We consider various classes of monoids of transformations on a finite chain, in particular transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested
in computing both their cardinals and the number of their idempotents.

In this note we present a short survey on these questions which have been approached by various authors and close the problem by computing the number of idempotents of those monoids not considered before. Fibonacci and Lucas numbers play an essential role in the last computations.

CEMAT - Center for Computational and Stochastic Mathematics