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This discovery yields elementary and direct derivations of several identities relating these operators at $t=1/q$ to the Rational Compositional Shuffle conjecture of [3]. In particular we show that if $m,n $ and $k$ are positive integers and $(m,n)$ is a coprime pair then $$ q^{(km-1)(kn-1)+k-1\\over 2} Q_{km,kn}(-1)^{kn}\\Big|_{t=1/q} \\,=\\, \\textstyle{[k]_q\\over [km]_q} e_{km}\\big[ X[km]_q\\big] $$ where as customarily, for any integer $s \\geq 0$ and "},"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":"1501.00631","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-04T04:28:19Z","cross_cats_sorted":[],"title_canon_sha256":"33f1272e34775d89b593a57599e2d564c79e6739572c1f115757026780867329","abstract_canon_sha256":"4817f00d3d8948abbcffa613dd63fa5bccaec6a0660327a8b07f8d39de8d3707"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:02.100145Z","signature_b64":"zi9QPHkgVwHntq7VeibbUbldebBhPI6YflWSUMUN6GKOAPt0pvYCQJ11bKUssxVTCcFreZz2JUxr0y+jSpSZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9aaf95862d78090471167651276e152a7e10fe4651e1804515c297499fa08a0b","last_reissued_at":"2026-05-18T02:30:02.099755Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:02.099755Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new Plethystic Symmetric Function Operator and The rational Compositional Shuffle Conjecture at t=1/q","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. 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In particular we show that if $m,n $ and $k$ are positive integers and $(m,n)$ is a coprime pair then $$ q^{(km-1)(kn-1)+k-1\\over 2} Q_{km,kn}(-1)^{kn}\\Big|_{t=1/q} \\,=\\, \\textstyle{[k]_q\\over [km]_q} e_{km}\\big[ X[km]_q\\big] $$ where as customarily, for any integer $s \\geq 0$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00631","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1501.00631","created_at":"2026-05-18T02:30:02.099814+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.00631v1","created_at":"2026-05-18T02:30:02.099814+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00631","created_at":"2026-05-18T02:30:02.099814+00:00"},{"alias_kind":"pith_short_12","alias_value":"TKXZLBRNPAEQ","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TKXZLBRNPAEQI4IW","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TKXZLBRN","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ","json":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ.json","graph_json":"https://pith.science/api/pith-number/TKXZLBRNPAEQI4IWOZISO3QVFJ/graph.json","events_json":"https://pith.science/api/pith-number/TKXZLBRNPAEQI4IWOZISO3QVFJ/events.json","paper":"https://pith.science/paper/TKXZLBRN"},"agent_actions":{"view_html":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ","download_json":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ.json","view_paper":"https://pith.science/paper/TKXZLBRN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.00631&json=true","fetch_graph":"https://pith.science/api/pith-number/TKXZLBRNPAEQI4IWOZISO3QVFJ/graph.json","fetch_events":"https://pith.science/api/pith-number/TKXZLBRNPAEQI4IWOZISO3QVFJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ/action/storage_attestation","attest_author":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ/action/author_attestation","sign_citation":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ/action/citation_signature","submit_replication":"https://pith.science/pith/TKXZLBRNPAEQI4IWOZISO3QVFJ/action/replication_record"}},"created_at":"2026-05-18T02:30:02.099814+00:00","updated_at":"2026-05-18T02:30:02.099814+00:00"}