  We study the conformal boundary conditions of the dilute O(n) model in two
dimensions. A pair of mutually dual solutions to the boundary Yang-Baxter
equations are found. They describe anisotropic special transitions, and can be
interpreted in terms of symmetry breaking interactions in the O(n) model. We
identify the corresponding boundary condition changing operators, Virasoro
characters, and conformally invariant partition functions. We compute the
entropies of the conformal boundary states, and organize the flows between the
various boundary critical points in a consistent phase diagram. The operators
responsible for the various flows are identified. Finally, we discuss the
relation to open boundary conditions in the O(n) model, and present new
crossing probabilities for Ising domain walls.
