{"paper":{"title":"The Eulerian distribution on involutions is indeed $\\gamma$-positive","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danielle Wang","submitted_at":"2018-08-25T23:14:27Z","abstract_excerpt":"Let $\\mathcal I_n$ and $\\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\\{1, \\dots, n\\}$, respectively, and let $\\text{des}(\\pi)$ denote the number of descents of the permutation $\\pi$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \\sum_{\\pi \\in \\mathcal I_n} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 1$ and $J_{2n}(t) := \\sum_{\\pi \\in \\mathcal J_{2n}} t^{\\text{des}(\\pi)}$ is $\\gamma$-positive for $n \\ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08481","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"}