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We study the enumeration of closed walks of given length and algebraic areaon the honeycomb lattice. Using an irreducible operator realization ofhoneycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonianand show that the generating function of closed walks maps to the grandpartition function of a system of particles with exclusion statistics of order$g=2$ and an appropriate spectrum, along the lines of a connection previouslyestablished by two of the authors. Reinterpreting the results in terms of thestandard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) and $g=2$exclusion whose physical meaning and properties require further elucidation. Inthis context we also obtain some unexpected Fibonacci sequences within theweights of the combinatorial factors appearing in the counting of walks.

Algebraic area enumeration of random walks on the honeycomb lattice