Title: Straightening rule for an $m'$-truncated polynomial ring,
Abstract: 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 $p$.