In this work we prove that a relationship exists between the elementar
y paths of the plane lattice Z(2) and those with growing altitude. For
tunately it is possible to make a more complete combinatorial study of
the Z(2) growing altitude elementary paths and in particular those wi
th barrier. From this combinatorial study we have established an impor
tant relationship permitting the calculation of the number of growing
altitude elementary paths with barrier and of given length.