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I thank the referee for helpful comments and for suggesting that I include the Murnaghan–Nakayama rule.
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Miller, A.R. The probability that a character value is zero for the symmetric group. Math. Z. 277, 1011–1015 (2014). https://doi.org/10.1007/s00209-014-1290-x
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DOI: https://doi.org/10.1007/s00209-014-1290-x