{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:LLI2XATIYGZIJMATNKTXFKJARC","short_pith_number":"pith:LLI2XATI","canonical_record":{"source":{"id":"1201.6576","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T15:34:25Z","cross_cats_sorted":[],"title_canon_sha256":"e17a3f6075c0afba724729b9fef86257a3382c2423e9ad1831bbabc0c6aeab5a","abstract_canon_sha256":"7feb99f9c1bc5163c15a28bd93e26a271cee3fb1a3e895977567ccaf4e2a26b7"},"schema_version":"1.0"},"canonical_sha256":"5ad1ab8268c1b284b0136aa772a92088a396402564897e27e30e2881607e9c7c","source":{"kind":"arxiv","id":"1201.6576","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.6576","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"arxiv_version","alias_value":"1201.6576v3","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.6576","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"pith_short_12","alias_value":"LLI2XATIYGZI","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"LLI2XATIYGZIJMAT","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"LLI2XATI","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:LLI2XATIYGZIJMATNKTXFKJARC","target":"record","payload":{"canonical_record":{"source":{"id":"1201.6576","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T15:34:25Z","cross_cats_sorted":[],"title_canon_sha256":"e17a3f6075c0afba724729b9fef86257a3382c2423e9ad1831bbabc0c6aeab5a","abstract_canon_sha256":"7feb99f9c1bc5163c15a28bd93e26a271cee3fb1a3e895977567ccaf4e2a26b7"},"schema_version":"1.0"},"canonical_sha256":"5ad1ab8268c1b284b0136aa772a92088a396402564897e27e30e2881607e9c7c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:01.757098Z","signature_b64":"2Z6B4iFBnt147xgvF1Nd52ybeHET0qGhfnHz1v7kpE4pVKX91RacvVtTUvLDGeEUOhbhJPcmaAeQjgsDJSjvCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ad1ab8268c1b284b0136aa772a92088a396402564897e27e30e2881607e9c7c","last_reissued_at":"2026-05-18T04:00:01.756569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:01.756569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.6576","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":"2026-05-18T04:00:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prfU2+Dsx7L56eBztwN9v0GKVxkpyn/iub6DY14krhjWQaLZfCjQzZSgIUcy9ZV4UdmAmyQcphpXg/0jHJxyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:01:25.984439Z"},"content_sha256":"fce179ec0c9e78d55f9057a8e7b6cf745c37596de885a714feecd87ebfbd7328","schema_version":"1.0","event_id":"sha256:fce179ec0c9e78d55f9057a8e7b6cf745c37596de885a714feecd87ebfbd7328"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:LLI2XATIYGZIJMATNKTXFKJARC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Statistics of blocks in k-divisible non-crossing partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Octavio Arizmendi","submitted_at":"2012-01-31T15:34:25Z","abstract_excerpt":"We derive a formula for the expected number of blocks of a given size from a non-crossing partition chosen uniformly at random. Moreover, we refine this result subject to the restriction of having a number of blocks given. Furthermore, we generalize to k-divisible partitions. In particular, we find that, asymptotically, the expected number of blocks of size t of a k-divisible non-crossing partition of nk elements chosen uniformly at random is (kn+1)/(k+1)^(t+1). Similar results are obtained for type B and type D k-divisible non-crossing partitions of Armstrong."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.6576","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":""},"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":"2026-05-18T04:00:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rqa1SxOz3VPl65SbleZS9dY7EhWDuygl5K2x09mDPnyI+Hg3kNcCzoQsaecFXvdnYVF2CZv62ICUt13osBe/Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:01:25.984824Z"},"content_sha256":"9bd250ee9b4e3e39e026d3d587f8dcfc0673d466059438918c64833a517af102","schema_version":"1.0","event_id":"sha256:9bd250ee9b4e3e39e026d3d587f8dcfc0673d466059438918c64833a517af102"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LLI2XATIYGZIJMATNKTXFKJARC/bundle.json","state_url":"https://pith.science/pith/LLI2XATIYGZIJMATNKTXFKJARC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LLI2XATIYGZIJMATNKTXFKJARC/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":"2026-06-11T06:01:25Z","links":{"resolver":"https://pith.science/pith/LLI2XATIYGZIJMATNKTXFKJARC","bundle":"https://pith.science/pith/LLI2XATIYGZIJMATNKTXFKJARC/bundle.json","state":"https://pith.science/pith/LLI2XATIYGZIJMATNKTXFKJARC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LLI2XATIYGZIJMATNKTXFKJARC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:LLI2XATIYGZIJMATNKTXFKJARC","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":"7feb99f9c1bc5163c15a28bd93e26a271cee3fb1a3e895977567ccaf4e2a26b7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T15:34:25Z","title_canon_sha256":"e17a3f6075c0afba724729b9fef86257a3382c2423e9ad1831bbabc0c6aeab5a"},"schema_version":"1.0","source":{"id":"1201.6576","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.6576","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"arxiv_version","alias_value":"1201.6576v3","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.6576","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"pith_short_12","alias_value":"LLI2XATIYGZI","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"LLI2XATIYGZIJMAT","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"LLI2XATI","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:9bd250ee9b4e3e39e026d3d587f8dcfc0673d466059438918c64833a517af102","target":"graph","created_at":"2026-05-18T04:00:01Z","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"},"paper":{"abstract_excerpt":"We derive a formula for the expected number of blocks of a given size from a non-crossing partition chosen uniformly at random. Moreover, we refine this result subject to the restriction of having a number of blocks given. Furthermore, we generalize to k-divisible partitions. In particular, we find that, asymptotically, the expected number of blocks of size t of a k-divisible non-crossing partition of nk elements chosen uniformly at random is (kn+1)/(k+1)^(t+1). Similar results are obtained for type B and type D k-divisible non-crossing partitions of Armstrong.","authors_text":"Octavio Arizmendi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T15:34:25Z","title":"Statistics of blocks in k-divisible non-crossing partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.6576","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:fce179ec0c9e78d55f9057a8e7b6cf745c37596de885a714feecd87ebfbd7328","target":"record","created_at":"2026-05-18T04:00:01Z","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":"7feb99f9c1bc5163c15a28bd93e26a271cee3fb1a3e895977567ccaf4e2a26b7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T15:34:25Z","title_canon_sha256":"e17a3f6075c0afba724729b9fef86257a3382c2423e9ad1831bbabc0c6aeab5a"},"schema_version":"1.0","source":{"id":"1201.6576","kind":"arxiv","version":3}},"canonical_sha256":"5ad1ab8268c1b284b0136aa772a92088a396402564897e27e30e2881607e9c7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ad1ab8268c1b284b0136aa772a92088a396402564897e27e30e2881607e9c7c","first_computed_at":"2026-05-18T04:00:01.756569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:01.756569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2Z6B4iFBnt147xgvF1Nd52ybeHET0qGhfnHz1v7kpE4pVKX91RacvVtTUvLDGeEUOhbhJPcmaAeQjgsDJSjvCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:01.757098Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.6576","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fce179ec0c9e78d55f9057a8e7b6cf745c37596de885a714feecd87ebfbd7328","sha256:9bd250ee9b4e3e39e026d3d587f8dcfc0673d466059438918c64833a517af102"],"state_sha256":"e6cd0a7556bac53fe0846b74f92fa6ae22ec69bd633062b82e612f2455b0d200"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y405ikqw4sg/ozoyQRwrYN2KtJdZqzM51jPxFh//xchqW2W17iOq2n2PzutBqVc6B5Bmf6kdk6LfmRbnYWDgAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:01:25.987721Z","bundle_sha256":"10cb10d1ca8ea600d7436d69e6d6a5448e4cc4498c1f044326a2f6bc19122abf"}}