Abstract :
[en] The starting point of this work is an equality between two
quantities $A$ and $B$ found in the literature, which involve
the {\em doubling-modulo-an-odd-integer} map, i.e.,
$x\in \N \mapsto 2x \bmod{(2n+1)}$ for some positive
integer $n$. More precisely, this doubling map defines a
permutation $\sigma_{2,n}$ and each of $A$ and $B$ counts
the number $C_2(n)$ of cycles of $\sigma_{2,n}$, hence $A=B$.
In the first part of this note, we give a direct proof of
this last equality. To do so, we consider and study a
generalized $(k,n)$-perfect shuffle permutation
$\sigma_{k,n}$, where we multiply by an integer $k\ge 2$
instead of $2$, and its number $C_k(n)$ of cycles.
The second part of this note lists some of the many
occurrences and applications of the doubling map and
its generalizations in the literature: in mathematics
(combinatorics of words, dynamical systems, number theory,
correcting algorithms), but also in card-shuffling,
juggling, bell-ringing, poetry, and music composition.
