Descent-restricted subsequences via RSK and evacuation

TL;DR

This paper generalizes the classical longest increasing subsequence problem by considering subsequences with specific descent restrictions, using RSK and Schützenberger involution to relate these statistics to the recording tableau.

Contribution

It introduces a framework for analyzing longest subsequences with descent constraints via RSK and Schützenberger involution, extending classical LIS results.

Findings

Statistics depend only on the recording tableau.

Extension of classical LIS results to descent-restricted subsequences.

Method applicable to various descent set restrictions.

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.