{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WQUVFX4GFLBQOUICNOP2KZYKTR","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":"5357416e0b5bb107c48f22cc140558d693f9922f14416127c87d0dc6418fa905","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-10T20:18:21Z","title_canon_sha256":"d31902cb39f74c562fb2522736fdc3a00e788c9363f89d8e14c6efc50d64a64c"},"schema_version":"1.0","source":{"id":"1502.03065","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.03065","created_at":"2026-05-18T02:25:04Z"},{"alias_kind":"arxiv_version","alias_value":"1502.03065v3","created_at":"2026-05-18T02:25:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.03065","created_at":"2026-05-18T02:25:04Z"},{"alias_kind":"pith_short_12","alias_value":"WQUVFX4GFLBQ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WQUVFX4GFLBQOUIC","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WQUVFX4G","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:7b8a323b10545d6e1ff34ab76ecb2dc83373819cb1b0b48e0b302ac4bf80855f","target":"graph","created_at":"2026-05-18T02:25:04Z","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":"The Pascal matrix, $P$, is an upper diagonal matrix whose entries are the binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies the beautiful relation $P=\\exp(H)$ in which $H$ has the numbers 1, 2, 3, etc. on its superdiagonal and zeros elsewhere. We generalize this identity to the incidence algebras $I(A^*)$ and $I(\\mathcal{S})$ of functions on words and permutations, respectively. In $I(A^*)$ the entries of $P$ and $H$ count subwords; in $I(\\mathcal{S})$ they count permutation patterns. Inspired by vincular permutation patterns we define what it means for a subword ","authors_text":"Anders Claesson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-10T20:18:21Z","title":"Subword counting and the incidence algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03065","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":"sha256:22a0942d612170920c6efbd2cd890a4c976b5122810228b1b04b3f2c995a989e","target":"record","created_at":"2026-05-18T02:25:04Z","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":"5357416e0b5bb107c48f22cc140558d693f9922f14416127c87d0dc6418fa905","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-10T20:18:21Z","title_canon_sha256":"d31902cb39f74c562fb2522736fdc3a00e788c9363f89d8e14c6efc50d64a64c"},"schema_version":"1.0","source":{"id":"1502.03065","kind":"arxiv","version":3}},"canonical_sha256":"b42952df862ac30751026b9fa5670a9c4a79ec3b42b2a5f63bc53f4de7737dcc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b42952df862ac30751026b9fa5670a9c4a79ec3b42b2a5f63bc53f4de7737dcc","first_computed_at":"2026-05-18T02:25:04.419241Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:04.419241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sOeTOQS0blcp2DXpLNgrxTpj7+AK0ZTJ1EjuRIHFwDUXprMVFBTKgUmnGDrhSOEFzhJrIQaE7F7nnJF5nVq/AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:04.419605Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.03065","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22a0942d612170920c6efbd2cd890a4c976b5122810228b1b04b3f2c995a989e","sha256:7b8a323b10545d6e1ff34ab76ecb2dc83373819cb1b0b48e0b302ac4bf80855f"],"state_sha256":"73aaa2b33c0ddd2b21de2c981a799730158e5ce3407b91d6a0b76554e2960f0a"}