{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:H25ROERDVJMPQCPWTWLZEKB5TT","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":"79b1d243adc172cde3e6a5e66b6bef6bd6086074d827434c4fa535ff1b34b188","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T10:35:57Z","title_canon_sha256":"e836b30d79b668dcbb8f923a617c51635f09bf90a8bc7bb8a89aacce00eaaf7b"},"schema_version":"1.0","source":{"id":"2606.09321","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.09321","created_at":"2026-06-09T02:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.09321v1","created_at":"2026-06-09T02:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.09321","created_at":"2026-06-09T02:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"H25ROERDVJMP","created_at":"2026-06-09T02:08:14Z"},{"alias_kind":"pith_short_16","alias_value":"H25ROERDVJMPQCPW","created_at":"2026-06-09T02:08:14Z"},{"alias_kind":"pith_short_8","alias_value":"H25ROERD","created_at":"2026-06-09T02:08:14Z"}],"graph_snapshots":[{"event_id":"sha256:7b104f39aa66056864c4696662b407d4ed00a43eb018ee0bc7733f72657da706","target":"graph","created_at":"2026-06-09T02:08:14Z","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/2606.09321/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Ballantine, Beck, Merca, and Sagan (arXiv:2409.11268) conjectured four identities, collectively Conjecture 19, relating the image of the map pre_k on integer partitions to four OEIS sequences. We prove parts (i) and (iii) unconditionally, prove part (iv) under the assumption that pre_2 is injective on partitions of n (Conjecture 1 of the same paper, and show this assumption is in fact equivalent to (iv)), and for part (ii) we prove the partition-theoretic half unconditionally and reduce the remaining content to a 2006 conjecture of Dean Hickerson on the OEIS concerning Huffman coding. We also ","authors_text":"Arnav Garg","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T10:35:57Z","title":"Proof of Conjecture 19 of Ballantine, Beck, Merca, and Sagan on Elementary Symmetric Partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09321","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:f9bce6f65eb1d4f5532f5d3b4efc45510827d3f36e95d76069227b2c54b0860e","target":"record","created_at":"2026-06-09T02:08:14Z","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":"79b1d243adc172cde3e6a5e66b6bef6bd6086074d827434c4fa535ff1b34b188","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T10:35:57Z","title_canon_sha256":"e836b30d79b668dcbb8f923a617c51635f09bf90a8bc7bb8a89aacce00eaaf7b"},"schema_version":"1.0","source":{"id":"2606.09321","kind":"arxiv","version":1}},"canonical_sha256":"3ebb171223aa58f809f69d9792283d9cc2ff4c147522ccb94d68e07d8346f209","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ebb171223aa58f809f69d9792283d9cc2ff4c147522ccb94d68e07d8346f209","first_computed_at":"2026-06-09T02:08:14.833692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:08:14.833692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AjvNcSYVBE4G8jtHC5gkDyjELG2LMzYX6/UKqaiyISWPFLgZDS55KnQxED19SJR3tn+QLdZULZ51ojd+GtseCQ==","signature_status":"signed_v1","signed_at":"2026-06-09T02:08:14.834604Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.09321","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f9bce6f65eb1d4f5532f5d3b4efc45510827d3f36e95d76069227b2c54b0860e","sha256:7b104f39aa66056864c4696662b407d4ed00a43eb018ee0bc7733f72657da706"],"state_sha256":"b65254b74a93a57606c8c9a05125fb901284f865212238966af684cd6e0b61ab"}