{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:NFWDCNVXBVUF2GIE5BWL4SAIMZ","short_pith_number":"pith:NFWDCNVX","canonical_record":{"source":{"id":"1407.1919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-08T00:57:10Z","cross_cats_sorted":[],"title_canon_sha256":"407f65ef1d65aa44efb000bccf4261df7e642347ed4a5bc1545c2181e31a898c","abstract_canon_sha256":"47b8ed6fb9bb19369f12e717ba00c87b278b91167a239d25e14ec1717d63b39d"},"schema_version":"1.0"},"canonical_sha256":"696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e","source":{"kind":"arxiv","id":"1407.1919","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.1919","created_at":"2026-05-18T02:48:08Z"},{"alias_kind":"arxiv_version","alias_value":"1407.1919v1","created_at":"2026-05-18T02:48:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1919","created_at":"2026-05-18T02:48:08Z"},{"alias_kind":"pith_short_12","alias_value":"NFWDCNVXBVUF","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"NFWDCNVXBVUF2GIE","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"NFWDCNVX","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:NFWDCNVXBVUF2GIE5BWL4SAIMZ","target":"record","payload":{"canonical_record":{"source":{"id":"1407.1919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-08T00:57:10Z","cross_cats_sorted":[],"title_canon_sha256":"407f65ef1d65aa44efb000bccf4261df7e642347ed4a5bc1545c2181e31a898c","abstract_canon_sha256":"47b8ed6fb9bb19369f12e717ba00c87b278b91167a239d25e14ec1717d63b39d"},"schema_version":"1.0"},"canonical_sha256":"696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:08.908180Z","signature_b64":"99oNhalufbSFKAlAZig88lR6nLDlI8STLtQ3cy7g8fzXMSi6SU9sSzHkue85G73V8qAJ3RMV/QXAOG+6QgjbCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e","last_reissued_at":"2026-05-18T02:48:08.907765Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:08.907765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.1919","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:48:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hlcgNEe8sEg+X/8f9jCn0nVlViqaVFcBzuHhWBAZ/7BI7W200pB8j/6Q+pbTXGU0lDe78sy6NGN365NyH6flAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:12:53.558219Z"},"content_sha256":"004c1723622917b463a5269d595737a92acf3e6b1b5fa5305ae0ed939583156e","schema_version":"1.0","event_id":"sha256:004c1723622917b463a5269d595737a92acf3e6b1b5fa5305ae0ed939583156e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:NFWDCNVXBVUF2GIE5BWL4SAIMZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"There are $(r+1)(r+2)(2r+3)(r^2+3r+5)$ Ways For the Four Teams of a World Cup Group to Each Have $r$ Goals For and $r$ Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1]","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Doron Zeilberger, Shalosh B. Ekhad","submitted_at":"2014-07-08T00:57:10Z","abstract_excerpt":"This short tribute to the guru of Enumerative and Algebraic Combinatorics started out when one the authors(DZ) attended the Stanely@70 conference, that took place at the same time as the preliminary stage of the 2014 World Cup. It states a surprising application of an analog of Richard Stanley's famous theorem about the enumeration of magic squares to the enumeration of possible outcomes in a World Cup Group."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1919","kind":"arxiv","version":1},"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":"2026-05-18T02:48:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/yNvnbj2TRv9woeUbTVV6mtR/0+xb6UlojMDQSbG83ASw6R5c/FaLN5o7bYqCfzoVxIEWWIUnVJjjeB8JzatDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:12:53.558571Z"},"content_sha256":"13a3e80ed0dcac543d037786104f6bc9f7c837977d328b4024b6e061cd4ca790","schema_version":"1.0","event_id":"sha256:13a3e80ed0dcac543d037786104f6bc9f7c837977d328b4024b6e061cd4ca790"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ/bundle.json","state_url":"https://pith.science/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ/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":"2026-05-30T13:12:53Z","links":{"resolver":"https://pith.science/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ","bundle":"https://pith.science/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ/bundle.json","state":"https://pith.science/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NFWDCNVXBVUF2GIE5BWL4SAIMZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:NFWDCNVXBVUF2GIE5BWL4SAIMZ","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":"47b8ed6fb9bb19369f12e717ba00c87b278b91167a239d25e14ec1717d63b39d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-08T00:57:10Z","title_canon_sha256":"407f65ef1d65aa44efb000bccf4261df7e642347ed4a5bc1545c2181e31a898c"},"schema_version":"1.0","source":{"id":"1407.1919","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.1919","created_at":"2026-05-18T02:48:08Z"},{"alias_kind":"arxiv_version","alias_value":"1407.1919v1","created_at":"2026-05-18T02:48:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1919","created_at":"2026-05-18T02:48:08Z"},{"alias_kind":"pith_short_12","alias_value":"NFWDCNVXBVUF","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"NFWDCNVXBVUF2GIE","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"NFWDCNVX","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:13a3e80ed0dcac543d037786104f6bc9f7c837977d328b4024b6e061cd4ca790","target":"graph","created_at":"2026-05-18T02:48:08Z","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":"This short tribute to the guru of Enumerative and Algebraic Combinatorics started out when one the authors(DZ) attended the Stanely@70 conference, that took place at the same time as the preliminary stage of the 2014 World Cup. It states a surprising application of an analog of Richard Stanley's famous theorem about the enumeration of magic squares to the enumeration of possible outcomes in a World Cup Group.","authors_text":"Doron Zeilberger, Shalosh B. Ekhad","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-08T00:57:10Z","title":"There are $(r+1)(r+2)(2r+3)(r^2+3r+5)$ Ways For the Four Teams of a World Cup Group to Each Have $r$ Goals For and $r$ Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1]"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1919","kind":"arxiv","version":1},"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":"sha256:004c1723622917b463a5269d595737a92acf3e6b1b5fa5305ae0ed939583156e","target":"record","created_at":"2026-05-18T02:48:08Z","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":"47b8ed6fb9bb19369f12e717ba00c87b278b91167a239d25e14ec1717d63b39d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-08T00:57:10Z","title_canon_sha256":"407f65ef1d65aa44efb000bccf4261df7e642347ed4a5bc1545c2181e31a898c"},"schema_version":"1.0","source":{"id":"1407.1919","kind":"arxiv","version":1}},"canonical_sha256":"696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"696c3136b70d685d1904e86cbe4808666e47d38764132ad69ff76610c009a64e","first_computed_at":"2026-05-18T02:48:08.907765Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:08.907765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"99oNhalufbSFKAlAZig88lR6nLDlI8STLtQ3cy7g8fzXMSi6SU9sSzHkue85G73V8qAJ3RMV/QXAOG+6QgjbCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:08.908180Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.1919","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:004c1723622917b463a5269d595737a92acf3e6b1b5fa5305ae0ed939583156e","sha256:13a3e80ed0dcac543d037786104f6bc9f7c837977d328b4024b6e061cd4ca790"],"state_sha256":"4b839529c42488d8d3d8fbf70345caf7ef84eec9f06ef5b896b393bc495504dd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LejQNt0WdVEdwexcfMcBJYHrYgjLYU7VLtSbl3l8SFHuft8aCoKVN7kj34ThkspTOpcZ6hCoz+M9fCSll4kPDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:12:53.560550Z","bundle_sha256":"ff5c1e72accdad74ce9caeb3386a830cfa921296f2db858a4ec5e8dc9c14b1f8"}}