Extending the parking space
The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2013-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2325/pdf |
| _version_ | 1797270305095286784 |
|---|---|
| author | Andrew Berget Brendon Rhoades |
| author_facet | Andrew Berget Brendon Rhoades |
| author_sort | Andrew Berget |
| collection | DOAJ |
| description | The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. |
| first_indexed | 2024-04-25T02:02:09Z |
| format | Article |
| id | doaj.art-bf2e5da0555a4929a31dbc3864238f50 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:09Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-bf2e5da0555a4929a31dbc3864238f502024-03-07T14:52:36ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.23252325Extending the parking spaceAndrew Berget0Brendon Rhoades1Department of Mathematics [Seattle]Department of Mathematics [Univ California San Diego]The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization.https://dmtcs.episciences.org/2325/pdfparking functionssymmetric groupdyck pathsrepresentationmatriod[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Andrew Berget Brendon Rhoades Extending the parking space Discrete Mathematics & Theoretical Computer Science parking functions symmetric group dyck paths representation matriod [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Extending the parking space |
| title_full | Extending the parking space |
| title_fullStr | Extending the parking space |
| title_full_unstemmed | Extending the parking space |
| title_short | Extending the parking space |
| title_sort | extending the parking space |
| topic | parking functions symmetric group dyck paths representation matriod [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2325/pdf |
| work_keys_str_mv | AT andrewberget extendingtheparkingspace AT brendonrhoades extendingtheparkingspace |