{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:X3CFBLULCXGMGUE5UMYUPZZQKW","short_pith_number":"pith:X3CFBLUL","schema_version":"1.0","canonical_sha256":"bec450ae8b15ccc3509da33147e73055a6061f94c46983eef83e381350f831f4","source":{"kind":"arxiv","id":"1401.7386","version":2},"attestation_state":"computed","paper":{"title":"Averaging algebras, Schr\\\"oder numbers, rooted trees and operads","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Jun Pei, Li Guo","submitted_at":"2014-01-29T01:51:22Z","abstract_excerpt":"In this paper, we study averaging operators from an algebraic and combinatorial point of view. We first construct free averaging algebras in terms of a class of bracketed words called averaging words. We next apply this construction to obtain one and two variable generating functions for subsets of averaging words when the averaging operator is taken to be idempotent. When the averaging algebra has an idempotent generator, the generating function in one variable is twice the generating function for large Schr\\\"oder numbers, leading us to give interpretations of large Schr\\\"oder numbers in term"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.7386","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-01-29T01:51:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d63312f218eef512bcffb0180201c7d47b7fa8432164b0a2beb4672283e6643e","abstract_canon_sha256":"2f69b99dfda600e1ca9e00a4ef85ee8d3486afa7501e5738d5fedbdfe5a1f391"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:15.660665Z","signature_b64":"EeMWLWGIfqfXVafzfNvcnx5XJCx5Vt6p3lvrzyvar3D/K8wbaYPrbxja/kQpDBTs8PCLDtRCn61QJthR9JXGAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bec450ae8b15ccc3509da33147e73055a6061f94c46983eef83e381350f831f4","last_reissued_at":"2026-05-18T01:30:15.660162Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:15.660162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Averaging algebras, Schr\\\"oder numbers, rooted trees and operads","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Jun Pei, Li Guo","submitted_at":"2014-01-29T01:51:22Z","abstract_excerpt":"In this paper, we study averaging operators from an algebraic and combinatorial point of view. We first construct free averaging algebras in terms of a class of bracketed words called averaging words. We next apply this construction to obtain one and two variable generating functions for subsets of averaging words when the averaging operator is taken to be idempotent. When the averaging algebra has an idempotent generator, the generating function in one variable is twice the generating function for large Schr\\\"oder numbers, leading us to give interpretations of large Schr\\\"oder numbers in term"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7386","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1401.7386","created_at":"2026-05-18T01:30:15.660227+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.7386v2","created_at":"2026-05-18T01:30:15.660227+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.7386","created_at":"2026-05-18T01:30:15.660227+00:00"},{"alias_kind":"pith_short_12","alias_value":"X3CFBLULCXGM","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"X3CFBLULCXGMGUE5","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"X3CFBLUL","created_at":"2026-05-18T12:28:54.890064+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW","json":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW.json","graph_json":"https://pith.science/api/pith-number/X3CFBLULCXGMGUE5UMYUPZZQKW/graph.json","events_json":"https://pith.science/api/pith-number/X3CFBLULCXGMGUE5UMYUPZZQKW/events.json","paper":"https://pith.science/paper/X3CFBLUL"},"agent_actions":{"view_html":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW","download_json":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW.json","view_paper":"https://pith.science/paper/X3CFBLUL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.7386&json=true","fetch_graph":"https://pith.science/api/pith-number/X3CFBLULCXGMGUE5UMYUPZZQKW/graph.json","fetch_events":"https://pith.science/api/pith-number/X3CFBLULCXGMGUE5UMYUPZZQKW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW/action/storage_attestation","attest_author":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW/action/author_attestation","sign_citation":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW/action/citation_signature","submit_replication":"https://pith.science/pith/X3CFBLULCXGMGUE5UMYUPZZQKW/action/replication_record"}},"created_at":"2026-05-18T01:30:15.660227+00:00","updated_at":"2026-05-18T01:30:15.660227+00:00"}