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 for the general case, but maybe there is in simple cases, like for the other hook partitions or for rectangle partition?

Thanks in advance!

Étienne