TITLE: Closed formulas for the character of the symmetric group
QUESTION [10 upvotes]: 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

REPLY [7 votes]: For generalizing the formulas in your question, see http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19 and Examples 1.7.13 and 1.7.14 in Macdonald's Symmetric Functions and Hall Polynomials, 2nd ed.
For a different formula, see https://www.researchgate.net/publication/227299451_Stanley%27s_Formula_for_Characters_of_the_Symmetric_Group.