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This observation leads to new limit formulae for $q$-analogues.\n  Keywords: juggling pattern; $q$-Stirling numb"},"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":"1310.2725","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-10T07:41:38Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"3187aa6d9ecbeae3653557c96a16d1f4befe29dbef0228d3f50b8a28ceed7664","abstract_canon_sha256":"a6c68ed2ecb8ad812391c1599559e583b3d3d719bd9d776bccbdce56b89cc0cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:01.069446Z","signature_b64":"s2EKBI8jtSMYc57NpZJtSef8YGuTu5eSksuq2tKdmLmiQLyobFgVV8ogadInBCWOpkxFKSCUMHZ4haHBV3OPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe23d1b19183c211661b8aa96030d00ad43e80bbc08e9699a702433140cf52c6","last_reissued_at":"2026-05-18T02:26:01.068819Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:01.068819Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric juggling with q-analogues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Alexander Engstr\\\"om, Harri Varpanen, Lasse Leskel\\\"a","submitted_at":"2013-10-10T07:41:38Z","abstract_excerpt":"We derive a combinatorial equilibrium for bounded juggling patterns with a random, $q$-geometric throw distribution. 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