Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three

Ross G. Pinsky

Let $S_n$ denote the set of permutations of $[n]$ and let $\ddot{I}ƒ=\ddot{I}{ƒ}_1\cdots \ddot{I}{ƒ}_n\in S_n$. For a subsequence $\{\ddot{I}{ƒ}_{i_j}{\}}_{j=1}^k$ of $\{\ddot{I}{ƒ}_i{\}}_{i=1}^n$ of length $k\ge 2$, construct th…