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Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of $[n]$.\n  It is known that the volume of the standard permutohedron equals the number of trees with $n$ labeled vertices multiplied by $\\sqrt{n}$. The number of integer points of the standard permutohedron equals the number of forests on $n$ labeled vertices.\n  In the paper we prove that the volume of the cyclopermutohedron also equals some weighted"},"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":"1505.00352","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-02T16:37:40Z","cross_cats_sorted":[],"title_canon_sha256":"789e4a0af01aaa4649525f29e45b190d1ee0ef19e37172475e3112c97653f703","abstract_canon_sha256":"edda44d116ec6502863aab23cf1e3b88b8e2896d279514cde3273dc0115661fc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:10.399888Z","signature_b64":"Mh/oTNyh02ZnINSMVI1XdPYaWUOvL9j3MF6n1I5UbZyQeGF6hamnBEQkBjoBam3Ekwai4xQrkqZcGxPt7YL5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f5b952d4cd921864e58a2ef81de7db79a18c81e6e1e71d503915e31fbd0ad91","last_reissued_at":"2026-05-18T02:17:10.399076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:10.399076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Volume and lattice points counting for the cyclopermutohedron","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Gaiane Panina, Ilya Nekrasov","submitted_at":"2015-05-02T16:37:40Z","abstract_excerpt":"The face lattice of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\\{1,...,n\\}$. 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