There is a tight lower bound for the triple product: $$LCS(\pi_1,\pi_2)LCS(\pi_2,\pi_3)LCS(\pi_3,\pi_1)\ge n$$

Thanks. I proved the CLT by Berry-Esseen Theorem. It is Theorem 3 on Page 111 by Petrov in the book "the sum of independent random variables". It is a generalized version of BE Theorem for independent random variables but not necessarily identical ones. I can have a error bound for centralized c.d.f. as of order $\frac{1}{\sqrt{n}}$ if assume $n-m=cn$, where $c$ does not depend on $n$.

Thanks for the link. That paper is very much related to my problem and it introduces me something about algebraic statistics on the topic of data ranking in which many functions of permutations have been applied to the table I mentioned. @ChristianStump

For now, I am using increasing order if you consider the dictionary as $\[n\]$. @DimaPasechnik

@GeoffRobinson Yes, if you prefer using permutation matrices to represent permutations.