{"paper":{"title":"Rook theory, normal ordering in the $q$-deformed Ore algebra and the polynomial generalization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Normal ordering coefficients in the q-deformed Ore algebra equal mixed rook and file placement numbers on the staircase and Laguerre boards.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matthias Schork","submitted_at":"2026-05-10T18:04:17Z","abstract_excerpt":"For words in the variables $X$ and $Y$ satisfying the commutation relation of the $q$-deformed generalized Ore algebra, $XY-qYX= \\mu I + \\nu Y$, we show that the corresponding normal ordering coefficients can be given an interpretation in terms of mixed placements of rooks and files. In particular, the associated $q$-deformed Ore-Stirling and Ore-Lah numbers are treated in detail. We show that the $q$-deformed Ore-Stirling numbers (resp., $q$-deformed Ore-Lah numbers) are given as mixed placement numbers of rooks and files on the staircase board (resp., Laguerre board). Using this combinatoria"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The q-deformed Ore-Stirling numbers (resp., q-deformed Ore-Lah numbers) are given as mixed placement numbers of rooks and files on the staircase board (resp., Laguerre board).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the algebraic normal-ordering coefficients admit a direct, weight-preserving bijection with the mixed rook-file placements on the named boards, allowing recurrence relations to be read off combinatorially.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"q-deformed Ore-Stirling numbers count mixed rook-file placements on staircase boards and q-deformed Ore-Lah numbers do the same on Laguerre boards, with the approach extended to polynomial commutation relations XY - qYX = f(Y).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Normal ordering coefficients in the q-deformed Ore algebra equal mixed rook and file placement numbers on the staircase and Laguerre boards.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"18f1dd89149828283f9bce265d511a1122f350311cb22cdda361aa55ee940560"},"source":{"id":"2605.09683","kind":"arxiv","version":2},"verdict":{"id":"0b97cc39-1ac0-4a0b-a1c0-15ab4b31d1e7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:46:50.477344Z","strongest_claim":"The q-deformed Ore-Stirling numbers (resp., q-deformed Ore-Lah numbers) are given as mixed placement numbers of rooks and files on the staircase board (resp., Laguerre board).","one_line_summary":"q-deformed Ore-Stirling numbers count mixed rook-file placements on staircase boards and q-deformed Ore-Lah numbers do the same on Laguerre boards, with the approach extended to polynomial commutation relations XY - qYX = f(Y).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the algebraic normal-ordering coefficients admit a direct, weight-preserving bijection with the mixed rook-file placements on the named boards, allowing recurrence relations to be read off combinatorially.","pith_extraction_headline":"Normal ordering coefficients in the q-deformed Ore algebra equal mixed rook and file placement numbers on the staircase and Laguerre boards."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.09683/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T16:37:56.502502Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T12:31:18.339195Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:00:45.070809Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"ba77f73e7b9339003a7f90e3684083b858e11340505d7da1aab8c441a874a583"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}