{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:5LVWHIJRQC7VF37JPBOBNITT62","short_pith_number":"pith:5LVWHIJR","canonical_record":{"source":{"id":"2305.05753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-05-09T20:21:16Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"361185fe330696b82e9aa3df0b8764ece56fed8d644175b41e9cb96566fa432d","abstract_canon_sha256":"fec88d15d39e4f9dcf01de371d9384875dc7dae8c31ad803c91472e7957807c2"},"schema_version":"1.0"},"canonical_sha256":"eaeb63a13180bf52efe9785c16a273f687f99b023b6a092dd3ba0baf5fe6c841","source":{"kind":"arxiv","id":"2305.05753","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2305.05753","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"arxiv_version","alias_value":"2305.05753v1","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.05753","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"pith_short_12","alias_value":"5LVWHIJRQC7V","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"pith_short_16","alias_value":"5LVWHIJRQC7VF37J","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"pith_short_8","alias_value":"5LVWHIJR","created_at":"2026-07-05T06:08:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:5LVWHIJRQC7VF37JPBOBNITT62","target":"record","payload":{"canonical_record":{"source":{"id":"2305.05753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-05-09T20:21:16Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"361185fe330696b82e9aa3df0b8764ece56fed8d644175b41e9cb96566fa432d","abstract_canon_sha256":"fec88d15d39e4f9dcf01de371d9384875dc7dae8c31ad803c91472e7957807c2"},"schema_version":"1.0"},"canonical_sha256":"eaeb63a13180bf52efe9785c16a273f687f99b023b6a092dd3ba0baf5fe6c841","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:08:44.916031Z","signature_b64":"LAEIIhBG2n1sbmVWeDoXEH75NsB9etn+yQ12GDNvwhZjpgiJAlMlkxPhPQhC+qTjT5asVUQ+Ir1e9tbHF/ZXDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eaeb63a13180bf52efe9785c16a273f687f99b023b6a092dd3ba0baf5fe6c841","last_reissued_at":"2026-07-05T06:08:44.915638Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:08:44.915638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2305.05753","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:08:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6m4P3skoEbknw5TRBvbqUAfRNhregrMh6IuxkKHCSr5licYOuNeDCimIwGT6aW7qoqZbYCLmCwejRG8aO5KtAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T06:18:00.171399Z"},"content_sha256":"fce56475113456484ea80066c95d3956bb66f4b4e8acb22838f8133c31c1217a","schema_version":"1.0","event_id":"sha256:fce56475113456484ea80066c95d3956bb66f4b4e8acb22838f8133c31c1217a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:5LVWHIJRQC7VF37JPBOBNITT62","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parts in $k$-indivisible Partitions Always Display Biases between Residue Classes","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":"2023-05-09T20:21:16Z","abstract_excerpt":"Let $k, t$ be coprime integers, and let $1 \\leq r \\leq t$. We let $D_k^\\times(r,t;n)$ denote the total number of parts among all $k$-indivisible partitions (i.e., those partitions where no part is divisible by $k$) of $n$ which are congruent to $r$ modulo $t$. In previous work of the authors, an asymptotic estimate for $D_k^\\times(r,t;n)$ was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture that there are no \"ties\" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.05753","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/2305.05753/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:08:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dxmlBTX2TI4NV4XO9xkiLxVuYNWmmigDDqAQxrKpaHRkizuVJQYvhHkPBoY9rQT1H0RK3TM6GGjzOeR1SNlIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T06:18:00.171782Z"},"content_sha256":"b58a89c1ce10eaa852c050ce3e3ec593ba3452d61317901a75e15360d16f0e33","schema_version":"1.0","event_id":"sha256:b58a89c1ce10eaa852c050ce3e3ec593ba3452d61317901a75e15360d16f0e33"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5LVWHIJRQC7VF37JPBOBNITT62/bundle.json","state_url":"https://pith.science/pith/5LVWHIJRQC7VF37JPBOBNITT62/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5LVWHIJRQC7VF37JPBOBNITT62/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-07T06:18:00Z","links":{"resolver":"https://pith.science/pith/5LVWHIJRQC7VF37JPBOBNITT62","bundle":"https://pith.science/pith/5LVWHIJRQC7VF37JPBOBNITT62/bundle.json","state":"https://pith.science/pith/5LVWHIJRQC7VF37JPBOBNITT62/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5LVWHIJRQC7VF37JPBOBNITT62/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:5LVWHIJRQC7VF37JPBOBNITT62","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":"fec88d15d39e4f9dcf01de371d9384875dc7dae8c31ad803c91472e7957807c2","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-05-09T20:21:16Z","title_canon_sha256":"361185fe330696b82e9aa3df0b8764ece56fed8d644175b41e9cb96566fa432d"},"schema_version":"1.0","source":{"id":"2305.05753","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2305.05753","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"arxiv_version","alias_value":"2305.05753v1","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.05753","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"pith_short_12","alias_value":"5LVWHIJRQC7V","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"pith_short_16","alias_value":"5LVWHIJRQC7VF37J","created_at":"2026-07-05T06:08:44Z"},{"alias_kind":"pith_short_8","alias_value":"5LVWHIJR","created_at":"2026-07-05T06:08:44Z"}],"graph_snapshots":[{"event_id":"sha256:b58a89c1ce10eaa852c050ce3e3ec593ba3452d61317901a75e15360d16f0e33","target":"graph","created_at":"2026-07-05T06:08:44Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2305.05753/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $k, t$ be coprime integers, and let $1 \\leq r \\leq t$. We let $D_k^\\times(r,t;n)$ denote the total number of parts among all $k$-indivisible partitions (i.e., those partitions where no part is divisible by $k$) of $n$ which are congruent to $r$ modulo $t$. In previous work of the authors, an asymptotic estimate for $D_k^\\times(r,t;n)$ was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture that there are no \"ties\" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe th","authors_text":"Faye Jackson, Misheel Otgonbayar","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-05-09T20:21:16Z","title":"Parts in $k$-indivisible Partitions Always Display Biases between Residue Classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.05753","kind":"arxiv","version":1},"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:fce56475113456484ea80066c95d3956bb66f4b4e8acb22838f8133c31c1217a","target":"record","created_at":"2026-07-05T06:08:44Z","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":"fec88d15d39e4f9dcf01de371d9384875dc7dae8c31ad803c91472e7957807c2","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-05-09T20:21:16Z","title_canon_sha256":"361185fe330696b82e9aa3df0b8764ece56fed8d644175b41e9cb96566fa432d"},"schema_version":"1.0","source":{"id":"2305.05753","kind":"arxiv","version":1}},"canonical_sha256":"eaeb63a13180bf52efe9785c16a273f687f99b023b6a092dd3ba0baf5fe6c841","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eaeb63a13180bf52efe9785c16a273f687f99b023b6a092dd3ba0baf5fe6c841","first_computed_at":"2026-07-05T06:08:44.915638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:08:44.915638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LAEIIhBG2n1sbmVWeDoXEH75NsB9etn+yQ12GDNvwhZjpgiJAlMlkxPhPQhC+qTjT5asVUQ+Ir1e9tbHF/ZXDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:08:44.916031Z","signed_message":"canonical_sha256_bytes"},"source_id":"2305.05753","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fce56475113456484ea80066c95d3956bb66f4b4e8acb22838f8133c31c1217a","sha256:b58a89c1ce10eaa852c050ce3e3ec593ba3452d61317901a75e15360d16f0e33"],"state_sha256":"6b0c2ada8f4c1009831a0ee1bf5deb8488de5c7d016ba0cb95ce835f737a5ba6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6low2XkCXC9W9kC6R19WNJiluPId7HpUcigrodBTgPIMz8t2HrswMVnvb8MVCRwRtafYRoshPYOzaNQ5Mo5JAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T06:18:00.173931Z","bundle_sha256":"81cd67ee79c6f15b96740650814835768899c0f52ec25ff3214122e9646ea7b0"}}