{"paper":{"title":"On plane permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian M. Reidys, Ricky X. F. Chen","submitted_at":"2014-11-20T13:51:55Z","abstract_excerpt":"In this paper we generalize permutations to plane permutations. We employ this framework to derive a combinatorial proof of a result of Zagier and Stanley, that enumerates the number of $n$-cycles $\\omega$, for which $\\omega(12\\cdots n)$ has exactly $k$ cycles. This quantity is $0$, if $n-k$ is odd and $\\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. The proof is facilitated by a natural transposition action on plane permutations which gives rise to various recurrences. Furthermore we study several distance problems of permutations. It tu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5552","kind":"arxiv","version":3},"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"}