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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"},"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":"1009.2474","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2010-09-13T18:32:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"46e6ed9d4e47d299d62c4bc6d7240f279b66331de8d1c8b728d64938b0b8e512","abstract_canon_sha256":"ba7214f8c772d4ad44f37bcd02a5b1198062e1737b7721630eadfc08c9806fe3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:56.943981Z","signature_b64":"0Ad6G86EhDD/VzhR4AI7h1w1TBpNOd4Xg0vVnZQIkGodxv8jWHXuCM6gk1l/vMQXCqH1fQK1nc+H57GT5+IlDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c7dc30f11b07f59ce5c3c29887537c2d9fe61a91b295be79d80ff8d06df3a67","last_reissued_at":"2026-05-18T04:40:56.943488Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:56.943488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Enumeration of $\\C{H}$-strata in quantum matrices with respect to dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.QA","authors_text":"Jason Bell, Karel Casteels, St\\'ephane Launois","submitted_at":"2010-09-13T18:32:17Z","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. 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