Symmetry of Narayana Numbers and Rowvacuation of Root Posets
For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2021-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509421000475/type/journal_article |
| _version_ | 1811156171545903104 |
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| author | Colin Defant Sam Hopkins |
| author_facet | Colin Defant Sam Hopkins |
| author_sort | Colin Defant |
| collection | DOAJ |
| description | For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality
$r-k$
. However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. |
| first_indexed | 2024-04-10T04:47:05Z |
| format | Article |
| id | doaj.art-4fc6ef7a39924ea2b610a8b19928e1a7 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:05Z |
| publishDate | 2021-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-4fc6ef7a39924ea2b610a8b19928e1a72023-03-09T12:34:52ZengCambridge University PressForum of Mathematics, Sigma2050-50942021-01-01910.1017/fms.2021.47Symmetry of Narayana Numbers and Rowvacuation of Root PosetsColin Defant0Sam Hopkins1Department of Mathematics, Princeton University, Princeton, NJ 08544 USA; E-mail:School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA; E-mail:For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers.https://www.cambridge.org/core/product/identifier/S2050509421000475/type/journal_article05E1805A1917B2220F55 |
| spellingShingle | Colin Defant Sam Hopkins Symmetry of Narayana Numbers and Rowvacuation of Root Posets Forum of Mathematics, Sigma 05E18 05A19 17B22 20F55 |
| title | Symmetry of Narayana Numbers and Rowvacuation of Root Posets |
| title_full | Symmetry of Narayana Numbers and Rowvacuation of Root Posets |
| title_fullStr | Symmetry of Narayana Numbers and Rowvacuation of Root Posets |
| title_full_unstemmed | Symmetry of Narayana Numbers and Rowvacuation of Root Posets |
| title_short | Symmetry of Narayana Numbers and Rowvacuation of Root Posets |
| title_sort | symmetry of narayana numbers and rowvacuation of root posets |
| topic | 05E18 05A19 17B22 20F55 |
| url | https://www.cambridge.org/core/product/identifier/S2050509421000475/type/journal_article |
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