The cycle notation of a permutation, say f = (137)(2)(4658)

allows it, to determine immediately the order of f:

The order of f is the l.c.m. of the cycle lengths of f

(here the product 3 x 1 x 4 = 12).

The Landau function G(n). G(n) is the maximal order of a permutation of an n element set, thus the maximum of the products n1 x ... x nt, where n1,...,nt are coprime numbers such that n1 + ... + nt = n.

Edmund Landau (1877-1938)

Example: If n = 10, then n! = 3628800,
whereas G(n) = 30 (= 2 · 3 · 5), thus a quite small number.