{"paper":{"title":"On some properties of the number of permutations being products of pairwise disjoint $d$-cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Maciej Ulas, Piotr Miska","submitted_at":"2019-04-06T09:23:21Z","abstract_excerpt":"Let $d\\geq 2$ be an integer. In this paper we study arithmetic properties of the sequence $(H_d(n))_{n\\in\\N}$, where $H_{d}(n)$ is the number of permutations in $S_{n}$ being products of pairwise disjoint cycles of a fixed length $d$. In particular we deal with periodicity modulo a given positive integer, behaviour of the $p$-adic valuations and various divisibility properties. Moreover, we introduce some related families of polynomials and study they properties. Among many results we obtain qualitative description of the $p$-adic valuation of the number $H_{d}(n)$ extending in this way earlie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03395","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":""},"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"}