The exponential generating function of Bernoulli number is $f(x)=x/(\exp(x)-1)$.
$x$ can be treated as species of 1-element set. Also $\exp(x)-1$ can be treated as nonempty set.
So, $f(x)*(\exp(x)-1)=x$ should have some combinatorial interpretation of exponential generating function.
Does anyone have any idea ?