= Solution
Each equivalence class has size $\prod_{j=1}^n a_j!$, because its $a_j$ rows of length $j$ may be permuted freely. Swapping two length-$j$ rows multiplies each of $\langle e(t),\omega\rangle$ and $\langle e(u),\omega\rangle$ by the same sign $(-1)^j$, so their product is constant on the class. Since
$$
\langle e(t),e(u)\rangle
=\sum_{\omega\in\Omega^\lambda}
\langle e(t),\omega\rangle\langle e(u),\omega\rangle,
$$
the contribution of every nonzero class is a multiple of $\prod_j a_j!$. The whole inner product is therefore such a multiple.
Back to article page
