  We study the structure of the asymptotic expansion of the probability that a
combinatorial object is connected. We show that the coefficients appearing in
those asymptotics are integers and can be interpreted as the counting sequences
of other derivative combinatorial classes. The general result applies to
rapidly growing combinatorial structures, which we call gargantuan, that also
admit a sequence decomposition. The result is then applied to several models of
graphs, of surfaces (square-tiled surfaces, combinatorial maps), and to
geometric models of higher dimension (constellations, graph encoded manifolds).
The corresponding derivative combinatorial classes are irreducible
(multi)tournaments, indecomposable (multi)permutations and indecomposable
perfect (multi)matchings.
