  An important family of structural constants in the theory of symmetric
functions and in the representation theory of symmetric groups and general
linear groups are the plethysm coefficients. In 1950, Foulkes observed that
they have some stability properties: certain sequences of plethysm coefficients
are eventually constant. Such stability properties were proven by Brion with
geometric techniques, and by Thibon and Carr\'e by means of vertex operators.
In this paper we present a new approach to prove such stability properties.Our
proofs are purely combinatorial and follow the same scheme. We decompose
plethysm coefficients in terms of other plethysm coefficients related to the
complete homogeneous basis of symmetric functions. We show that these other
plethysm coefficients count integer points in polytopes and we prove stability
for them by exhibiting bijections between the corresponding sets of integer
points of each polytope.
