Alternating Catalan numbers and cover with triple ramification

The Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, t...

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Autores: Farkas, Gavril, Moschetti, Riccardo, Naranjo del Val, Juan Carlos, Pirola, Gian Pietro
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2021
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/190456
Acceso en línea:https://hdl.handle.net/2445/190456
Access Level:acceso abierto
Palabra clave:Corbes algebraiques
Geometria algebraica
Teoria de grups
Combinatòria (Matemàtica)
Algebraic curves
Algebraic geometry
Group theory
Combinations
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spelling Alternating Catalan numbers and cover with triple ramificationFarkas, GavrilMoschetti, RiccardoNaranjo del Val, Juan CarlosPirola, Gian PietroCorbes algebraiquesGeometria algebraicaTeoria de grupsCombinatòria (Matemàtica)Algebraic curvesAlgebraic geometryGroup theoryCombinationsThe Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, the Catalan number $C_n$ counts the covers $C \rightarrow \mathbb{P}^1$ of minimal degree $n+1$ from a general curve $C$ of genus $2 n$. Each such cover has simple ramification and its monodromy group equals $S_{n+1}$. By degenerating $C$ to a rational $g$-nodal curve, it was already known to Castelnuovo $[\mathrm{C}]$ that the number of such covers coincides with the degree of the Grassmannian $G(2, n+2)$ in its Plücker embedding, which is well-known to equal $C_n$.Centro Edizioni Scuola Normale Superiore di Pisa2021info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2445/190456Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésVersió postprint del document publicat a: https://doi.org/10.2422/2036-2145.201909_009Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2021, vol. XXII, num. 2, p. 665-690https://doi.org/10.2422/2036-2145.201909_009(c) Centro Edizioni Scuola Normale Superiore di Pisa, 2021info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1904562026-05-27T06:46:51Z
dc.title.none.fl_str_mv Alternating Catalan numbers and cover with triple ramification
title Alternating Catalan numbers and cover with triple ramification
spellingShingle Alternating Catalan numbers and cover with triple ramification
Farkas, Gavril
Corbes algebraiques
Geometria algebraica
Teoria de grups
Combinatòria (Matemàtica)
Algebraic curves
Algebraic geometry
Group theory
Combinations
title_short Alternating Catalan numbers and cover with triple ramification
title_full Alternating Catalan numbers and cover with triple ramification
title_fullStr Alternating Catalan numbers and cover with triple ramification
title_full_unstemmed Alternating Catalan numbers and cover with triple ramification
title_sort Alternating Catalan numbers and cover with triple ramification
dc.creator.none.fl_str_mv Farkas, Gavril
Moschetti, Riccardo
Naranjo del Val, Juan Carlos
Pirola, Gian Pietro
author Farkas, Gavril
author_facet Farkas, Gavril
Moschetti, Riccardo
Naranjo del Val, Juan Carlos
Pirola, Gian Pietro
author_role author
author2 Moschetti, Riccardo
Naranjo del Val, Juan Carlos
Pirola, Gian Pietro
author2_role author
author
author
dc.subject.none.fl_str_mv Corbes algebraiques
Geometria algebraica
Teoria de grups
Combinatòria (Matemàtica)
Algebraic curves
Algebraic geometry
Group theory
Combinations
topic Corbes algebraiques
Geometria algebraica
Teoria de grups
Combinatòria (Matemàtica)
Algebraic curves
Algebraic geometry
Group theory
Combinations
description The Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, the Catalan number $C_n$ counts the covers $C \rightarrow \mathbb{P}^1$ of minimal degree $n+1$ from a general curve $C$ of genus $2 n$. Each such cover has simple ramification and its monodromy group equals $S_{n+1}$. By degenerating $C$ to a rational $g$-nodal curve, it was already known to Castelnuovo $[\mathrm{C}]$ that the number of such covers coincides with the degree of the Grassmannian $G(2, n+2)$ in its Plücker embedding, which is well-known to equal $C_n$.
publishDate 2021
dc.date.none.fl_str_mv 2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/190456
url https://hdl.handle.net/2445/190456
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.2422/2036-2145.201909_009
Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2021, vol. XXII, num. 2, p. 665-690
https://doi.org/10.2422/2036-2145.201909_009
dc.rights.none.fl_str_mv (c) Centro Edizioni Scuola Normale Superiore di Pisa, 2021
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Centro Edizioni Scuola Normale Superiore di Pisa, 2021
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Centro Edizioni Scuola Normale Superiore di Pisa
publisher.none.fl_str_mv Centro Edizioni Scuola Normale Superiore di Pisa
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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