Straightening rule for an $m'$-truncated polynomial ring

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime $p$ the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition $\lambda$, we discovered the number of compositions $\delta$ such that $\delta$ can be rearranged to $\lambda$ and whose partial sums of $\delta$ are not divisible by $p$.