{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:423CEAH4O47AGAN3H3XVBAKFEC","short_pith_number":"pith:423CEAH4","canonical_record":{"source":{"id":"2606.13274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T12:27:41Z","cross_cats_sorted":[],"title_canon_sha256":"d75f71c69dd203845a2ec43c47b74aed1946c6f6bd887d0b0393de7586a14d4b","abstract_canon_sha256":"3fa92ad42dc70cf77b340a4b5af7cee6be4cd981ff1421f773d932eb22e73655"},"schema_version":"1.0"},"canonical_sha256":"e6b62200fc773e0301bb3eef50814520b39819a0e7f10af7dc22fd860a82c378","source":{"kind":"arxiv","id":"2606.13274","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13274","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13274v1","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13274","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"pith_short_12","alias_value":"423CEAH4O47A","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"pith_short_16","alias_value":"423CEAH4O47AGAN3","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"pith_short_8","alias_value":"423CEAH4","created_at":"2026-06-12T01:09:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:423CEAH4O47AGAN3H3XVBAKFEC","target":"record","payload":{"canonical_record":{"source":{"id":"2606.13274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T12:27:41Z","cross_cats_sorted":[],"title_canon_sha256":"d75f71c69dd203845a2ec43c47b74aed1946c6f6bd887d0b0393de7586a14d4b","abstract_canon_sha256":"3fa92ad42dc70cf77b340a4b5af7cee6be4cd981ff1421f773d932eb22e73655"},"schema_version":"1.0"},"canonical_sha256":"e6b62200fc773e0301bb3eef50814520b39819a0e7f10af7dc22fd860a82c378","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:50.031565Z","signature_b64":"B8/EJgVqddxLjE64vBxParQnHuMBv57ncT+iMocInDhNpjktmdB4JR70HA1rihjzYCPbUOGme6a9L8cLjfQ5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6b62200fc773e0301bb3eef50814520b39819a0e7f10af7dc22fd860a82c378","last_reissued_at":"2026-06-12T01:09:50.030727Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:50.030727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.13274","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":"2026-06-12T01:09:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YjiX1QtTeWQgsNhkK7s4nKFxmM1STypiXh+jsjr9MP4exHFc7lCzxS8LA9GbYixW750dc7cl0odU5eIVkyjjDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:36:43.879666Z"},"content_sha256":"3edd32d74c067276aba2fe02055ffc0d9eb8312809d00ab87458e7d406c66564","schema_version":"1.0","event_id":"sha256:3edd32d74c067276aba2fe02055ffc0d9eb8312809d00ab87458e7d406c66564"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:423CEAH4O47AGAN3H3XVBAKFEC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Monotonicity of the rank functions for concave compositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nian Hong Zhou","submitted_at":"2026-06-11T12:27:41Z","abstract_excerpt":"A (strongly) concave composition of an integer $n$ is a sequence of positive integers that is (strictly) decreasing to a point and then (strictly) increasing thereafter, such that the sum of the entries equals $n$. The value at the low point is called the center part. The difference between the number of entries before and after the low point of the sequence is referred to as the rank of the (strongly) concave composition. The rank functions $V_d(m,n)$ and $V(m,n)$ are defined as the number of concave compositions and strongly concave compositions, respectively, of $n$ with rank $m$. By constr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13274","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/2606.13274/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":"2026-06-12T01:09:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"szYOHlGgRsrwdcQKKRuzJVV1uhqhq5Ipp7tfY/E78I+mHVHTgtsSU6d1dx1CcEhhkHZ23FJBBA7JKQa5NmktCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:36:43.880161Z"},"content_sha256":"1c25df0d3651efc8ec516393029c3bbd1e1a124528038f54cfb4b949c534745f","schema_version":"1.0","event_id":"sha256:1c25df0d3651efc8ec516393029c3bbd1e1a124528038f54cfb4b949c534745f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/423CEAH4O47AGAN3H3XVBAKFEC/bundle.json","state_url":"https://pith.science/pith/423CEAH4O47AGAN3H3XVBAKFEC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/423CEAH4O47AGAN3H3XVBAKFEC/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":"2026-08-12T08:36:43Z","links":{"resolver":"https://pith.science/pith/423CEAH4O47AGAN3H3XVBAKFEC","bundle":"https://pith.science/pith/423CEAH4O47AGAN3H3XVBAKFEC/bundle.json","state":"https://pith.science/pith/423CEAH4O47AGAN3H3XVBAKFEC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/423CEAH4O47AGAN3H3XVBAKFEC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:423CEAH4O47AGAN3H3XVBAKFEC","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":"3fa92ad42dc70cf77b340a4b5af7cee6be4cd981ff1421f773d932eb22e73655","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T12:27:41Z","title_canon_sha256":"d75f71c69dd203845a2ec43c47b74aed1946c6f6bd887d0b0393de7586a14d4b"},"schema_version":"1.0","source":{"id":"2606.13274","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13274","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13274v1","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13274","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"pith_short_12","alias_value":"423CEAH4O47A","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"pith_short_16","alias_value":"423CEAH4O47AGAN3","created_at":"2026-06-12T01:09:50Z"},{"alias_kind":"pith_short_8","alias_value":"423CEAH4","created_at":"2026-06-12T01:09:50Z"}],"graph_snapshots":[{"event_id":"sha256:1c25df0d3651efc8ec516393029c3bbd1e1a124528038f54cfb4b949c534745f","target":"graph","created_at":"2026-06-12T01:09:50Z","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/2606.13274/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A (strongly) concave composition of an integer $n$ is a sequence of positive integers that is (strictly) decreasing to a point and then (strictly) increasing thereafter, such that the sum of the entries equals $n$. The value at the low point is called the center part. The difference between the number of entries before and after the low point of the sequence is referred to as the rank of the (strongly) concave composition. The rank functions $V_d(m,n)$ and $V(m,n)$ are defined as the number of concave compositions and strongly concave compositions, respectively, of $n$ with rank $m$. By constr","authors_text":"Nian Hong Zhou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T12:27:41Z","title":"Monotonicity of the rank functions for concave compositions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13274","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":"sha256:3edd32d74c067276aba2fe02055ffc0d9eb8312809d00ab87458e7d406c66564","target":"record","created_at":"2026-06-12T01:09:50Z","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":"3fa92ad42dc70cf77b340a4b5af7cee6be4cd981ff1421f773d932eb22e73655","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T12:27:41Z","title_canon_sha256":"d75f71c69dd203845a2ec43c47b74aed1946c6f6bd887d0b0393de7586a14d4b"},"schema_version":"1.0","source":{"id":"2606.13274","kind":"arxiv","version":1}},"canonical_sha256":"e6b62200fc773e0301bb3eef50814520b39819a0e7f10af7dc22fd860a82c378","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6b62200fc773e0301bb3eef50814520b39819a0e7f10af7dc22fd860a82c378","first_computed_at":"2026-06-12T01:09:50.030727Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-12T01:09:50.030727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B8/EJgVqddxLjE64vBxParQnHuMBv57ncT+iMocInDhNpjktmdB4JR70HA1rihjzYCPbUOGme6a9L8cLjfQ5Aw==","signature_status":"signed_v1","signed_at":"2026-06-12T01:09:50.031565Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.13274","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3edd32d74c067276aba2fe02055ffc0d9eb8312809d00ab87458e7d406c66564","sha256:1c25df0d3651efc8ec516393029c3bbd1e1a124528038f54cfb4b949c534745f"],"state_sha256":"8ecca8eba859b1b10b7f730fc306c4851f95eb56708a8e873a27a678109a7271"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aKnhV+/IVm1SZ+UlSicmRPHK8zjUpoExEGiYX4Ofdfub5RlOohfdxYfb64R8JFR/TPKo6A66/mxpJ+TnucBPAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:36:43.884998Z","bundle_sha256":"833e96b4583f404028514bc9ab0ddea506ee8412310ca777314b8957ae239468"}}