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Decreasing subsequences and Viennot for oscillating tableaux

Elijah Bodish, Ben Elias, David E. V. Rose, Logan Tatham

TL;DR

The paper extends Viennot's shadow-line framework to oscillating tableaux and uses this to derive a Type $C$ analogue of Schensted's theorem for longest decreasing subsequences. It then connects these combinatorics to Sundaram–Stanley's type $C$ bijection and to representation theory of $\mathfrak{sp}_{2n}$, showing that the dimension of the invariant subspace in $V^{\otimes 2k}$ matches the count of $(n+1)$-avoiding matchings of $2k$ points. The core method is a diagrammatic, timeline-based extension of Viennot diagrams that encodes the bumping process and the evolution of oscillating tableaux. This yields a direct combinatorial proof and a unified perspective linking oscillating tableaux, pattern-avoidance in matchings, and Brauer–Schur–Weyl duality in type $C$. The results thereby bridge classical symmetric-function combinatorics with Lie-theoretic invariants in a concrete, diagrammatic framework.

Abstract

We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type $C$ analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type $C$ webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding matchings of $2k$ points.

Decreasing subsequences and Viennot for oscillating tableaux

TL;DR

The paper extends Viennot's shadow-line framework to oscillating tableaux and uses this to derive a Type analogue of Schensted's theorem for longest decreasing subsequences. It then connects these combinatorics to Sundaram–Stanley's type bijection and to representation theory of , showing that the dimension of the invariant subspace in matches the count of -avoiding matchings of points. The core method is a diagrammatic, timeline-based extension of Viennot diagrams that encodes the bumping process and the evolution of oscillating tableaux. This yields a direct combinatorial proof and a unified perspective linking oscillating tableaux, pattern-avoidance in matchings, and Brauer–Schur–Weyl duality in type . The results thereby bridge classical symmetric-function combinatorics with Lie-theoretic invariants in a concrete, diagrammatic framework.

Abstract

We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a -fold tensor product of the vector representation of equals the number of -avoiding matchings of points.

Paper Structure

This paper contains 5 sections, 10 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The Sundaram--Stanley bijection
  3. Decreasing sequences and patterns
  4. Viennot's geometric construction and decreasing subsequences
  5. Oscillating Viennot diagrams and patterns

Key Result

Theorem 1.1

Let $w \in \mathfrak{S}_k$. The length of the longest increasing (resp. decreasing) subsequence of $w$ is equal to the number of columnsHere we use synecdoche: if $\mathop{\mathrm{RS}}\nolimits(w) = (P,Q,\lambda)$ then by "the number of columns in $\mathop{\mathrm{RS}}\nolimits(w)$" we mean the numb

Theorems & Definitions (34)

  • Theorem 1.1: Schensted
  • Example 1.2
  • Example 1.3
  • Remark 1.4: Relation to $\mathfrak{gl}_{n}$ representation theory
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8: Relation to $\mathfrak{sp}_{2n}$ representation theory
  • Definition 2.1: Viennot's geometric construction
  • Example 2.2
  • ...and 24 more