On parity and characters of symmetric groups

Abstract

We present a conjectural parity bias in the character values of the symmetric group. The main conjecture says that a character value chosen uniformly at random from the character table of Sn is congruent to 0mod2 with probability →1 as n→∞. A more general conjecture says that the same is true for all primes p, not only p=2. We relate these conjectures to zeros, give generating functions for computing lower bounds, and present some computational data in support of the main conjecture.