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

Abstract

Let $S_n$ denote the set of permutations of $[n]$ and let $\sigma=\sigma_1\cdots\sigma_n\in S_n$. For a subsequence $\{\sigma_{i_j}\}_{j=1}^k$ of $\{\sigma_i\}_{i=1}^n$ of length $k\ge2$, construct the ``up/down'' sequence $V_1\cdots V_{k-1}$ defined by $$ V_j=\begin{cases} U,\ \text{if}\ \sigma_{i_j+1}-\sigma_{i_j}>0;\\ D,\ \text{if}\ \sigma_{i_j+1}-\sigma_{i_j}<0.\end{cases} $$ Consider now a fixed up/down pattern: $V_1\cdots V_l$, where $l\in\mathbb{N}$ and $V_j\in\{U, D\},\ j\in[l]$. Given a permutation $\sigma\in S_n$, consider the length of the longest subsequence of $\sigma$ that repeats this pattern. For example, consider $l=3$ and $V_1V_2V_3=UUD$. Then for the permutation $342617985\in S_9$, the length of the longest subsequence that repeats the pattern $UUD$ is 7; it is obtained by 3461798 and 3461785. The above framework includes two well-known cases. The pattern $U$ is the celebrated case of the longest increasing subsequence. The pattern $UD$ (or $DU$) is the case of the longest alternating subsequence. These have been studied both under the uniform distribution on $S_n$ as well as under the uniform distribution on those permutations in $S_n$ which avoid a particular pattern of length three. In this paper, we consider the patterns $UUD$ and $UUUD$ under the uniform distribution on those permutations in $S_n$ which avoid the pattern $132$. We prove that the expected value of the longest increasing subsequence following the pattern $UUD$ is asymptotic to $\frac37n$ and the expected value of the longest increasing subsequence following the pattern $UUUD$ is asymptotic to $\frac4{11}n$. (For $UD$ (alternating subsequences) it is known to be $\frac12n$.) This leads directly to appropriate corresponding results for permutations avoiding any particular pattern of length three.