{"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 ${\\bf r}\\in\\mathscr{R}(\\Delta)$. It is proved that the se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1955","kind":"arxiv","version":1},"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"}