Ordered Bell number
where these three formulations correspond to the three weak orderings on two elements. In general, in a multivariate integral, the ordering in which the variables may be grouped into a sequence of nested integrals forms a weak ordering.
Because log 2 is less than one, the form of this approximation shows that the ordered Bell numbers exceed the corresponding factorials by an exponential factor. The asymptotic convergence of this approximation may be expressed as