{"paper":{"title":"Most principal permutation classes have nonrational generating functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mikl\\'os B\\'ona","submitted_at":"2019-01-24T17:03:07Z","abstract_excerpt":"We prove that for any fixed $n$, and for most permutation patterns $q$, the number $\\textup{Av}_{n,\\ell}(q)$ of $q$-avoiding permutations of length $n$ that consist of $\\ell$ skew blocks is a monotone decreasing function of $\\ell$. We then show that this implies that for most patterns $q$, the generating function $\\sum_{n\\geq 0} \\textup{Av}_n(q)z^n$ of the sequence $\\textup{Av}_n(q)$ of the numbers of $q$-avoiding permutations is not rational. Placing our results in a broader context, we show that for rational power series $F(z)$ and $G(z)$ with nonnegative real coefficients, the relation $F(z"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.08506","kind":"arxiv","version":4},"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"}