{"paper":{"title":"A minimum-change version of the Chung-Feller theorem for Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Christoph Standke, Torsten M\\\"utze, Veit Wiechert","submitted_at":"2016-03-08T14:11:54Z","abstract_excerpt":"A Dyck path with $2k$ steps and $e$ flaws is a path in the integer lattice that starts at the origin and consists of $k$ many $\\nearrow$-steps and $k$ many $\\searrow$-steps that change the current coordinate by $(1,1)$ or $(1,-1)$, respectively, and that has exactly $e$ many $\\searrow$-steps below the line $y=0$. Denoting by $D_{2k}^e$ the set of Dyck paths with $2k$ steps and $e$ flaws, the Chung-Feller theorem asserts that the sets $D_{2k}^0,D_{2k}^1,\\ldots,D_{2k}^k$ all have the same cardinality $\\frac{1}{k+1}\\binom{2k}{k}=C_k$, the $k$-th Catalan number. The standard combinatorial proof of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02525","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":""},"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"}