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In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples.\n  In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is th"},"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":"1701.06954","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-24T16:09:03Z","cross_cats_sorted":[],"title_canon_sha256":"bfb1713fc353e10b033e3f013cc1538c5e7fa62f47615f2cd2eb858e2d6f2e66","abstract_canon_sha256":"7623962a50e66ac52a8ff9d3e0b340a1eac17dd28c137964d04454fc7d013811"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:43.616975Z","signature_b64":"CqCixPbGQt/ohhQ3k03vvbmZaKtZfPlXZHmUuYE6/45ti9LKhluyM0wZ8CXG2tjdp5E93eGAUuKnRWqe/yGzCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32971529aeab14abb08bb4df8dcccbc14f9797234030307592d05197db72bd3d","last_reissued_at":"2026-05-17T23:44:43.616534Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:43.616534Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The cycle polynomial of a permutation group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jason Semeraro, Peter J. 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