{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3ETC5HIRI6DWYKNI3GQERFM4QL","short_pith_number":"pith:3ETC5HIR","canonical_record":{"source":{"id":"1704.02618","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-09T15:37:35Z","cross_cats_sorted":[],"title_canon_sha256":"c25ae916a49371fc23bfcd3ed6d317f926aa64dcdf1c397ab7849040617df74e","abstract_canon_sha256":"93745e341dff1eea13716738111115ec1de3c22bbdb725e4eed6054e09a275f3"},"schema_version":"1.0"},"canonical_sha256":"d9262e9d1147876c29a8d9a048959c82de4525168383eb3733188e43bac0eada","source":{"kind":"arxiv","id":"1704.02618","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02618","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02618v1","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02618","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"pith_short_12","alias_value":"3ETC5HIRI6DW","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3ETC5HIRI6DWYKNI","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3ETC5HIR","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3ETC5HIRI6DWYKNI3GQERFM4QL","target":"record","payload":{"canonical_record":{"source":{"id":"1704.02618","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-09T15:37:35Z","cross_cats_sorted":[],"title_canon_sha256":"c25ae916a49371fc23bfcd3ed6d317f926aa64dcdf1c397ab7849040617df74e","abstract_canon_sha256":"93745e341dff1eea13716738111115ec1de3c22bbdb725e4eed6054e09a275f3"},"schema_version":"1.0"},"canonical_sha256":"d9262e9d1147876c29a8d9a048959c82de4525168383eb3733188e43bac0eada","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:44.956934Z","signature_b64":"sIdituJzImJLzLkb0PEI4RLgir32D0vEPF3rqI8unGkbjo37HnMzcpP7lBYaZgOjFeIQN6aXAhnhYdt0ayQVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9262e9d1147876c29a8d9a048959c82de4525168383eb3733188e43bac0eada","last_reissued_at":"2026-05-18T00:46:44.956284Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:44.956284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.02618","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":"2026-05-18T00:46:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P7+PFFtG1d7/+fnxcXslXxxy+AFSTF7VyxXO01KgsaA50BBYbdE5pPuXT8Ptv/4aH5UGblugMiv06cmDCpUOCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T05:39:29.380037Z"},"content_sha256":"a92d3a39db682ebb9697fed52f401214c5b3167087a0b374be053cab5d1671ef","schema_version":"1.0","event_id":"sha256:a92d3a39db682ebb9697fed52f401214c5b3167087a0b374be053cab5d1671ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3ETC5HIRI6DWYKNI3GQERFM4QL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tail positive words and generalized coinvariant algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Timothy Wilson, Brendon Rhoades","submitted_at":"2017-04-09T15:37:35Z","abstract_excerpt":"Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\\mathbb{Q}[x_1, \\dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \\geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {\\em tail positive words}. We calcul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02618","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":"2026-05-18T00:46:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x0Xo6N9C2xJPrONdtIydPsZSz2F3g5VwrULgObjQFm5bHyY01KhsMa9RpVqSia7OIqhgMCpzLYMgJl8db+FRAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T05:39:29.380406Z"},"content_sha256":"9591858bfad526fc5416e983221911a4ec0bafb5875f9cd5894c42a67de65262","schema_version":"1.0","event_id":"sha256:9591858bfad526fc5416e983221911a4ec0bafb5875f9cd5894c42a67de65262"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3ETC5HIRI6DWYKNI3GQERFM4QL/bundle.json","state_url":"https://pith.science/pith/3ETC5HIRI6DWYKNI3GQERFM4QL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3ETC5HIRI6DWYKNI3GQERFM4QL/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":"2026-07-04T05:39:29Z","links":{"resolver":"https://pith.science/pith/3ETC5HIRI6DWYKNI3GQERFM4QL","bundle":"https://pith.science/pith/3ETC5HIRI6DWYKNI3GQERFM4QL/bundle.json","state":"https://pith.science/pith/3ETC5HIRI6DWYKNI3GQERFM4QL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3ETC5HIRI6DWYKNI3GQERFM4QL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3ETC5HIRI6DWYKNI3GQERFM4QL","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":"93745e341dff1eea13716738111115ec1de3c22bbdb725e4eed6054e09a275f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-09T15:37:35Z","title_canon_sha256":"c25ae916a49371fc23bfcd3ed6d317f926aa64dcdf1c397ab7849040617df74e"},"schema_version":"1.0","source":{"id":"1704.02618","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02618","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02618v1","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02618","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"pith_short_12","alias_value":"3ETC5HIRI6DW","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3ETC5HIRI6DWYKNI","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3ETC5HIR","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:9591858bfad526fc5416e983221911a4ec0bafb5875f9cd5894c42a67de65262","target":"graph","created_at":"2026-05-18T00:46:44Z","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":"Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\\mathbb{Q}[x_1, \\dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \\geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {\\em tail positive words}. We calcul","authors_text":"Andrew Timothy Wilson, Brendon Rhoades","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-09T15:37:35Z","title":"Tail positive words and generalized coinvariant algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02618","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:a92d3a39db682ebb9697fed52f401214c5b3167087a0b374be053cab5d1671ef","target":"record","created_at":"2026-05-18T00:46:44Z","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":"93745e341dff1eea13716738111115ec1de3c22bbdb725e4eed6054e09a275f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-09T15:37:35Z","title_canon_sha256":"c25ae916a49371fc23bfcd3ed6d317f926aa64dcdf1c397ab7849040617df74e"},"schema_version":"1.0","source":{"id":"1704.02618","kind":"arxiv","version":1}},"canonical_sha256":"d9262e9d1147876c29a8d9a048959c82de4525168383eb3733188e43bac0eada","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9262e9d1147876c29a8d9a048959c82de4525168383eb3733188e43bac0eada","first_computed_at":"2026-05-18T00:46:44.956284Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:44.956284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sIdituJzImJLzLkb0PEI4RLgir32D0vEPF3rqI8unGkbjo37HnMzcpP7lBYaZgOjFeIQN6aXAhnhYdt0ayQVCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:44.956934Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.02618","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a92d3a39db682ebb9697fed52f401214c5b3167087a0b374be053cab5d1671ef","sha256:9591858bfad526fc5416e983221911a4ec0bafb5875f9cd5894c42a67de65262"],"state_sha256":"dad5a952f131d0b70f47ef91d55758994e00f230bb021c955a6e6665f7c0652f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6zPJRLYMtfYCPfkUTAxutm1i/So3sfVv1Ci4/8tL45P/JvgBE3kkz3VqjI+qCHzxKyvMrxBII5ynT7sH2Ki7DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T05:39:29.382525Z","bundle_sha256":"3c081b035d8c9afde6f6af5359b95cb1c83771f8a1e0bcd2a8eebe408b68064e"}}