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The classical Chung-Feller theorem says that the total number of $(n,m)$-Dyck path is independent of $m$ and is equal to the $n$-th Catalan number $C_n=\\frac{1}{n+1}{2n \\choose n}$. For any integer $k$ with $1 \\leq k \\leq n$, let $p_{n,m,k}$ be the total number of $(n,m)$-Dyck paths with $k"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1711.08584","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-23T06:16:13Z","cross_cats_sorted":[],"title_canon_sha256":"23bc0f8205c165a474503875b4fc54f623ef7b65578aa2eca4aec7bd12d08a51","abstract_canon_sha256":"560f730ff0252e42e14474fb8df484c36e190e17cab437863005a5fcefdb6f2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:59.375634Z","signature_b64":"tH2a7uhQ2S4Z/+Ks0BgCpAy1l7j0KTBAuNZy1PeFEBcoIaDKG/DiDcsx+Q65PmKsZEakOQfZsjdiBwly6epnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad6467854ec97940f49cd8a3942a47b6cf6d06604982490dacfdc37b269f56e0","last_reissued_at":"2026-05-18T00:24:59.375256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:59.375256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Refinements of two identities on $(n,m)$-Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kuo Yu, Rosena R. X. Du","submitted_at":"2017-11-23T06:16:13Z","abstract_excerpt":"For integers $n, m$ with $n \\geq 1$ and $0 \\leq m \\leq n$, an $(n,m)$-Dyck path is a lattice path in the integer lattice $\\mathbb{Z} \\times \\mathbb{Z}$ using up steps $(0,1)$ and down steps $(1,0)$ that goes from the origin $(0,0)$ to the point $(n,n)$ and contains exactly $m$ up steps below the line $y=x$. The classical Chung-Feller theorem says that the total number of $(n,m)$-Dyck path is independent of $m$ and is equal to the $n$-th Catalan number $C_n=\\frac{1}{n+1}{2n \\choose n}$. 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