  We are interested in the asymptotics of the number of standard Young tableaux
$f^{\lambda/\mu}$ of a given skew shape $\lambda/\mu$. We mainly restrict
ourselves to the case where both diagrams are balanced, but investigate all
growth regimes of $|\mu|$ compared to $|\lambda|$, from $|\mu|$ fixed to
$|\mu|$ of order $|\lambda|$. When $|\mu|=o(|\lambda|^{1/3})$, we get an
asymptotic expansion to any order. When $|\mu|=o(|\lambda|^{1/2})$, we get a
sharp upper bound. For bigger $|\mu|$, we prove a weaker bound and give a
conjecture on what we believe to be the correct order of magnitude.
  Our results are obtained by expressing $f^{\lambda/\mu}$ in terms of
irreducible character values of the symmetric group and applying known upper
bounds on characters.
