{"paper":{"title":"Parking functions, Fubini rankings, and Boolean intervals in the weak order of $\\mathfrak{S}_n$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jan Kretschmann, J. Carlos Mart\\'inez Mori, Jennifer Elder, Pamela E. Harris","submitted_at":"2023-06-26T14:37:10Z","abstract_excerpt":"Let $\\mathfrak{S}_n$ denote the symmetric group and let $W(\\mathfrak{S}_n)$ denote the weak order of $\\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W(\\mathfrak{S}_n)$ and the total number of Boolean intervals of rank $k$ in $W(\\mathfrak{S}_n)$. Furthermore, for any $\\pi\\in\\mathfrak{S}_n$, we establish that the number of Boolean intervals in $W(\\mathfrak{S}_n)$ with minimal element $\\pi$ is a product of Fibonacci numbers. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.14734","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2306.14734/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}