In the study of protein folding, our understanding of the protein structures is limited. In this paper we find one way to characterize the compact structures of lattice protein model. A quantity called "Partnum" is given to each compact structure. The Partnum is compared with the "Designability" of protein structures. It is shown that the highly designable structures have, on average, an atypical number of local degree of freedom. The statistical property of Partnum and its dependence on sequence length is also studied. (C) 2000 American Institute of Physics. [S0021-9606(00)50313-4].