id: 05192675
dt: j
an: 2007b.00419
au: Thorup, Anders
ti: The Josefus permutation. (Om Josefus’ permutation.)
so: Normat. Nord. Mat. Tidskr. 55, No. 1, 25-36 (2007).
py: 2007
pu: Nationellt Centrum för Matematikutbildning, Göteborg
la: DA
cc: K20 N70
ut: combinatorics; fixed points; cycle types
ci:
li:
ab: Summary: It is well-known that any permutation can be presented as a
composition of cycles. In fact every permutation has a unique
representation, up to order, as a product of disjoint (and hence
commuting) cycles. It turns out that the composition
$c_n,c_{n-1},\dots,c_2$ where $c_k = (1, 2, 3, \dots, k)$ has a nice
and easily found such representation, but surprisingly if the order of
the cycles is reversed the problem becomes almost intractable. The
elementary nature of the problem gives a good exposure to students, and
it has in fact been studied by students over the years. The article is
in the nature of a progress report, and the cases of $n = 2^m, 2m-1$
are treated in detail. Surprising connections to other parts of
combinatorics are highlighted.
rv: