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Motivated by a problem arising out of DNA origami, we give a general countingframework and enumeration formulas for various cellular embeddings of bouquetsand dipoles under different kinds of symmetries. Our algebraic framework can beused constructively to generate desired symmetry classes, and we use Burnside'sLemma with various symmetry groups to derive the enumeration formulas. Ourresults assimilate several existing formulas into this unified framework.Furthermore, we provide new formulas for bouquets with colored edges (and thusfor bouquets in nonorientable surfaces) as well as for directed embeddings ofdirected bouquets. We also enumerate vertex-labeled dipole embeddings. Sincedipole embeddings may be represented by permutations, the formulas also applyto certain equivalence classes of permutations and permutation matrices. Theresulting bouquet and dipole symmetry formulas enumerate structures relevant toa wide variety of areas in addition to DNA origami, including RNA secondarystructures, Feynman diagrams, and topological graph theory. For uncoloredobjects we catalog 58 distinct sequences, of which 43 have not, as far as weknow, been described previously.

A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings Under Symmetries