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Guo","submitted_at":"2014-05-19T14:49:15Z","abstract_excerpt":"We prove that, for any prime $p$ and positive integer $r$ with $p^r>2$, the number of multinomial coefficients such that $$ {k\\choose k_1,k_2,\\ldots,k_n}=p^r,\\quad \\text{and}\\quad k_1+2k_2+\\cdots+nk_n=n, $$ is given by $$ \\delta_{p^r,\\,k}\\left(\\left\\lfloor\\frac{n-1}{p^r-1}\\right\\rfloor -\\delta_{0,\\,n\\bmod p^r} \\right), $$ where $\\delta_{i,j}$ is the Kronecker delta and $\\lfloor x\\rfloor$ stands for the largest integer not exceeding $x$. This confirms a recent conjecture of Mircea Merca."},"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":"1405.4755","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-19T14:49:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d4018092aade63d58216a6cffd04d263af1447873b379adc4b5a59f5b5bb810c","abstract_canon_sha256":"983fa6f92dea51ff614ba557c45716e7387818e600939d93a2c79a056a4d8735"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:34.733871Z","signature_b64":"UJ32iYxIjNzdVAIPdfrFLWSTuk6r9H0iiQRl+dQryk0vV7MDOxeJDqmaS+6io+Di39OIhE66DVgxRnCnsYU1AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04851ec79c013dce07bb74d7681fd751385c0915f7b6a0702b2855d251e12dc3","last_reissued_at":"2026-05-18T02:51:34.733435Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:34.733435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a conjecture of Mircea Merca","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2014-05-19T14:49:15Z","abstract_excerpt":"We prove that, for any prime $p$ and positive integer $r$ with $p^r>2$, the number of multinomial coefficients such that $$ {k\\choose k_1,k_2,\\ldots,k_n}=p^r,\\quad \\text{and}\\quad k_1+2k_2+\\cdots+nk_n=n, $$ is given by $$ \\delta_{p^r,\\,k}\\left(\\left\\lfloor\\frac{n-1}{p^r-1}\\right\\rfloor -\\delta_{0,\\,n\\bmod p^r} \\right), $$ where $\\delta_{i,j}$ is the Kronecker delta and $\\lfloor x\\rfloor$ stands for the largest integer not exceeding $x$. 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