I cannot understand the definition of polynomials $s_I$ in the Milnor and Stasheff, Characteristic classes, page $188.$
In Milnor and Stasheff, Characteristic classes page $188$, the polynomials $S_I$ are defined as follows.
Let $t_1,...,t_n$ be indeterminates.
Now for any partition $I=i_1,...,i_r$ of $k$, define a polynomial $S_I$ in $k$ variables as follows. Choose $n \geq k$ so that the elementary symmetric functions $\sigma_1,...,\sigma_k$ of $t_1,...t_n$ are algebraically independent and let $s_I(\sigma_1,....,\sigma_k)= \sum t_1^{i_1}...t_r^{i_r}$.
The summation $s_I(\sigma_1,....,\sigma_k)= \sum t_1^{i_1}...t_r^{i_r}$ is taken over all monomials transformed from $t_1^{i_1}...t_r^{i_r}$ by the permuation group acting on the set $\{t_1,....,t_n\}$.
$s() = 1$
$s_1(\sigma_1) = \sigma_1$
$s_{2}(\sigma_1,\sigma_2) = \sigma_1^2-2\sigma_2$
$s_{1,1}(\sigma_1,\sigma_2) = \sigma_2$
$s_{3}(\sigma_1,\sigma_2,\sigma_3) = \sigma_1^3 - 3 \sigma_1 \sigma_2 + 3\sigma_3$
$s_{1,2}(\sigma_1,\sigma_2,\sigma_3) = \sigma_1 \sigma_2 - 3\sigma_3$
$s_{1,1,1}(\sigma_1,\sigma_2,\sigma_3) = \sigma_3$
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Edit after the answer of Eric
I can understand the above equations. For example, consider the following equation.
$s_{3}(\sigma_1,\sigma_2,\sigma_3) = \sigma_1^3 - 3 \sigma_1 \sigma_2 + 3\sigma_3$
This equation means the following.
$$s_3(t_1,t_2,t_3) := (t_1)^3+(t_2)^3+(t_3)^3 \\ = (t_1+t_2+t_3)^3 -3(t_1+t_2+t_3)(t_1t_2+t_2t_3+t_3t_1) -3t_1t_2t_3 \\ = \sigma_1^3 - 3 \sigma_1 \sigma_2 - 3\sigma_3$$
The last sign is minus instead of plus ... I guess the book misprinted?