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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

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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.

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Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by Michel Lassalle : https://link.springer.com/article/10.1007/s00208-007-0156-5.

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