I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:

$$\chi_{n}(\sigma) = 1$$ $$\chi_{11...1}(\sigma) = sgn(\sigma)$$ $$\chi_{n-1,1}(\sigma) = fix(\sigma)-1$$ $$\chi_{21...1}(\sigma) = sgn(\sigma)(fix(\sigma) - 1)$$

Are they any other simple formulas like these? I know that the answer is no, but maybe there is in simple cases, like for the others hook partitions or for rectangle partition?

Thanks in advance!

Étienne