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Motivated by understanding new columns of decomposition matrices, we prove that if $\\lambda$ is an $e$-regular partition such that $d$ divides the arm length of any rim hook of $\\lambda$ of size divisible by $e$, then $m_e(\\lambda)'$ is a partition such that the arm length of any of its rim hooks of size divisible by $e$ is congruent to $-1$ modulo $d$. 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