  In this note we study numerically the combinatorics of curves and geodesics
on the torus with one boundary component. A potential computational difficulty
is avoided by counting inside specific orbits of the mapping class group up to
a certain length, either geometric or combinatorial. Some cases are rigurolosly
determined and the Euler totient function emerges. More complicated orbits are
computed to suggest an array of formulae continuing to involve the Euler
totient function. We formulate six precise conjectures. These include a novel
study of an "inverse" function of the Mirzakhani's asymptotics. The geometric
part of our study was motivated by the rationality aspect of these asymptotics.
