  Motivated by a problem arising out of DNA origami, we give a general counting
framework and enumeration formulas for various cellular embeddings of bouquets
and dipoles under different kinds of symmetries. Our algebraic framework can be
used constructively to generate desired symmetry classes, and we use Burnside's
Lemma with various symmetry groups to derive the enumeration formulas. Our
results assimilate several existing formulas into this unified framework.
Furthermore, we provide new formulas for bouquets with colored edges (and thus
for bouquets in nonorientable surfaces) as well as for directed embeddings of
directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since
dipole embeddings may be represented by permutations, the formulas also apply
to certain equivalence classes of permutations and permutation matrices. The
resulting bouquet and dipole symmetry formulas enumerate structures relevant to
a wide variety of areas in addition to DNA origami, including RNA secondary
structures, Feynman diagrams, and topological graph theory. For uncolored
objects we catalog 58 distinct sequences, of which 43 have not, as far as we
know, been described previously.
