{"paper":{"title":"The ascent lattice on Dyck paths","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jean-Luc Baril, Mehdi Naima, Mireille Bousquet-M\\'elou, Sergey Kirgizov","submitted_at":"2024-09-24T11:27:42Z","abstract_excerpt":"In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\\`ere, where cover relations replace a factor $DU^k D$ by $U^kD^2$. By relating this poset to another poset recently defined by Nadeau and Tewari, we prove that this still yields a lattice, which we call the ascent lattice, $L_n$. We then count intervals in $L_n$. Their generating function is found to be algebraic of degree $3$. The proof is based on a recursive decomposition of intervals in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.15982","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/2409.15982/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"}