{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-steep-bounce-zeta-map-in-parabolic","title":"The Steep-Bounce Zeta Map in Parabolic Cataland","arxiv_id":"1903.08515","date":"2019-03-20","proceeding":null,"authors":["Cesar Ceballos","Wenjie Fang","Henri Mühle"],"abstract":"As a classical object, the Tamari lattice has many generalizations, including $\\nu$-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to $\\nu$-Tamari lattices for bounce paths $\\nu$. We then introduce a new combinatorial object called `left-aligned colorable tree', and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in $q,t$-Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.","url_abs":"http://arxiv.org/abs/1903.08515v3","url_pdf":"http://arxiv.org/pdf/1903.08515v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-steep-bounce-zeta-map-in-parabolic","repo_url":"https://github.com/fwjmath/assorted-tamari","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}