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Using the circle method, we obtain the asymptotic\n  \\[\n  D_{k}(r,t;n) = \\frac{3^{\\frac{1}{4}}e^{\\pi\\sqrt{\\frac{2Kn}{3}}}}{\\pi t 2^{\\frac{3}{4}}K^{\\frac{1}{4}}n^{\\frac{1}{4}}\\sqrt{k}}\\left(\\log k + \\left(\\frac{3\\sqrt{K}\\log k}{8\\sqrt{6}\\pi} - \\frac{t\\pi(k-1)K^{\\frac{1}{2}}}{2\\sqrt{6}}\\left(\\frac{r}{t}- \\frac{1}{2}\\right)\\right)n^{-\\frac{1}{2}} + O(n^{-1})\\righ"},"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":"2207.04352","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-07-10T00:38:37Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7226035ca2c5dc62bea2e5a46e2c25a2dcc3e02579c010e95ff5b63ddede2f16","abstract_canon_sha256":"055e366b18c02576a1b86ced907b6253d7414a5b13ac5e9f8f02b2fcbf344d34"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:38:58.441762Z","signature_b64":"GnAyvQRIlWJY+Uj7Qv3sxQzl/a64gMs6nkX+N8IugqNVQ1vCSLtODrnFEsGFqhutrhu4E3fWJqyL7lW0AdxdCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d65ca5d8b2f6061ffb0eae893249bf4bbbbc3c7961eceaafaa76d56db0a2ba61","last_reissued_at":"2026-07-05T04:38:58.441404Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:38:58.441404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Biases among Congruence Classes for Parts in k-regular Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Faye Jackson, Misheel Otgonbayar","submitted_at":"2022-07-10T00:38:37Z","abstract_excerpt":"For integers $k,t \\geq 2$ and $1\\leq r \\leq t$ let $D_k(r,t;n)$ be the number of parts among all $k$-regular partitions (i.e., partitions of $n$ where all parts have multiplicity less than $k$) of $n$ that are congruent to $r$ modulo $t$. Using the circle method, we obtain the asymptotic\n  \\[\n  D_{k}(r,t;n) = \\frac{3^{\\frac{1}{4}}e^{\\pi\\sqrt{\\frac{2Kn}{3}}}}{\\pi t 2^{\\frac{3}{4}}K^{\\frac{1}{4}}n^{\\frac{1}{4}}\\sqrt{k}}\\left(\\log k + \\left(\\frac{3\\sqrt{K}\\log k}{8\\sqrt{6}\\pi} - \\frac{t\\pi(k-1)K^{\\frac{1}{2}}}{2\\sqrt{6}}\\left(\\frac{r}{t}- \\frac{1}{2}\\right)\\right)n^{-\\frac{1}{2}} + O(n^{-1})\\righ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.04352","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.04352/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2207.04352","created_at":"2026-07-05T04:38:58.441461+00:00"},{"alias_kind":"arxiv_version","alias_value":"2207.04352v1","created_at":"2026-07-05T04:38:58.441461+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.04352","created_at":"2026-07-05T04:38:58.441461+00:00"},{"alias_kind":"pith_short_12","alias_value":"2ZOKLWFS6YDB","created_at":"2026-07-05T04:38:58.441461+00:00"},{"alias_kind":"pith_short_16","alias_value":"2ZOKLWFS6YDB76YO","created_at":"2026-07-05T04:38:58.441461+00:00"},{"alias_kind":"pith_short_8","alias_value":"2ZOKLWFS","created_at":"2026-07-05T04:38:58.441461+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/2ZOKLWFS6YDB76YOV2ETESN7JO","json":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO.json","graph_json":"https://pith.science/api/pith-number/2ZOKLWFS6YDB76YOV2ETESN7JO/graph.json","events_json":"https://pith.science/api/pith-number/2ZOKLWFS6YDB76YOV2ETESN7JO/events.json","paper":"https://pith.science/paper/2ZOKLWFS"},"agent_actions":{"view_html":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO","download_json":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO.json","view_paper":"https://pith.science/paper/2ZOKLWFS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2207.04352&json=true","fetch_graph":"https://pith.science/api/pith-number/2ZOKLWFS6YDB76YOV2ETESN7JO/graph.json","fetch_events":"https://pith.science/api/pith-number/2ZOKLWFS6YDB76YOV2ETESN7JO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO/action/storage_attestation","attest_author":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO/action/author_attestation","sign_citation":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO/action/citation_signature","submit_replication":"https://pith.science/pith/2ZOKLWFS6YDB76YOV2ETESN7JO/action/replication_record"}},"created_at":"2026-07-05T04:38:58.441461+00:00","updated_at":"2026-07-05T04:38:58.441461+00:00"}