Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 4 / b

The coefficient $⟨e(t),\omega ⟩$ is nonzero exactly when every column of $t$ meets every row of $\omega$ in at most one entry. Permuting equal-length rows of $\omega$ preserves this condition, and the same holds with $u$ in place of $t$. Therefore membership in $A_{tu}$ is constant on each class of the stated equivalence relation, so $A_{tu}$ is a union of equivalence classes.

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 $⟨e(t),\omega ⟩$ and $⟨e(u),\omega ⟩$ by the same sign $(-1{)}^j$, so their product is constant on the class. Sincethe contribution of every nonzero class is a multiple of ${\prod }_j{a}_j!$. The whole inner product is therefore such a multiple.