{"paper":{"title":"Some Sufficient Conditions for Finding a Nesting of the Normalized Matching Posets of Rank 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wei-Tian Li, Yu-Lun Chang","submitted_at":"2017-09-06T11:17:47Z","abstract_excerpt":"Given a graded poset $P$, consider a chain decomposition $\\mathcal{C}$ of $P$. If $|C_1|\\le |C_2|$ implies that the set of the ranks of elements in $C_1$ is a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\\in \\mathcal{C}$, then we say $\\mathcal{C}$ is a nested chain decomposition (or nesting, for short) of $P$, and $P$ is said to be nested. In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset is nested. This conjecture is proved to be true only for all posets of rank 2 [W:05], some posets of rank 3 [HLS:09,ENSST:11], and the very special cases for h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.01768","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"}