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Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...

In the title, $LIS$ stands for the length of longest increasing subsequence and Greek letters stand for permutations from symmetric group $S_n$. Considering some cases such as $\pi^{-1}=\sigma=...