  We evaluate combinatorially certain connection coefficients of the symmetric
group that count the number of factorizations of a long cycle as a product of
three permutations. Such factorizations admit an important topological
interpretation in terms of unicellular constellations on orientable surfaces.
Algebraic computation of these coefficients was first done by Jackson using
irreducible characters of the symmetric group. However, bijective computations
of these coefficients are so far limited to very special cases. Thanks to a new
bijection that refines the work of Schaeffer and Vassilieva, and Vassilieva, we
give an explicit closed form evaluation of the generating series for these
coefficients. The main ingredient in the bijection is a modified oriented
tricolored tree tractable to enumerate. Finally, reducing this bijection to
factorizations of a long cycle into two permutations, we get the analogue
formula for the corresponding generating series.
