{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:KTFZACWJK5BVZYDEGVL5SO5OLD","short_pith_number":"pith:KTFZACWJ","canonical_record":{"source":{"id":"1305.2206","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-09T20:13:43Z","cross_cats_sorted":[],"title_canon_sha256":"900a01510e98add48fc50dbdc9e74463ae4269402da60b1f3e78eb92a5af1f36","abstract_canon_sha256":"03e5a4668918194cdca61ed6f0e12147d20368be6f3a51c821aa91ea335493fc"},"schema_version":"1.0"},"canonical_sha256":"54cb900ac957435ce0643557d93bae58e1a019850e7b0896203af93942ce14d8","source":{"kind":"arxiv","id":"1305.2206","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2206","created_at":"2026-05-18T01:26:03Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2206v2","created_at":"2026-05-18T01:26:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2206","created_at":"2026-05-18T01:26:03Z"},{"alias_kind":"pith_short_12","alias_value":"KTFZACWJK5BV","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"KTFZACWJK5BVZYDE","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"KTFZACWJ","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:KTFZACWJK5BVZYDEGVL5SO5OLD","target":"record","payload":{"canonical_record":{"source":{"id":"1305.2206","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-09T20:13:43Z","cross_cats_sorted":[],"title_canon_sha256":"900a01510e98add48fc50dbdc9e74463ae4269402da60b1f3e78eb92a5af1f36","abstract_canon_sha256":"03e5a4668918194cdca61ed6f0e12147d20368be6f3a51c821aa91ea335493fc"},"schema_version":"1.0"},"canonical_sha256":"54cb900ac957435ce0643557d93bae58e1a019850e7b0896203af93942ce14d8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:03.209892Z","signature_b64":"fMYIzMFXKbf9B3qOivNhOeEp5SBZdVueONtwMbUR5TNviKB6bz8p3oq7Ai4iUqOsvAE9JtKF58yWhjseSerYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54cb900ac957435ce0643557d93bae58e1a019850e7b0896203af93942ce14d8","last_reissued_at":"2026-05-18T01:26:03.209473Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:03.209473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.2206","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:26:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"acoSNShhFbWO4ikFXPB1sZo2OUic9X50Gy2Fu505HZsoTF/72YgWgCHKk0fdW4r0A+0GkNvfZ5cU+O/5KgNcAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:10:44.543953Z"},"content_sha256":"543562da85812a26604aebd6da4a310586a255943519d7590f1201fef360be41","schema_version":"1.0","event_id":"sha256:543562da85812a26604aebd6da4a310586a255943519d7590f1201fef360be41"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:KTFZACWJK5BVZYDEGVL5SO5OLD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Symmetries of statistics on lattice paths between two boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Martin Rubey, Sergi Elizalde","submitted_at":"2013-05-09T20:13:43Z","abstract_excerpt":"We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with T' have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is inde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2206","kind":"arxiv","version":2},"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":"2026-05-18T01:26:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PStw+2BRN7ub6cojaOdVjBxKEqIemAQqOUzZt27UOiaSgp2y+5B41pT8HQ5fJbREzgy8QIoG43y6gzwpBAX3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:10:44.544738Z"},"content_sha256":"63a8bbfee48ec7694f129508d62c2426063d0301a95206546746f3daeaa396fe","schema_version":"1.0","event_id":"sha256:63a8bbfee48ec7694f129508d62c2426063d0301a95206546746f3daeaa396fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KTFZACWJK5BVZYDEGVL5SO5OLD/bundle.json","state_url":"https://pith.science/pith/KTFZACWJK5BVZYDEGVL5SO5OLD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KTFZACWJK5BVZYDEGVL5SO5OLD/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":"2026-06-10T01:10:44Z","links":{"resolver":"https://pith.science/pith/KTFZACWJK5BVZYDEGVL5SO5OLD","bundle":"https://pith.science/pith/KTFZACWJK5BVZYDEGVL5SO5OLD/bundle.json","state":"https://pith.science/pith/KTFZACWJK5BVZYDEGVL5SO5OLD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KTFZACWJK5BVZYDEGVL5SO5OLD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KTFZACWJK5BVZYDEGVL5SO5OLD","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":"03e5a4668918194cdca61ed6f0e12147d20368be6f3a51c821aa91ea335493fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-09T20:13:43Z","title_canon_sha256":"900a01510e98add48fc50dbdc9e74463ae4269402da60b1f3e78eb92a5af1f36"},"schema_version":"1.0","source":{"id":"1305.2206","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2206","created_at":"2026-05-18T01:26:03Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2206v2","created_at":"2026-05-18T01:26:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2206","created_at":"2026-05-18T01:26:03Z"},{"alias_kind":"pith_short_12","alias_value":"KTFZACWJK5BV","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"KTFZACWJK5BVZYDE","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"KTFZACWJ","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:63a8bbfee48ec7694f129508d62c2426063d0301a95206546746f3daeaa396fe","target":"graph","created_at":"2026-05-18T01:26:03Z","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 prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with T' have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is inde","authors_text":"Martin Rubey, Sergi Elizalde","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-09T20:13:43Z","title":"Symmetries of statistics on lattice paths between two boundaries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2206","kind":"arxiv","version":2},"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":"sha256:543562da85812a26604aebd6da4a310586a255943519d7590f1201fef360be41","target":"record","created_at":"2026-05-18T01:26:03Z","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":"03e5a4668918194cdca61ed6f0e12147d20368be6f3a51c821aa91ea335493fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-09T20:13:43Z","title_canon_sha256":"900a01510e98add48fc50dbdc9e74463ae4269402da60b1f3e78eb92a5af1f36"},"schema_version":"1.0","source":{"id":"1305.2206","kind":"arxiv","version":2}},"canonical_sha256":"54cb900ac957435ce0643557d93bae58e1a019850e7b0896203af93942ce14d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54cb900ac957435ce0643557d93bae58e1a019850e7b0896203af93942ce14d8","first_computed_at":"2026-05-18T01:26:03.209473Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:03.209473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fMYIzMFXKbf9B3qOivNhOeEp5SBZdVueONtwMbUR5TNviKB6bz8p3oq7Ai4iUqOsvAE9JtKF58yWhjseSerYDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:03.209892Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.2206","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:543562da85812a26604aebd6da4a310586a255943519d7590f1201fef360be41","sha256:63a8bbfee48ec7694f129508d62c2426063d0301a95206546746f3daeaa396fe"],"state_sha256":"bd461795021b5d2ec3321c30d709618f2a1d3504445c836fcfc92e1bdee1fe87"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1eAQi4qGCNELaPcYxqZq5bTY+jooPAklxiKolQw8JBYBQttyCCpZaDd/6DS1kfAU6LR5ITF3acM1OZwKYQk8Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T01:10:44.549052Z","bundle_sha256":"e85183f39eb2dee79019cdfc900767c526a92a56ac4f21f24df5e4237e3f836f"}}