{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:SKPQN4MJZKKITJ5VBGAWXZ5MDB","short_pith_number":"pith:SKPQN4MJ","schema_version":"1.0","canonical_sha256":"929f06f189ca9489a7b509816be7ac18417b18ce3776bf216a93d5fc057781e9","source":{"kind":"arxiv","id":"1512.05937","version":2},"attestation_state":"computed","paper":{"title":"A combinatorial Hopf algebra for the boson normal ordering problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Ali Chouria, Imad Eddine Bousbaa, Jean-Gabriel Luque","submitted_at":"2015-12-18T13:14:55Z","abstract_excerpt":"In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \\emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\\bf{r,s}}(k)$ appearing in the identity $(a^\\dag)^{r_n}a^{s_n}\\cdots(a^\\dag)^{r_1}a^{s_1}=(a^\\dag)^\\alpha\\displaystyle\\sum S_{\\bf{r,s}}(k)(a^\\dag)^k a^k$, where $\\alpha$ is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which specializes to the enveloping algebra of the Heisenberg Lie a"},"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":"1512.05937","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-18T13:14:55Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a98bff1502bc766ef3bf0881314f86b7c3a418cb6edecbb501450e5331afb574","abstract_canon_sha256":"00c1b7849dcc6b699154c87fdbb3898f9d77a2874e8ed056ef4647bdda883bd7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:04.696979Z","signature_b64":"ufFSLmNYVtVVmsTB+Dc9NbcaOTRlEIj+NcIRWym07aZ/6QGjO+Zhw6GhVleBUHJJnhS28XLchI+1No7Ckxx4Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"929f06f189ca9489a7b509816be7ac18417b18ce3776bf216a93d5fc057781e9","last_reissued_at":"2026-05-18T00:24:04.696405Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:04.696405Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A combinatorial Hopf algebra for the boson normal ordering problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Ali Chouria, Imad Eddine Bousbaa, Jean-Gabriel Luque","submitted_at":"2015-12-18T13:14:55Z","abstract_excerpt":"In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \\emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\\bf{r,s}}(k)$ appearing in the identity $(a^\\dag)^{r_n}a^{s_n}\\cdots(a^\\dag)^{r_1}a^{s_1}=(a^\\dag)^\\alpha\\displaystyle\\sum S_{\\bf{r,s}}(k)(a^\\dag)^k a^k$, where $\\alpha$ is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which specializes to the enveloping algebra of the Heisenberg Lie a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05937","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":"1512.05937","created_at":"2026-05-18T00:24:04.696518+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.05937v2","created_at":"2026-05-18T00:24:04.696518+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05937","created_at":"2026-05-18T00:24:04.696518+00:00"},{"alias_kind":"pith_short_12","alias_value":"SKPQN4MJZKKI","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"SKPQN4MJZKKITJ5V","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"SKPQN4MJ","created_at":"2026-05-18T12:29:42.218222+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/SKPQN4MJZKKITJ5VBGAWXZ5MDB","json":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB.json","graph_json":"https://pith.science/api/pith-number/SKPQN4MJZKKITJ5VBGAWXZ5MDB/graph.json","events_json":"https://pith.science/api/pith-number/SKPQN4MJZKKITJ5VBGAWXZ5MDB/events.json","paper":"https://pith.science/paper/SKPQN4MJ"},"agent_actions":{"view_html":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB","download_json":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB.json","view_paper":"https://pith.science/paper/SKPQN4MJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.05937&json=true","fetch_graph":"https://pith.science/api/pith-number/SKPQN4MJZKKITJ5VBGAWXZ5MDB/graph.json","fetch_events":"https://pith.science/api/pith-number/SKPQN4MJZKKITJ5VBGAWXZ5MDB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB/action/storage_attestation","attest_author":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB/action/author_attestation","sign_citation":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB/action/citation_signature","submit_replication":"https://pith.science/pith/SKPQN4MJZKKITJ5VBGAWXZ5MDB/action/replication_record"}},"created_at":"2026-05-18T00:24:04.696518+00:00","updated_at":"2026-05-18T00:24:04.696518+00:00"}