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Our main result is an asymptotic for the number of solutions to $\\pi_1 + \\pi_2 + \\pi_3 = f$ with $\\pi_1,\\pi_2,\\pi_3\\in S$, where $f:\\{1,\\dots,n\\}\\to G$ is an arbitary function satisfying $\\sum_{i=1}^n f(i) = \\sum G$. This extends recent work of Manners, Mrazovi\\'c, and the author. Using the same method we also prove a less interesting asymptotic for solutions to $\\pi_1 + \\pi_2 + \\pi_3 + \\pi_4 = f$, and we also s"},"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":"1704.02407","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-08T00:18:38Z","cross_cats_sorted":["cs.CR"],"title_canon_sha256":"d8980d9e25d242c5b168e83dea82ffadbd28092e713d07d2140282fbceb60d06","abstract_canon_sha256":"be88e61fe3fa079a508d33e366a383eaf8b74aa14876b1b376631d7000c089d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:45.677981Z","signature_b64":"jwdIkChOzDXS2h11f8VYcEIBwkvDId0XGOCb+u+gXsZiglGmm0VnU/5y16yEKZ+3YE5mPgmi5T0R5eT5HOQfAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6de5566549e2bc1cf5c2962642d6f6512cf1995c5d93f13a2b423894e15ce5c1","last_reissued_at":"2026-05-18T00:46:45.677355Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:45.677355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"More on additive triples of bijections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR"],"primary_cat":"math.CO","authors_text":"Sean Eberhard","submitted_at":"2017-04-08T00:18:38Z","abstract_excerpt":"We study additive properties of the set $S$ of bijections (or permutations) $\\{1,\\dots,n\\}\\to G$, thought of as a subset of $G^n$, where $G$ is an arbitrary abelian group of order $n$. 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