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The front representation of $\\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block whose smallest integer is $i$. Using the front representation, we find a recurrence relation for the number of $12... k12$-avoiding partitions for $k\\geq2$. Similarly, we find a recurrence relatio"},"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":"0907.1485","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-07-09T10:21:56Z","cross_cats_sorted":[],"title_canon_sha256":"6b15ad1d00563c9950593731672611e2d0b5472f54bc1a55850aa96ec936f048","abstract_canon_sha256":"6dfba4805207394ababb0983b70ed80a1fea49816b1e857d783e22352116f3d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:38.310734Z","signature_b64":"515WtFoktx5SpXBzOZ+6/kBmbzS3QhRqLhk9HsbtTeJvlekfZ4QWkISK7hc/8HOHYcrbouUe76f+c6xcSNPlDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1fc8775d4f9e19693f3f9f53dab825cbbabb59c5c31ddebfbfc9e4bd90d1d67","last_reissued_at":"2026-05-18T04:14:38.310289Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:38.310289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Front representation of set partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jang Soo Kim","submitted_at":"2009-07-09T10:21:56Z","abstract_excerpt":"Let $\\pi$ be a set partition of $[n]=\\{1,2,...,n\\}$. The standard representation of $\\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block which does not contain any integer between $i$ and $j$. The front representation of $\\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block whose smallest integer is $i$. Using the front representation, we find a recurrence relation for the number of $12... k12$-avoiding partitions for $k\\geq2$. 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