  We introduce a family of sequence transformations, defined via partial Bell
polynomials, that may be used for a systematic study of a wide variety of
problems in enumerative combinatorics. This family includes some of the
transformations listed in the paper by Bernstein & Sloane, now seen as
transformations under the umbrella of partial Bell polynomials. Our goal is to
describe these transformations from the algebraic and combinatorial points of
view. We provide functional equations satisfied by the generating functions,
derive inverse relations, and give a convolution formula. While the full range
of applications remains unexplored, in this paper we show a glimpse of the
versatility of Bell transformations by discussing the enumeration of several
combinatorial configurations, including rational Dyck paths, rooted planar
maps, and certain classes of permutations.
