{"paper":{"title":"Counting Connected Set Partitions of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Frank Simon, Martin Trinks, Peter Tittmann","submitted_at":"2010-05-11T06:44:44Z","abstract_excerpt":"Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\\pi=\\{V_1, ..., V_k\\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\\pi$ is connected for $1\\le j\\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is then $Q(G, x)=\\sum_{i=0}^n q_{i}(G)x^i$. This paper presents a splitting approach to the partition polynomial on a separating vertex set $X$ in $G$ and summarizes some properties of the bond lattice. Furthermore the bivariate partiti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1726","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"}