Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}=\delta_\text{odd}(n).$$
I like to ask:
QUESTION 1. Is there a formula for the enumeration (cardinality) of $a_n=\#\{\lambda\vdash n: \chi_{(n)}^{\lambda}=0\}$? How about $b_n=\#\{\lambda\vdash n: \chi_{(n)}^{\lambda}=1\}$ or $c_n=\#\{\lambda\vdash n: \chi_{(n)}^{\lambda}=-1\}$?
Encouraged by Mark Wildom's prompt/neat reply, I like to update the question to counting all $0$'s.
QUESTION 2. How about the cardinality (or generating function) of the below?
$$d_n=\#\{(\lambda,\mu): \lambda\vdash n, \mu\vdash n, \chi_{\mu}^{\lambda}=0\}.$$
Best Answer
It is an open problem to determine how many of the entries of the character table are zero — or, indeed whether the proportion of zeros tends to a positive constant, or to zero. One should be careful about the exact problem considered: if a character is picked at random, and a group element is picked at random (so $p(n)$ choices for the character and $n!$ choices for the group element) then almost all entries are zero. This is the partial result mentioned in Stanley's comment to the question. However this is a very different statistic from picking the character at random and a conjugacy class at random (the problem in the question). Miller (On parity and characters of symmetric groups) has done some numerics on this (see Table 3), and conjectured that almost all values were multiples of any given number $d$. This last conjecture has been established for all prime numbers $d$, in the work of Peluse (On even entries in the character table of the symmetric group) and Peluse and Soundararajan (Almost all entries in the character table of the symmetric group are multiples of any given prime) (consequently, $\pm 1$ — or indeed any fixed non-zero integer — appears with density zero in the character table). As for zero entries, the best lower bound that I know (which follows from these ideas of using Murnaghan—Nakayama and the structure of random partitions) is that at least a proportion $c/\log n$ of the entries are zero.