{"paper":{"title":"Note on the number of antichains in generalizations of the Boolean lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jinyoung Park, Michail Sarantis, Prasad Tetali","submitted_at":"2023-05-25T22:55:07Z","abstract_excerpt":"We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\\Bigl(\\frac{n}{\\log n}\\Bigr)$, the number $\\alpha([t]^n)$ of antichains of the poset $[t]^n$ is at most \\[\\exp_2\\Bigl(1+O\\Bigl(\\Bigl(\\frac{t\\log^3 n}{n}\\Bigr)^{1/2}\\Bigr)\\Bigr)N(t,n)\\,,\\] where $N(t,n)$ is the size of a largest level of $[t]^n$. This, in particular, says that if $t \\ll n/\\log^3 n$ as $n \\rightarrow \\infty$, then $\\log\\alpha([t]^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t, n$ in this range.\n  Particularly for $t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.16520","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.16520/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}