{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/explicit-formulas-for-permutation-pattern","title":"Explicit formulas for permutation pattern character polynomials","arxiv_id":"2310.18798","date":"2023-10-28","proceeding":null,"authors":["Jonas Iskander"],"abstract":"Given permutations $\\pi \\in S_n$ and $\\sigma \\in S_k$, let $N_\\sigma(\\pi)$ denote the number of occurrences of $\\sigma$ in $\\pi$. While pattern avoidance and the distribution of pattern occurrences in permutations have been extensively studied, their interactions with the group structure on $S_n$ are still poorly understood. Gaetz and Ryba showed that the expected value of $\\chi^{\\lambda[n]}(\\pi)N_\\sigma(\\pi)$ for $\\pi \\in S_n$ is given by a polynomial $a_\\sigma^\\lambda(n)$. More recently, Gaetz and Pierson derived explicit formulas for $a_{\\mathrm{id}_k}^\\lambda(n)$ when $\\lvert\\lambda\\rvert \\le 2$, which led them to conjecture that the polynomials $a_{\\mathrm{id}_k}^\\lambda(n)$ are real-rooted and nonnegative for $n \\ge k$. We show that for all partitions $\\lambda$, the polynomials $a_{\\mathrm{id}_k}^\\lambda(n)$ admit explicit closed forms in $n$ and $k$. These formulas allow us to exhibit counterexamples to Gaetz and Pierson's real-rootedness conjecture as well as to prove special cases of their nonnegativity conjecture. Our results imply that the expected value of $f \\cdot N_{\\mathrm{id}_k}$ on $S_n$ admits a closed form whenever $f$ is a permutation statistic expressible as a polynomial in the functions $m_j \\colon \\bigsqcup_{n \\ge 0} S_n \\to \\mathbb{Z}$ which count $j$-cycles in their inputs.","url_abs":"https://arxiv.org/abs/2310.18798v1","url_pdf":"https://arxiv.org/pdf/2310.18798v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"explicit-formulas-for-permutation-pattern","repo_url":"https://github.com/jonasiskander/character-polynomials","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}