20.25 (2022)

Let $n = k^2$. Let $\sigma \in S_n$ be the product of $k$ disjoint cycles, of lengths $1, 3, 5, \dots, 2k - 1$. Find the limiting proportion of odd entries in the $\sigma$-column of the character table of $S_n$.

For some background, see pages 1005–1006 in (D. Gluck, Proc. Amer. Math. Soc., 147 (2019), 1005–1011).