{"paper":{"title":"Generalized Macaulay representations and the flag $f$-vectors of generalized colored complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Fong Ernest Chong","submitted_at":"2013-06-07T17:36:07Z","abstract_excerpt":"A colored complex of type $\\mathbf{a} = (a_1, \\dots, a_n)$ is a simplicial complex ${\\Delta}$ on a vertex set $V$, together with an ordered partition $(V_1, \\dots, V_n)$ of $V$, such that every face $F$ of ${\\Delta}$ satisfies $|F \\cap V_i| \\leq a_i$. For each $\\mathbf{b} = (b_1, \\dots, b_n) \\leq \\mathbf{a}$, let $f_{\\mathbf{b}}$ be the number of faces $F$ of ${\\Delta}$ such that $|F \\cap V_i| = b_i$. The array of integers $\\{f_{\\mathbf{b}}\\}_{\\mathbf{b} \\leq \\mathbf{a}}$ is called the fine $f$-vector of ${\\Delta}$, and it is a refinement of the $f$-vector of ${\\Delta}$. In this paper, we gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1787","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"}