Find the character $\chi^{(n-3, 1^3)}_{\dots 3^{\alpha_3} 2^{\alpha_2} 1^{\alpha_1}}$ of irreducible representation of symmetric group $S_n$ for $\alpha_i > 3$.

I know, that I somehow should use Murnaghan–Nakayama rule or Frobenius formula for this problem. May somebody help me? I don't know what to start with. Could somebody share examples or some appropriate literature. Thanks in advance.

I have just found that this character may be found from exercise 4.15 from Fullton, Harris; they ask to prove the following $$ \chi^{(d-2, 1^2)}_{\dots, 3^{i_3}, 2^{i_2}, 1^{i_1}} = \frac{1}{2}(i_1 - 1)(i_1 - 2) - i_2. $$ For example, if I prove exercise, may then the previous one be $$ \chi^{(n-3, 1^3)}_{\dots 3^{\alpha_3} 2^{\alpha_2} 1^{\alpha_1}} = \frac{1}{6}(\alpha_1 - 1)(\alpha_1 - 2)(\alpha_1 - 3) - \frac 1 2 (\alpha_2 - 1)(\alpha_2 - 2) + (\alpha_3 - 1)? $$


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