit="part-chapter-count">5</biblScope></imprint></monogr><series><title level="s" type="main" xml:lang="en">Lecture Notes in Computer Science</title><editor><persName><forename type="first">David</forename><surname>Hutchison</surname></persName><affiliation><orgName type="institution">Lancaster University</orgName><address><settlement>Lancaster</settlement><country key="GB">UNITED KINGDOM</country></address></affiliation></editor><editor><persName><forename type="first">Takeo</forename><surname>Kanade</surname></persName><affiliation><orgName type="institution">Carnegie Mellon University</orgName><address><settlement>Pittsburgh</settlement><region>PA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Josef</forename><surname>Kittler</surname></persName><affiliation><orgName type="institution">University of Surrey</orgName><address><settlement>Guildford</settlement><country key="GB">UNITED KINGDOM</country></address></affiliation></editor><editor><persName><forename type="first">Jon</forename><forename type="first">M.</forename><surname>Kleinberg</surname></persName><affiliation><orgName type="institution">Cornell University</orgName><address><settlement>Ithaca</settlement><region>NY</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Friedemann</forename><surname>Mattern</surname></persName><affiliation><orgName type="institution">ETH Zurich</orgName><address><settlement>Zurich</settlement><country key="CH">SWITZERLAND</country></address></affiliation></editor><editor><persName><forename type="first">John</forename><forename type="first">C.</forename><surname>Mitchell</surname></persName><affiliation><orgName type="institution">Stanford University</orgName><address><settlement>Stanford</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Moni</forename><surname>Naor</surname></persName><affiliation><orgName type="institution">Weizmann Institute of Science</orgName><address><settlement>Rehovot</settlement><country key="IL">ISRAEL</country></address></affiliation></editor><editor><persName><forename type="first">Oscar</forename><surname>Nierstrasz</surname></persName><affiliation><orgName type="institution">University of Bern</orgName><address><settlement>Bern</settlement><country key="CH">SWITZERLAND</country></address></affiliation></editor><editor><persName><forename type="first">C.</forename><surname>Pandu Rangan</surname></persName><affiliation><orgName type="institution">Indian Institute of Technology</orgName><address><settlement>Madras</settlement><country key="IN">INDIA</country></address></affiliation></editor><editor><persName><forename type="first">Bernhard</forename><surname>Steffen</surname></persName><affiliation><orgName type="institution">University of Dortmund</orgName><address><settlement>Dortmund</settlement><country key="DE">GERMANY</country></address></affiliation></editor><editor><persName><forename type="first">Madhu</forename><surname>Sudan</surname></persName><affiliation><orgName type="institution">Massachusetts Institute of Technology</orgName><address><region>MA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Demetri</forename><surname>Terzopoulos</surname></persName><affiliation><orgName type="institution">University of California</orgName><address><settlement>Los Angeles</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Doug</forename><surname>Tygar</surname></persName><affiliation><orgName type="institution">University of California</orgName><address><settlement>Berkeley</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Moshe</forename><forename type="first">Y.</forename><surname>Vardi</surname></persName><affiliation><orgName type="institution">Rice University</orgName><address><settlement>Houston</settlement><region>TX</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Gerhard</forename><surname>Weikum</surname></persName><affiliation><orgName type="institution">Max-Planck Institute of Computer Science</orgName><address><settlement>Saarbrücken</settlement><country key="DE">GERMANY</country></address></affiliation></editor><idno type="pISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="seriesID">558</idno></series></biblStruct></sourceDesc></fileDesc><profileDesc><abstract xml:lang="en"><head>Abstract</head><p>A <hi rend="italic">permutomino</hi> is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular <hi rend="italic">convex permutominoes</hi> have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of <hi rend="italic">pseudo-square</hi> polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as <formula xml:id="IEq1" notation="TEX"><media mimeType="image" url=""></media>$XY\overline{X} \,\overline{Y}$
</formula>.</p><p>In this paper we consider 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 <hi rend="italic">X</hi>, each word <hi rend="italic">Y</hi> such that <formula xml:id="IEq2" notation="TEX"><media mimeType="image" url=""></media>$XY\overline{X} \,\overline{Y}$
</formula> is pseudo-square is prefix of an infinite word <hi rend="italic">Y</hi><hi rend="subscript"> ∞ </hi> with period 4 |<hi rend="italic">X</hi>|<hi rend="subscript"><hi rend="italic">N</hi></hi> |<hi rend="italic">X</hi>|<hi rend="subscript"><hi rend="italic">E</hi></hi>.</p><p>Some conjectures obtained through exhaustive search are also presented and discussed in the final section.</p></abstract><textClass ana="subject"><keywords scheme="book-subject-collection"><list><label>SUCO11645</label><item><term>Computer Science</term></item></list></keywords></textClass><textClass ana="subject"><keywords scheme="book-subject"><list><label>I</label><item><term type="Primary">Computer Science</term></item><label>I22013</label><item><term type="Secondary" subtype="priority-1">Computer Graphics</term></item><label>I22021</label><item><term type="Secondary" subtype="priority-2">Image Processing and Computer Vision</term></item><label>I22005</label><item><term type="Secondary" 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DisplayOrder="Western"><GivenName>Gerhard</GivenName><FamilyName>Weikum</FamilyName></EditorName></Editor><Affiliation ID="Aff1"><OrgName>Lancaster University</OrgName><OrgAddress><City>Lancaster</City><Country>UK</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Carnegie Mellon University</OrgName><OrgAddress><City>Pittsburgh</City><State>PA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgName>University of Surrey</OrgName><OrgAddress><City>Guildford</City><Country>UK</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgName>Cornell University</OrgName><OrgAddress><City>Ithaca</City><State>NY</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff5"><OrgName>ETH Zurich</OrgName><OrgAddress><City>Zurich</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff6"><OrgName>Stanford University</OrgName><OrgAddress><City>Stanford</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff7"><OrgName>Weizmann Institute of Science</OrgName><OrgAddress><City>Rehovot</City><Country>Israel</Country></OrgAddress></Affiliation><Affiliation ID="Aff8"><OrgName>University of Bern</OrgName><OrgAddress><City>Bern</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff9"><OrgName>Indian Institute of Technology</OrgName><OrgAddress><City>Madras</City><Country>India</Country></OrgAddress></Affiliation><Affiliation ID="Aff10"><OrgName>University of Dortmund</OrgName><OrgAddress><City>Dortmund</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff11"><OrgName>Massachusetts Institute of Technology</OrgName><OrgAddress><State>MA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff12"><OrgName>University of California</OrgName><OrgAddress><City>Los Angeles</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff13"><OrgName>University of California</OrgName><OrgAddress><City>Berkeley</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff14"><OrgName>Rice University</OrgName><OrgAddress><City>Houston</City><State>TX</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff15"><OrgName>Max-Planck Institute of Computer Science</OrgName><OrgAddress><City>Saarbrücken</City><Country>Germany</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Proceedings" ContainsESM="No" Language="En" MediaType="eBook" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-642-19867-0</BookID><BookTitle>Discrete Geometry for Computer Imagery</BookTitle><BookSubTitle>16th IAPR International Conference, DGCI 2011, Nancy, France, April 6-8, 2011. Proceedings</BookSubTitle><BookVolumeNumber>6607</BookVolumeNumber><BookSequenceNumber>6607</BookSequenceNumber><BookDOI>10.1007/978-3-642-19867-0</BookDOI><BookTitleID>217662</BookTitleID><BookPrintISBN>978-3-642-19866-3</BookPrintISBN><BookElectronicISBN>978-3-642-19867-0</BookElectronicISBN><BookChapterCount>43</BookChapterCount><BookCopyright><CopyrightHolderName>Springer Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2011</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I22013" Priority="1" Type="Secondary">Computer Graphics</BookSubject><BookSubject Code="I22021" Priority="2" Type="Secondary">Image Processing and Computer Vision</BookSubject><BookSubject Code="I22005" Priority="3" Type="Secondary">Computer Imaging, Vision, Pattern Recognition and Graphics</BookSubject><BookSubject Code="I2203X" Priority="4" Type="Secondary">Pattern Recognition</BookSubject><BookSubject Code="I17028" Priority="5" Type="Secondary">Discrete Mathematics in Computer Science</BookSubject><BookSubject Code="I16021" Priority="6" Type="Secondary">Algorithm Analysis and Problem Complexity</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Isabelle</GivenName><FamilyName>Debled-Rennesson</FamilyName></EditorName><Contact><Email>isabelle.debled-rennesson@loria.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Eric</GivenName><FamilyName>Domenjoud</FamilyName></EditorName><Contact><Email>eric.domenjoud@loria.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Bertrand</GivenName><FamilyName>Kerautret</FamilyName></EditorName><Contact><Email>bertrand.kerautret@loria.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Philippe</GivenName><FamilyName>Even</FamilyName></EditorName><Contact><Email>philippe.even@loria.fr</Email></Contact></Editor><Affiliation ID="Aff16"><OrgName>LORIA, Equipe ADAGIo</OrgName><OrgAddress><Street>Campus Scientifique, BP 239</Street><Postcode>54506</Postcode><City>Vandœuvre-lès-Nancy Cedex</City><Country>France</Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Part ID="Part6"><PartInfo TocLevels="0"><PartID>6</PartID><PartSequenceNumber>6</PartSequenceNumber><PartTitle>Discrete Tomography</PartTitle><PartChapterCount>5</PartChapterCount><PartContext><SeriesID>558</SeriesID><BookTitle>Discrete Geometry for Computer Imagery</BookTitle></PartContext></PartInfo><Chapter ID="Chap32" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingDepth="2" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>32</ChapterID><ChapterDOI>10.1007/978-3-642-19867-0_32</ChapterDOI><ChapterSequenceNumber>32</ChapterSequenceNumber><ChapterTitle Language="En">Tiling the Plane with Permutations</ChapterTitle><ChapterFirstPage>381</ChapterFirstPage><ChapterLastPage>393</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2011</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><PartID>6</PartID><BookID>978-3-642-19867-0</BookID><BookTitle>Discrete Geometry for Computer Imagery</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff17"><AuthorName DisplayOrder="Western"><GivenName>Alexandre</GivenName><FamilyName>Blondin Massé</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff18"><AuthorName DisplayOrder="Western"><GivenName>Andrea</GivenName><FamilyName>Frosini</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff19"><AuthorName DisplayOrder="Western"><GivenName>Simone</GivenName><FamilyName>Rinaldi</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff17"><AuthorName DisplayOrder="Western"><GivenName>Laurent</GivenName><FamilyName>Vuillon</FamilyName></AuthorName></Author><Affiliation ID="Aff17"><OrgDivision>Laboratoire de Mathématiques</OrgDivision><OrgName>Université de Savoie, CNRS UMR 5127</OrgName><OrgAddress><Postcode>73376</Postcode><City>Le Bourget du Lac</City></OrgAddress></Affiliation><Affiliation ID="Aff18"><OrgDivision>Dipartimento di Sistemi e Informatica</OrgDivision><OrgName>Università di Firenze</OrgName><OrgAddress><Street>viale Morgagni 65</Street><Postcode>50134</Postcode><City>Firenze</City></OrgAddress></Affiliation><Affiliation ID="Aff19"><OrgDivision>Dipartimento di Matematica e Informatica</OrgDivision><OrgName>Università di Siena</OrgName><OrgAddress><Street>Pian dei Mantellini 44</Street><Postcode>53100</Postcode><City>Siena</City></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>A <Emphasis Type="Italic">permutomino</Emphasis> is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular <Emphasis Type="Italic">convex permutominoes</Emphasis> have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of <Emphasis Type="Italic">pseudo-square</Emphasis> polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject FileRef="978-3-642-19867-0_32_Chapter_TeX2GIF_IEq1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$XY\overline{X} \,\overline{Y}$</EquationSource></InlineEquation>.</Para><Para>In this paper we consider 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 <Emphasis Type="Italic">X</Emphasis>, each word <Emphasis Type="Italic">Y</Emphasis> such that <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject FileRef="978-3-642-19867-0_32_Chapter_TeX2GIF_IEq2.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$XY\overline{X} \,\overline{Y}$</EquationSource></InlineEquation> is pseudo-square is prefix of an infinite word <Emphasis Type="Italic">Y</Emphasis><Subscript> ∞ </Subscript> with period 4 |<Emphasis Type="Italic">X</Emphasis>|<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript> |<Emphasis Type="Italic">X</Emphasis>|<Subscript><Emphasis Type="Italic">E</Emphasis></Subscript>.</Para><Para>Some conjectures obtained through exhaustive search are also presented and discussed in the final section.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Part></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Tiling the Plane with Permutations</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Tiling the Plane with Permutations</title></titleInfo><name type="personal"><namePart type="given">Alexandre</namePart><namePart type="family">Blondin Massé</namePart><affiliation>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Andrea</namePart><namePart type="family">Frosini</namePart><affiliation>Dipartimento di Sistemi e Informatica, Università di Firenze, viale Morgagni 65, 50134, Firenze</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Simone</namePart><namePart type="family">Rinaldi</namePart><affiliation>Dipartimento di Matematica e Informatica, Università di Siena, Pian dei Mantellini 44, 53100, Siena</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Laurent</namePart><namePart type="family">Vuillon</namePart><affiliation>Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376, Le Bourget du Lac</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="conference" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-BFHXPBJJ-3">conference</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2011</dateIssued><copyrightDate encoding="w3cdtf">2011</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: A permutomino is a polyomino uniquely determined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2,3]. On the other side, Beauquier and Nivat [5] introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. polyominoes that tile the plane by translation: a polyomino is called pseudo-square if its boundary word may be factorized as $XY\overline{X} \,\overline{Y}$ . In this paper we consider 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 $XY\overline{X} \,\overline{Y}$ is pseudo-square is prefix of an infinite word Y ∞ with period 4 |X| N |X| E . Some conjectures obtained through exhaustive search are also presented and discussed in the final section.</abstract><relatedItem type="host"><titleInfo><title>Discrete Geometry for Computer Imagery</title><subTitle>16th IAPR International Conference, DGCI 2011, Nancy, France, April 6-8, 2011. Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Isabelle</namePart><namePart type="family">Debled-Rennesson</namePart><affiliation>LORIA, Equipe ADAGIo, Campus Scientifique, BP 239, 54506, Vandœuvre-lès-Nancy Cedex, France</affiliation><affiliation>E-mail: isabelle.debled-rennesson@loria.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Eric</namePart><namePart type="family">Domenjoud</namePart><affiliation>LORIA, Equipe ADAGIo, Campus Scientifique, BP 239, 54506, Vandœuvre-lès-Nancy Cedex, France</affiliation><affiliation>E-mail: eric.domenjoud@loria.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Bertrand</namePart><namePart type="family">Kerautret</namePart><affiliation>LORIA, Equipe ADAGIo, Campus Scientifique, BP 239, 54506, Vandœuvre-lès-Nancy Cedex, France</affiliation><affiliation>E-mail: 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type="given">Takeo</namePart><namePart type="family">Kanade</namePart><affiliation>Carnegie Mellon University, Pittsburgh, PA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Josef</namePart><namePart type="family">Kittler</namePart><affiliation>University of Surrey, Guildford, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jon</namePart><namePart type="given">M.</namePart><namePart type="family">Kleinberg</namePart><affiliation>Cornell University, Ithaca, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Friedemann</namePart><namePart type="family">Mattern</namePart><affiliation>ETH Zurich, Zurich, Switzerland</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">John</namePart><namePart 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type="family">Steffen</namePart><affiliation>University of Dortmund, Dortmund, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Madhu</namePart><namePart type="family">Sudan</namePart><affiliation>Massachusetts Institute of Technology, MA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Demetri</namePart><namePart type="family">Terzopoulos</namePart><affiliation>University of California, Los Angeles, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Doug</namePart><namePart type="family">Tygar</namePart><affiliation>University of California, Berkeley, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moshe</namePart><namePart type="given">Y.</namePart><namePart type="family">Vardi</namePart><affiliation>Rice University, Houston, TX, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Gerhard</namePart><namePart type="family">Weikum</namePart><affiliation>Max-Planck Institute of Computer Science, Saarbrücken, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2011</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="SeriesID">558</identifier><recordInfo><recordOrigin>Springer Berlin Heidelberg, 2011</recordOrigin></recordInfo></relatedItem><identifier type="istex">3C7B117A1CD87393EE2957754EE0B53E2C5BDB8F</identifier><identifier type="ark">ark:/67375/HCB-ZSQV3PRX-N</identifier><identifier type="DOI">10.1007/978-3-642-19867-0_32</identifier><identifier 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