{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:J6N4UHLG5MCEM6S56PEMTCXA6Q","short_pith_number":"pith:J6N4UHLG","canonical_record":{"source":{"id":"1003.1362","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-06T08:29:54Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"31c9546da58ace53601a09cbc26fa006ce12f52dbc9154b941cc8c5f735c40bc","abstract_canon_sha256":"0f140879bc740a67c85512b5f351d0e2fbc854024b45a037fb65397e952bea15"},"schema_version":"1.0"},"canonical_sha256":"4f9bca1d66eb04467a5df3c8c98ae0f40425cfb34742fe2a3d3b2469b407bcb7","source":{"kind":"arxiv","id":"1003.1362","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.1362","created_at":"2026-05-18T03:55:37Z"},{"alias_kind":"arxiv_version","alias_value":"1003.1362v3","created_at":"2026-05-18T03:55:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.1362","created_at":"2026-05-18T03:55:37Z"},{"alias_kind":"pith_short_12","alias_value":"J6N4UHLG5MCE","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"J6N4UHLG5MCEM6S5","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"J6N4UHLG","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:J6N4UHLG5MCEM6S56PEMTCXA6Q","target":"record","payload":{"canonical_record":{"source":{"id":"1003.1362","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-06T08:29:54Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"31c9546da58ace53601a09cbc26fa006ce12f52dbc9154b941cc8c5f735c40bc","abstract_canon_sha256":"0f140879bc740a67c85512b5f351d0e2fbc854024b45a037fb65397e952bea15"},"schema_version":"1.0"},"canonical_sha256":"4f9bca1d66eb04467a5df3c8c98ae0f40425cfb34742fe2a3d3b2469b407bcb7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:37.698808Z","signature_b64":"IW/iHBtMUYRfzswJ3yc/RjuONC9DTrwtF20vOD6DGrNkWQ/zVfCf+cjOprZrSKGJBg6VHXjuq3ylgfBn5+ktBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f9bca1d66eb04467a5df3c8c98ae0f40425cfb34742fe2a3d3b2469b407bcb7","last_reissued_at":"2026-05-18T03:55:37.698178Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:37.698178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.1362","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":"2026-05-18T03:55:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B2j54kfPn6W9g4KizmSA2zkqJjNXVEayfRdVNRDtrPOFtwL+oS3Vql5MY1MalzdATwfxyPkU/wb9VjCeoTa6Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:18:28.772171Z"},"content_sha256":"6376f3a860fe2340e05200d870f60c560162116b50d0236f386bca09ad9e2d94","schema_version":"1.0","event_id":"sha256:6376f3a860fe2340e05200d870f60c560162116b50d0236f386bca09ad9e2d94"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:J6N4UHLG5MCEM6S56PEMTCXA6Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting walks in a quadrant: a unified approach via boundary value problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Kilian Raschel","submitted_at":"2010-03-06T08:29:54Z","abstract_excerpt":"The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.1362","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":"2026-05-18T03:55:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bD734gv8O5K5U0HkwGUpkP8xmvI+AuGiQVlCiUPvEUSqN4pHViOPAJV8oAjX1GdXXUv3DYIGnmvBHteiA6HaAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:18:28.772536Z"},"content_sha256":"3c68b4ed99b153cd519a5cb5271d76e7b8a88e44241a9a1486dcd634cd6236ee","schema_version":"1.0","event_id":"sha256:3c68b4ed99b153cd519a5cb5271d76e7b8a88e44241a9a1486dcd634cd6236ee"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q/bundle.json","state_url":"https://pith.science/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q/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":"2026-05-28T08:18:28Z","links":{"resolver":"https://pith.science/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q","bundle":"https://pith.science/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q/bundle.json","state":"https://pith.science/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J6N4UHLG5MCEM6S56PEMTCXA6Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:J6N4UHLG5MCEM6S56PEMTCXA6Q","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":"0f140879bc740a67c85512b5f351d0e2fbc854024b45a037fb65397e952bea15","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-06T08:29:54Z","title_canon_sha256":"31c9546da58ace53601a09cbc26fa006ce12f52dbc9154b941cc8c5f735c40bc"},"schema_version":"1.0","source":{"id":"1003.1362","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.1362","created_at":"2026-05-18T03:55:37Z"},{"alias_kind":"arxiv_version","alias_value":"1003.1362v3","created_at":"2026-05-18T03:55:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.1362","created_at":"2026-05-18T03:55:37Z"},{"alias_kind":"pith_short_12","alias_value":"J6N4UHLG5MCE","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"J6N4UHLG5MCEM6S5","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"J6N4UHLG","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:3c68b4ed99b153cd519a5cb5271d76e7b8a88e44241a9a1486dcd634cd6236ee","target":"graph","created_at":"2026-05-18T03:55:37Z","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 aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk","authors_text":"Kilian Raschel","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-06T08:29:54Z","title":"Counting walks in a quadrant: a unified approach via boundary value problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.1362","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":"sha256:6376f3a860fe2340e05200d870f60c560162116b50d0236f386bca09ad9e2d94","target":"record","created_at":"2026-05-18T03:55:37Z","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":"0f140879bc740a67c85512b5f351d0e2fbc854024b45a037fb65397e952bea15","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-06T08:29:54Z","title_canon_sha256":"31c9546da58ace53601a09cbc26fa006ce12f52dbc9154b941cc8c5f735c40bc"},"schema_version":"1.0","source":{"id":"1003.1362","kind":"arxiv","version":3}},"canonical_sha256":"4f9bca1d66eb04467a5df3c8c98ae0f40425cfb34742fe2a3d3b2469b407bcb7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4f9bca1d66eb04467a5df3c8c98ae0f40425cfb34742fe2a3d3b2469b407bcb7","first_computed_at":"2026-05-18T03:55:37.698178Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:37.698178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IW/iHBtMUYRfzswJ3yc/RjuONC9DTrwtF20vOD6DGrNkWQ/zVfCf+cjOprZrSKGJBg6VHXjuq3ylgfBn5+ktBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:37.698808Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.1362","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6376f3a860fe2340e05200d870f60c560162116b50d0236f386bca09ad9e2d94","sha256:3c68b4ed99b153cd519a5cb5271d76e7b8a88e44241a9a1486dcd634cd6236ee"],"state_sha256":"4d67e1686c4fb59331dff69dc51cc2e9beff9a9aaacf774a1dde7d37a16365b5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hr0J+JN78rPhMQlDVBRQsLQ7XWMGPU7L+0uoLqR0dmMEHVYynzHVGg3MbrxgFfb8tdM1D/jSMPT7hO/VCUENCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T08:18:28.774554Z","bundle_sha256":"5774388afa7c41f3d1d37845d7dd2ff1855bce9caf3270139f8027cc69de2d8e"}}