On the Longest Common Pattern Contained in Two or More Random   Permutations
  We provide upper and lower bounds for the expected length $\mathbb E(L_{n,m})$ of the longest common pattern contained in $m$ random permutations of length $n$. We also address the tightness of the concentration of $L_{n,m}$ around $\mathbb E(L_{n,m})$. 
Michael Earnest, Anant Godbole and Yevgeniy Rudoy
math.CO