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Descent-restricted subsequences via RSK and evacuation

Krishna Menon, Anurag Singh

TL;DR

The paper addresses longest subsequences of a permutation under descent-set restrictions, generalizing the classical LIS problem. It develops a unified, tableau-centric approach via the Robinson--Schensted--Knuth correspondence and the Sch\\"utzenberger evacuation to express descent-restricted statistics \\mathsf{ls}_D(\\pi) in terms of the recording tableau \\ Q(\\pi)$ and its evacuation, enabling explicit extraction from growth diagrams. Key contributions include bounds and exact formulas for the case of a single descent, plus an inductive, diagrammatic framework for general descent sets \\mathsf{ls}_D(\\pi)$, with algorithms to compute them from \\ Q(\\pi)$ and the evacuation diagram; these results establish that the descent-restricted statistics depend only on the recording tableau and provide a deeper tableau-theoretic understanding of subsequences with prescribed descent behavior. The work also discusses equivalence between permutations under these statistics, counting identities via SYT data, and several promising directions for distributions, pattern avoidance, and pattern-sets, highlighting the practical and theoretical impact of a RSK-based approach to descent-restricted subsequences.

Abstract

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.

Descent-restricted subsequences via RSK and evacuation

TL;DR

The paper addresses longest subsequences of a permutation under descent-set restrictions, generalizing the classical LIS problem. It develops a unified, tableau-centric approach via the Robinson--Schensted--Knuth correspondence and the Sch\\"utzenberger evacuation to express descent-restricted statistics \\mathsf{ls}_D(\\pi) in terms of the recording tableau \\ Q(\\pi), with algorithms to compute them from \\ Q(\\pi)$ and the evacuation diagram; these results establish that the descent-restricted statistics depend only on the recording tableau and provide a deeper tableau-theoretic understanding of subsequences with prescribed descent behavior. The work also discusses equivalence between permutations under these statistics, counting identities via SYT data, and several promising directions for distributions, pattern avoidance, and pattern-sets, highlighting the practical and theoretical impact of a RSK-based approach to descent-restricted subsequences.

Abstract

The length of a longest increasing subsequence in a permutation 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 , 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.
Paper Structure (9 sections, 14 theorems, 17 equations, 4 figures, 1 algorithm)

This paper contains 9 sections, 14 theorems, 17 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The RSK correspondence
  4. The Schützenberger involution
  5. Restricting the number of descents
  6. Restricting the descent set
  7. Exactly one descent
  8. The general case
  9. Future directions

Key Result

Lemma 3.3

For any permutation $\pi$, we have $\mathsf{ls}_d(\pi) \geq d + 1$ for all $d \leq \operatorname{des}(\pi)$ and $\mathsf{ls}_d(\pi) = 0$ for all $d > \operatorname{des}(\pi)$. Moreover, for any $d_1 < d_2 \leq \operatorname{des}(\pi)$, we have $\mathsf{ls}_{d_1}(\pi) < \mathsf{ls}_{d_2}(\pi)$.

Figures (4)

  • Figure 1: Evacuation growth diagram for $Q$ defined in \ref{['evacex']}.
  • Figure 2: If the diagram is the evacuation growth diagram of $\pi$, then the three red triangles are the evacuation growth diagrams of the factors $\pi_{[1, 2]}$, $\pi_{[3, 6]}$, and $\pi(7)$.
  • Figure 3: Extracting $\mathsf{ls}_{\{2, 3\}}(\pi) = 5$ from the $a_{i, j}(\pi)$ triangle.
  • Figure 4: Showing $\mathsf{ls}_{\{1, 2, 4, 5\}}(\pi) = 0$ using the $a_{i, j}(\pi)$ triangle.

Theorems & Definitions (40)

  • Remark 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 30 more