{"paper":{"title":"Few Sequence Pairs Suffice: Representing All Rectangle Placements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jannik Silvanus, Jens Vygen","submitted_at":"2017-08-31T15:40:40Z","abstract_excerpt":"We consider representations of general non-overlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every placement of $n$ rectangles.\n  We prove a new upper bound of $\\mathcal{O}(\\frac{n!}{n^6} \\cdot (\\frac{11+5 \\sqrt 5}{2})^n)$ and a new lower bound of $\\Omega(\\frac{n!}{n^4} \\cdot (4 + 2 \\sqrt2)^n)$ on the minimum cardinality of complete sets of representations. A key concept in the proofs of these results are pattern-avoiding permutations.\n  The new upper bound di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.09779","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"}