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I'm not sure if you will consider this nontrivial enough, but from the character table you can very quickly show that the number of conjugacy classes of even permutations is always greater than or equal to the number of conjugacy classes of odd permutations.

One just applies the general fact that the sum of the entries in any row of a character table (not weighted by the size of the conjugacy class) is a nonnegative integer (because the character $$\chi ( \pi ) = \frac{\\# S_n}{\\# C_\pi},$$ where $C_\pi$ is the set of conjugates of the permutation $\pi$, corresponds to an actual representation, namely, take a vector space with basis $\{ e_\pi \mid \pi \in S_n\}$ and define $g(e_\pi) = e_{g \pi g^{-1}}$ for $g \in S_n$) to the sign character.

I'm not sure if you will consider this nontrivial, but from the character table you can very quickly show that the number of conjugacy classes of even permutations is always greater than or equal to the number of conjugacy classes of odd permutations.

One just applies the general fact that the sum of the entries in any row of a character table (not weighted by the size of the conjugacy class) is a nonnegative integer (because the character $\chi ( \pi ) = \frac{\# S_n}{\# C_\pi}$, where $C_\pi$ is the set of conjugates of the permutation $\pi$, corresponds to an actual representation, namely, take a vector space with basis $\{ e_\pi \mid \pi \in S_n\}$ and define $g(e_\pi) = e_{g \pi g^{-1}}$ for $g \in S_n$) to the sign character.

I'm not sure if you will consider this nontrivial enough, but from the character table you can very quickly show that the number of conjugacy classes of even permutations is always greater than or equal to the number of conjugacy classes of odd permutations.

One just applies the general fact that the sum of the entries in any row of a character table (not weighted by the size of the conjugacy class) is a nonnegative integer (because the character $$\chi ( \pi ) = \frac{\# S_n}{\# C_\pi},$$ where $C_\pi$ is the set of conjugates of the permutation $\pi$, corresponds to an actual representation, namely, take a vector space with basis $\{ e_\pi \mid \pi \in S_n\}$ and define $g(e_\pi) = e_{g \pi g^{-1}}$ for $g \in S_n$) to the sign character.

I'm not sure if you will consider this nontrivial enough, but from the character table you can very quickly show that the number of conjugacy classes of even permutations is always greater than or equal to the number of conjugacy classes of odd permutations.

One just applies the general fact that the sum of the entries in any row of a character table (not weighted by the size of the conjugacy class) is a nonnegative integer (because the character $$\chi ( \pi ) = \frac{\\# S_n}{\\# C_\pi},$$ where $C_\pi$ is the set of conjugates of the permutation $\pi$, corresponds to an actual representation, namely, take a vector space with basis $\{ e_\pi \mid \pi \in S_n\}$ and define $g(e_\pi) = e_{g \pi g^{-1}}$ for $g \in S_n$) to the sign character.