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Here the notation ${\\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\\bf r}\\geq {\\bf r}'$ means that $r_i\\geq r'_i$ for every $i$. Let $\\mathscr{R}(\\Delta)$ be the set of vectors ${\\bf r}$ such that ${\\bf r}>0$ and ${\\bf r}\\Delta\\geq 0$. In this paper, $(\\Delta,{\\bf r})$-parking functions are defined for any ${\\bf r}\\in\\mathscr{R}(\\Delta)$. It is proved that the se"},"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":"1407.1955","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-08T05:42:12Z","cross_cats_sorted":[],"title_canon_sha256":"d0b840f508d41d701ac08302e82133b46ea2d0be0a14f6806814a8332dfaee5f","abstract_canon_sha256":"91625c52370cda6cc685d48d17adc367bcb69320c12d93f043ee19800c0371d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:08.798202Z","signature_b64":"YhVd7+zrUo1USfNBQq+HVUw2EL079rHWhtZU37MLZbScTpmuAJ0+nalaU43xwd9o8PXXgj0o3kGFit0YtVnkAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e8f08c352e4498c3009a698232ea0fdc6de5ddc7dc04d3501ab3b866996d3a30","last_reissued_at":"2026-05-18T02:48:08.797536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:08.797536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parking functions on toppling matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Ma, Yeong-Nan Yeh","submitted_at":"2014-07-08T05:42:12Z","abstract_excerpt":"Let $\\Delta$ be an integer $n \\times n$-matrix which satisfies the\n  conditions: $\\det \\Delta\\neq 0$, $\\Delta_{ij}\\leq 0\\text{ for }i\\neq j,$ and\n  there exists a vector ${\\bf r}=(r_1,\\ldots,r_n)>0$ such that ${\\bf r}\\Delta \\geq 0$. Here the notation ${\\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\\bf r}\\geq {\\bf r}'$ means that $r_i\\geq r'_i$ for every $i$. Let $\\mathscr{R}(\\Delta)$ be the set of vectors ${\\bf r}$ such that ${\\bf r}>0$ and ${\\bf r}\\Delta\\geq 0$. In this paper, $(\\Delta,{\\bf r})$-parking functions are defined for any ${\\bf r}\\in\\mathscr{R}(\\Delta)$. 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