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:



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> **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 $c\_n=\#\{\lambda\vdash n: \chi\_{(n)}^{\lambda}=-1\}$?
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Encouraged by Mark Wildom's prompt/neat [reply](https://mathoverflow.net/questions/403791/counting-pm-1-and-0s-in-the-character-tables#comment1034199_403791), I like to update the question to counting all $0$'s.



> 
> **QUESTION 2.** How about the cardinality (or generating function) of the below?
> $$d\_n=\#\{(\lambda,\mu): \lambda\vdash n, \mu\vdash n, \chi\_{\mu}^{\lambda}=0\}.$$
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Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$
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](https://mathoverflow.net/questions/403791/counting-pm-1-and-0s-in-the-character-tables#comment1034207_403791) 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](https://www.armiller.org/Parity.pdf)) 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](https://arxiv.org/abs/2007.06652)) and Peluse and Soundararajan ([Almost all entries in the character table of the symmetric group are multiples of any given prime](https://arxiv.org/abs/2010.12410)) (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.


