Ordered Bell number
13 plane trees with ordered leaves and equal-length root-leaf paths, with the gaps between adjacent leaves labeled by the height above the leaves of the nearest common ancestor. These labels induce a weak ordering on the gaps, showing that the trees of this type are counted by the ordered Bell numbers.
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