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Then a sequence of numbers $\\{r_n({\\sf Q}_n)\\}_{n\\ge 1}$ is defined by the equation \\begin{eqnarray*} \\sum_{n\\ge 1}r_n({\\sf Q}_n)\\frac{z^n}{n!\\,M(n)}=-\\log(\\sum_{n\\ge 0}(-1)^n\\frac{z^n}{n!\\,M(n)}). \\end{eqnarray*} Let $\\bar{{\\sf Q}}_n$ denote the poset ${\\sf Q}_n$ with a $\\hat{0}$ adjoined and let $\\hat{1}$ denote the unique maximal element in the poset ${\\sf Q}_n$. Furthermore, let $\\mu_{{\\sf Q}_n}$ be the M\\\"{o}bius function on the poset $\\"},"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":"1407.0242","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-01T13:55:38Z","cross_cats_sorted":[],"title_canon_sha256":"6cc0df972d40e9b66dee45e4f3c19e7a8c512e29498b3258a22d8a788ef7c4ab","abstract_canon_sha256":"c2cd662d9260eacfda33494c855a60ef49677d9d4812ae35e2afb70327bb44cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:38.652166Z","signature_b64":"uDBZwbCZtTz5a6C+5cgcV3DX9qnNWlYKKc11WcFu/LjrQcbbmLBz39+AEADKxEO2+X8jzUOVjm0c0bp9s6yyCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b7fbcf61ba0a9c45d0c9f3669ff142a1e0911da40c45335c69255ec7fdc827f","last_reissued_at":"2026-05-18T01:24:38.651640Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:38.651640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heaps and Two Exponential Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emma Yu Jin","submitted_at":"2014-07-01T13:55:38Z","abstract_excerpt":"Take ${\\sf Q}=({\\sf Q}_1,{\\sf Q}_2,\\ldots)$ to be an exponential structure and $M(n)$ to be the number of minimal elements of ${\\sf Q}_n$ where $M(0)=1$. Then a sequence of numbers $\\{r_n({\\sf Q}_n)\\}_{n\\ge 1}$ is defined by the equation \\begin{eqnarray*} \\sum_{n\\ge 1}r_n({\\sf Q}_n)\\frac{z^n}{n!\\,M(n)}=-\\log(\\sum_{n\\ge 0}(-1)^n\\frac{z^n}{n!\\,M(n)}). \\end{eqnarray*} Let $\\bar{{\\sf Q}}_n$ denote the poset ${\\sf Q}_n$ with a $\\hat{0}$ adjoined and let $\\hat{1}$ denote the unique maximal element in the poset ${\\sf Q}_n$. 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