{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:BR64GDYRWB7VTTS4HQUYQ5JXYL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ba7214f8c772d4ad44f37bcd02a5b1198062e1737b7721630eadfc08c9806fe3","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2010-09-13T18:32:17Z","title_canon_sha256":"46e6ed9d4e47d299d62c4bc6d7240f279b66331de8d1c8b728d64938b0b8e512"},"schema_version":"1.0","source":{"id":"1009.2474","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2474","created_at":"2026-05-18T04:40:56Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2474v2","created_at":"2026-05-18T04:40:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2474","created_at":"2026-05-18T04:40:56Z"},{"alias_kind":"pith_short_12","alias_value":"BR64GDYRWB7V","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"BR64GDYRWB7VTTS4","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"BR64GDYR","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:c016826dbf91a744a2c25259fe8a44f36ef0b37bfcf96455044959f2cb31566e","target":"graph","created_at":"2026-05-18T04:40:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We present a combinatorial method to determine the dimension of $\\C{H}$-strata in the algebra of $m\\times n$ quantum matrices $\\Oq$ as follows. To a given $\\C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $\\C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $\\C{H}$-strata in $\\Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures propose","authors_text":"Jason Bell, Karel Casteels, St\\'ephane Launois","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2010-09-13T18:32:17Z","title":"Enumeration of $\\C{H}$-strata in quantum matrices with respect to dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2474","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:27ed9d4dec0825ffbf54e6cf9599b5ea7588109e8cd0d108e9868bafdcffe2b0","target":"record","created_at":"2026-05-18T04:40:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ba7214f8c772d4ad44f37bcd02a5b1198062e1737b7721630eadfc08c9806fe3","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2010-09-13T18:32:17Z","title_canon_sha256":"46e6ed9d4e47d299d62c4bc6d7240f279b66331de8d1c8b728d64938b0b8e512"},"schema_version":"1.0","source":{"id":"1009.2474","kind":"arxiv","version":2}},"canonical_sha256":"0c7dc30f11b07f59ce5c3c29887537c2d9fe61a91b295be79d80ff8d06df3a67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c7dc30f11b07f59ce5c3c29887537c2d9fe61a91b295be79d80ff8d06df3a67","first_computed_at":"2026-05-18T04:40:56.943488Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:56.943488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0Ad6G86EhDD/VzhR4AI7h1w1TBpNOd4Xg0vVnZQIkGodxv8jWHXuCM6gk1l/vMQXCqH1fQK1nc+H57GT5+IlDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:56.943981Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2474","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27ed9d4dec0825ffbf54e6cf9599b5ea7588109e8cd0d108e9868bafdcffe2b0","sha256:c016826dbf91a744a2c25259fe8a44f36ef0b37bfcf96455044959f2cb31566e"],"state_sha256":"4bc688165d8c472ba3313a6b42484a8d2c8b426cf473b06a2b4a46d32dd57d5e"}