This problem can be solved using the method of inserting spaces. Since no two dance programs can be adjacent, we can consider the space before and after each of the 6 singing programs as a potential position for inserting a dance program, resulting in 7 spaces. We need to arrange the dance programs into these spaces to ensure they are not adjacent. There are $P_{7}^{4}$ ways to do this. After arranging the dance programs, there are $A_{6}^{6}$ ways to arrange the singing programs.

Therefore, the total number of arrangements is $P_{7}^{4} \times A_{6}^{6}$.

The answer is $\boxed{P_{7}^{4} \times A_{6}^{6}}$.