{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:XHW7JASAYFUPPC3J3PV2WDCYDH","short_pith_number":"pith:XHW7JASA","canonical_record":{"source":{"id":"2410.22486","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-10-29T19:23:24Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"12dfa79e708c1e50fd377939c0ad1f4156dab2a563a533fc072144e638e9cb15","abstract_canon_sha256":"757bd6693a9f68c3cae8b9a80d6378291eb0163a12ad3782c12f5ac2ac0711cd"},"schema_version":"1.0"},"canonical_sha256":"b9edf48240c168f78b69dbebab0c5819e058480ce0e8cc967479224b8bfa8ba7","source":{"kind":"arxiv","id":"2410.22486","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.22486","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"arxiv_version","alias_value":"2410.22486v3","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.22486","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"pith_short_12","alias_value":"XHW7JASAYFUP","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"pith_short_16","alias_value":"XHW7JASAYFUPPC3J","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"pith_short_8","alias_value":"XHW7JASA","created_at":"2026-07-05T09:56:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:XHW7JASAYFUPPC3J3PV2WDCYDH","target":"record","payload":{"canonical_record":{"source":{"id":"2410.22486","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-10-29T19:23:24Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"12dfa79e708c1e50fd377939c0ad1f4156dab2a563a533fc072144e638e9cb15","abstract_canon_sha256":"757bd6693a9f68c3cae8b9a80d6378291eb0163a12ad3782c12f5ac2ac0711cd"},"schema_version":"1.0"},"canonical_sha256":"b9edf48240c168f78b69dbebab0c5819e058480ce0e8cc967479224b8bfa8ba7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:56:22.643524Z","signature_b64":"IA14dM+pvscw9oB0MaI0tC9wTDS24OdnP6cMgB/6bMy8u8GZxJndZ1vvVGanC6qByBYTFGn7fz3CLJ1bycMWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9edf48240c168f78b69dbebab0c5819e058480ce0e8cc967479224b8bfa8ba7","last_reissued_at":"2026-07-05T09:56:22.642987Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:56:22.642987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.22486","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:56:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I0v1ct4zv0MyU/6TLU7OulLgq/OZpJ6n24TrRQPdFMtt/IqHC279I0R1BLhFBDsl2+BQanjcm8jv7iuiisNtBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T01:11:48.107566Z"},"content_sha256":"cf4afad70e38e32eac5bfd5e0ac9a0db79046403c9d05c3967d265b23a802edf","schema_version":"1.0","event_id":"sha256:cf4afad70e38e32eac5bfd5e0ac9a0db79046403c9d05c3967d265b23a802edf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:XHW7JASAYFUPPC3J3PV2WDCYDH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multifold Convolutions, Generating Functions and 1d Random Walks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Shannon Starr, Timothy Li","submitted_at":"2024-10-29T19:23:24Z","abstract_excerpt":"We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\\infty}$: namely, for each $k \\in \\N$ the $k$-fold convolution is $\\mathcal{M}^{(k)}_n(\\boldsymbol{a}) = \\sum_{j_1+\\dots+j_k=n} a_{j_1} \\cdots a_{j_k}$. Let $C_n$ be the Catalan numbers, and let $B_n$ be the central binomial coefficients. Then for random Dyck paths or simple random walk bridges, the multifold convolutions give moments of returns to the origin, using the stars-and-bars problem. There are well-known explicit formulas for the multifold convolutions of $C_n$ and $B_n$. But even for combinatorial sequences"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.22486","kind":"arxiv","version":3},"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/2410.22486/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":"2026-07-05T09:56:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cq/lZfylz/Xh/DZNi0xb4KsuL2XCwFeE0cea+SqcUOo94Djg5UhocFV2PbueXO7KxIdCXpZ15nAsCwK6LB54CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T01:11:48.108186Z"},"content_sha256":"f33a081e00072fccd3b9dcc53c26e0ffb6d8b50ad0c8bb97df212afb8c39f956","schema_version":"1.0","event_id":"sha256:f33a081e00072fccd3b9dcc53c26e0ffb6d8b50ad0c8bb97df212afb8c39f956"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XHW7JASAYFUPPC3J3PV2WDCYDH/bundle.json","state_url":"https://pith.science/pith/XHW7JASAYFUPPC3J3PV2WDCYDH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XHW7JASAYFUPPC3J3PV2WDCYDH/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":"2026-08-21T01:11:48Z","links":{"resolver":"https://pith.science/pith/XHW7JASAYFUPPC3J3PV2WDCYDH","bundle":"https://pith.science/pith/XHW7JASAYFUPPC3J3PV2WDCYDH/bundle.json","state":"https://pith.science/pith/XHW7JASAYFUPPC3J3PV2WDCYDH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XHW7JASAYFUPPC3J3PV2WDCYDH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:XHW7JASAYFUPPC3J3PV2WDCYDH","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":"757bd6693a9f68c3cae8b9a80d6378291eb0163a12ad3782c12f5ac2ac0711cd","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-10-29T19:23:24Z","title_canon_sha256":"12dfa79e708c1e50fd377939c0ad1f4156dab2a563a533fc072144e638e9cb15"},"schema_version":"1.0","source":{"id":"2410.22486","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.22486","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"arxiv_version","alias_value":"2410.22486v3","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.22486","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"pith_short_12","alias_value":"XHW7JASAYFUP","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"pith_short_16","alias_value":"XHW7JASAYFUPPC3J","created_at":"2026-07-05T09:56:22Z"},{"alias_kind":"pith_short_8","alias_value":"XHW7JASA","created_at":"2026-07-05T09:56:22Z"}],"graph_snapshots":[{"event_id":"sha256:f33a081e00072fccd3b9dcc53c26e0ffb6d8b50ad0c8bb97df212afb8c39f956","target":"graph","created_at":"2026-07-05T09:56:22Z","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/2410.22486/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\\infty}$: namely, for each $k \\in \\N$ the $k$-fold convolution is $\\mathcal{M}^{(k)}_n(\\boldsymbol{a}) = \\sum_{j_1+\\dots+j_k=n} a_{j_1} \\cdots a_{j_k}$. Let $C_n$ be the Catalan numbers, and let $B_n$ be the central binomial coefficients. Then for random Dyck paths or simple random walk bridges, the multifold convolutions give moments of returns to the origin, using the stars-and-bars problem. There are well-known explicit formulas for the multifold convolutions of $C_n$ and $B_n$. But even for combinatorial sequences","authors_text":"Shannon Starr, Timothy Li","cross_cats":["math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-10-29T19:23:24Z","title":"Multifold Convolutions, Generating Functions and 1d Random Walks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.22486","kind":"arxiv","version":3},"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:cf4afad70e38e32eac5bfd5e0ac9a0db79046403c9d05c3967d265b23a802edf","target":"record","created_at":"2026-07-05T09:56:22Z","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":"757bd6693a9f68c3cae8b9a80d6378291eb0163a12ad3782c12f5ac2ac0711cd","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-10-29T19:23:24Z","title_canon_sha256":"12dfa79e708c1e50fd377939c0ad1f4156dab2a563a533fc072144e638e9cb15"},"schema_version":"1.0","source":{"id":"2410.22486","kind":"arxiv","version":3}},"canonical_sha256":"b9edf48240c168f78b69dbebab0c5819e058480ce0e8cc967479224b8bfa8ba7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9edf48240c168f78b69dbebab0c5819e058480ce0e8cc967479224b8bfa8ba7","first_computed_at":"2026-07-05T09:56:22.642987Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:56:22.642987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IA14dM+pvscw9oB0MaI0tC9wTDS24OdnP6cMgB/6bMy8u8GZxJndZ1vvVGanC6qByBYTFGn7fz3CLJ1bycMWAA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:56:22.643524Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.22486","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf4afad70e38e32eac5bfd5e0ac9a0db79046403c9d05c3967d265b23a802edf","sha256:f33a081e00072fccd3b9dcc53c26e0ffb6d8b50ad0c8bb97df212afb8c39f956"],"state_sha256":"538ade0d86827e100b0290e462b9168a98d557ab399626520e0474b0daeeab91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rxav8Hf4s3vyuKVPsItqjrBADB/3awPyS9yFPlI91xv0T3G5VTyQYVRIrJV2GtnlxDQXqptb3zhwtY30qpRICg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T01:11:48.113318Z","bundle_sha256":"8e8ae8dc4b374fcea488b8ecdc88354b871079cc67697eac7aa3c1c72227465b"}}