Recent technology mappers for LUT based FPGAs employ cut enumeration. Although many cuts are often needed to find good network, enumerating all cuts with large size consumes run-time very much. The number of cuts exponentially increases with the size of cuts, which causes long runtime. Furthermore, an inefficiency of bottom-up merging in existing algorithms makes the run-time much longer. This paper presents a novel cut enumeration. The proposed algorithm is efficient because it enumerates cuts without bottomup merging. Our algorithm has two modes; exhaustive enumeration and partial enumeration. Exhaustive enumeration enumerates all cuts. Partial enumeration enumerates partial cuts with a guarantee that a depth-minimum network can be constructed. The experimental results show that exhaustive enumeration runs about 3 times and 8 times faster than existing bottom-up algorithm   for K = 8, 9, respectively. The quality of network are the same. Furthermore, partial enumeration runs about 6 times and 18 times faster than bottom-up algorithm for K = 8, 9, respectively. Area of network derived by the set of cuts enumerated by partial enumeration is only 4 % larger than that derived by exhaustive enumeration on average, and the depth is the same.