Remixed Eulerian numbers
Remixed Eulerian numbers are a polynomial q-deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia–Remmel’s q-hit numbers. W...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000579/type/journal_article |
| _version_ | 1797768821288730624 |
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| author | Philippe Nadeau Vasu Tewari |
| author_facet | Philippe Nadeau Vasu Tewari |
| author_sort | Philippe Nadeau |
| collection | DOAJ |
| description | Remixed Eulerian numbers are a polynomial q-deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia–Remmel’s q-hit numbers. We study their combinatorics in more depth. As polynomials in q, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At
$q=1$
, the former recovers Postnikov’s interpretation whereas the latter recovers Liu’s interpretation, both of which were obtained via methods different from ours. |
| first_indexed | 2024-03-12T20:58:52Z |
| format | Article |
| id | doaj.art-462bb8689db94cdeb8cb4988463c2641 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-12T20:58:52Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-462bb8689db94cdeb8cb4988463c26412023-07-31T09:20:07ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.57Remixed Eulerian numbersPhilippe Nadeau0https://orcid.org/0000-0002-7230-755XVasu Tewari1Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France; E-mail:Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI 96822, USA; E-mail:Remixed Eulerian numbers are a polynomial q-deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia–Remmel’s q-hit numbers. We study their combinatorics in more depth. As polynomials in q, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At $q=1$ , the former recovers Postnikov’s interpretation whereas the latter recovers Liu’s interpretation, both of which were obtained via methods different from ours.https://www.cambridge.org/core/product/identifier/S2050509423000579/type/journal_article05A1552B05 |
| spellingShingle | Philippe Nadeau Vasu Tewari Remixed Eulerian numbers Forum of Mathematics, Sigma 05A15 52B05 |
| title | Remixed Eulerian numbers |
| title_full | Remixed Eulerian numbers |
| title_fullStr | Remixed Eulerian numbers |
| title_full_unstemmed | Remixed Eulerian numbers |
| title_short | Remixed Eulerian numbers |
| title_sort | remixed eulerian numbers |
| topic | 05A15 52B05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000579/type/journal_article |
| work_keys_str_mv | AT philippenadeau remixedeuleriannumbers AT vasutewari remixedeuleriannumbers |