English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $d\geq 2$ be an integer. In this paper we study arithmetic properties ofthe sequence $(H_d(n))_{n\in\N}$, where $H_{d}(n)$ is the number ofpermutations in $S_{n}$ being products of pairwise disjoint cycles of a fixedlength $d$. In particular we deal with periodicity modulo a given positiveinteger, behaviour of the $p$-adic valuations and various divisibilityproperties. Moreover, we introduce some related families of polynomials andstudy they properties. Among many results we obtain qualitative description ofthe $p$-adic valuation of the number $H_{d}(n)$ extending in this way earlierresults of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida.

On some properties of the number of permutations being products of pairwise disjoint $d$-cycles