The following may be known, but I didn't find anything in the literature.

**Background:**

The irreducible representations of $S_n$ correspond to shapes of Young tableaux with $n$ elements. Let $\lambda$ be a shape of a Young tablaux. The corresponding irreducible representation is generated by the set of all $e_T$, where $T$ is a Young tableaux of shape $\lambda$, and $e_T$ is the polytabloid corresponding to $T$. A basis for the irreducible representation is given by all standard Young tableaux $T_1,\ldots,T_m$ of shape $\lambda$. They in fact span it as a lattice. That it, for any tableaux $T$ of shape $\lambda$, $$ e_T = c_1 e_{T_1} + \ldots + c_m e_{T_m} $$ where $c_1,\ldots,c_m \in \mathbb{Z}$.

**Question:**

How large are the coefficients $c_1,\ldots,c_m$ (in absolute value)?

A simple analysis of the straightening algorithm (see for example en.wikipedia.org/wiki/Garnir_relations, or the book of Sagan) gives a bound of $n^{O(n^2)}$, but I would be surprised if this is tight. Can it be that the coefficients are bounded by $n^{O(n)}$ or even $\exp(n)$?