Descent-restricted subsequences via RSK and evacuation

The length $\mathsf{is}(π)$ of a longest increasing subsequence in a permutation $π$ 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 $\mathsf{is}(π)$, we show how these statistics can be obtained using the RSK correspondence and the Schützenberger involution. In particular, these statistics only depend on the recording tableau of the permutation.

💡 Research Summary

The paper “Descent‑restricted subsequences via RSK and evacuation” studies natural generalizations of the classical longest increasing subsequence (LIS) statistic for permutations. While the LIS length is(π) counts the longest subsequence with no descents, the authors introduce two families of statistics that control the number and the exact positions of descents in a subsequence.

1. Fixed‑descent‑count statistic ls₍d₎(π).
For a permutation π and a non‑negative integer d, define
 ls₍d₎(π)=max{|I| : des(π_I)=d}.
Thus ls₍0₎(π)=is(π). Basic properties are proved: ls₍d₎(π)≥d+1 for d≤des(π), ls₍d₎(π)=0 for d>des(π), and the values are strictly increasing in d. The authors establish two-sided bounds
 is(π)+d ≤ ls₍d₎(π) ≤ min{ is_{d+1}(π), asc(π)+d+1 }.
The lower bound is attained exactly when the number of ascents equals is(π)−1; in this case the permutation has the minimal possible ascent count compatible with its LIS length, and ls₍d₎(π)=is(π)+d. When asc(π)=is(π) the upper bound is(π)+d+1 holds, but the converse fails in general (e.g. π=563412).

2. Fixed‑descent‑set statistic ls_D(π).
For a subset D⊆