Descent-restricted subsequences via RSK and evacuation

Abstract

The length $\mathsf{is}(\pi)$ of a longest increasing subsequence in a permutation $\pi$ has been extensively studied. An increasing subsequence is one that has no descents. We study generalizations of this statistic by finding longest subsequences with other descent restrictions. We first consider the statistic which encodes the longest length of a subsequence with a given number of descents. We then generalize this to restrict the descent set of the subsequence. Extending the classical result for $\mathsf{is}(\pi)$, we show how these statistics can be obtained using the RSK correspondence and the Sch\"utzenberger involution. In particular, these statistics only depend on the recording tableau of the permutation.