  Many Riordan arrays play a significant role in algebraic combinatorics. We
explore the inversion of Riordan arrays in this context. We give a general
construct for the inversion of a Riordan array, and study this in the case of
various subgroups of the Riordan group. For instance, we show that the
inversion of an ordinary Bell matrix is an exponential Riordan array in the
associated subgroup. Examples from combinatorics and algebraic combinatorics
illustrate the usefulness of such inversions. We end with a brief look at the
inversion of exponential Riordan arrays. A final example places Airey's
convergent factor in the context of a simple exponential Riordan array.
