A $q,t-$analogue of Narayana numbers
We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbe...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2013-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2329/pdf |
| _version_ | 1797270307802710016 |
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| author | Jean-Christophe Aval Michele D'Adderio Mark Dukes Angela Hicks Yvan Le Borgne |
| author_facet | Jean-Christophe Aval Michele D'Adderio Mark Dukes Angela Hicks Yvan Le Borgne |
| author_sort | Jean-Christophe Aval |
| collection | DOAJ |
| description | We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics. |
| first_indexed | 2024-04-25T02:02:12Z |
| format | Article |
| id | doaj.art-c84224da3384492e85b807461fdd2b72 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:12Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-c84224da3384492e85b807461fdd2b722024-03-07T14:52:36ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.23292329A $q,t-$analogue of Narayana numbersJean-Christophe Aval0Michele D'AdderioMark Dukes1https://orcid.org/0000-0002-2779-2680Angela Hicks2Yvan Le Borgne3Laboratoire Bordelais de Recherche en InformatiqueDepartment of Computer and Information Sciences [Univ Strathclyde]Department of Mathematics [Univ California San Diego]Laboratoire Bordelais de Recherche en InformatiqueWe study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics.https://dmtcs.episciences.org/2329/pdfqt-narayanarectangular polyominoesparking functions.[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Jean-Christophe Aval Michele D'Adderio Mark Dukes Angela Hicks Yvan Le Borgne A $q,t-$analogue of Narayana numbers Discrete Mathematics & Theoretical Computer Science q t-narayana rectangular polyominoes parking functions. [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | A $q,t-$analogue of Narayana numbers |
| title_full | A $q,t-$analogue of Narayana numbers |
| title_fullStr | A $q,t-$analogue of Narayana numbers |
| title_full_unstemmed | A $q,t-$analogue of Narayana numbers |
| title_short | A $q,t-$analogue of Narayana numbers |
| title_sort | q t analogue of narayana numbers |
| topic | q t-narayana rectangular polyominoes parking functions. [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2329/pdf |
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