  The symmetry constraints imposing on the quantum states of a dot with 13
electrons has been investigated. Based on this study, the favorable structures
(FSs) of each state has been identified. Numerical calculations have been
performed to inspect the role played by the FSs. It was found that, if a
first-state has a remarkably competitive FS, this FS would be pursued and the
state would be crystal-like and have a specific core-ring structure associated
with the FS. The magic numbers are found to be closely related to the FSs.
