{"paper":{"title":"The Bruhat order on conjugation-invariant sets of involutions in the symmetric group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mikael Hansson","submitted_at":"2015-02-12T10:56:54Z","abstract_excerpt":"Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \\subseteq \\{0,1,\\ldots,n\\}$, let \\[ F_n^A=\\{\\sigma \\in I_n \\mid \\text{$\\sigma$ has $a$ fixed points for some $a \\in A$}\\}. \\] We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\\{1\\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03598","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"}