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RSK via local transformations

Sam Hopkins

TL;DR

This work presents a local-toggle (per-box) description of the Robinson–Schensted–Knuth correspondence, recasting RSK as a bijection $A \mapsto \widehat{A}$ by inserting corner boxes and toggling along diagonals to produce a reverse plane partition, while preserving transpose symmetry. It develops a complete combinatorial framework, including an inverse procedure, independence from insertion order, and a connection to the octahedron recurrence, and uses this to derive hook-length formulas and refined weighted identities for reverse plane partitions. The approach also yields a Greene–Kleitman invariant, establishing the equivalence with classical RSK on matrices and linking to broader topics such as dual RSK and $d$-complete posets. Overall, the toggle formulation provides a transparent, local, and structurally rich viewpoint on RSK with direct combinatorial consequences and avenues for generalization.

Abstract

We explain how to define the Robinson-Schensted-Knuth (RSK) correspondence in terms of local transformations called "toggles." (This note, which is not intended for publication and which is based on presentations of Alex Postnikov, was written in 2014 and has been circulating since then. We are finally posting it to the arXiv for preservation purposes.)

RSK via local transformations

TL;DR

This work presents a local-toggle (per-box) description of the Robinson–Schensted–Knuth correspondence, recasting RSK as a bijection by inserting corner boxes and toggling along diagonals to produce a reverse plane partition, while preserving transpose symmetry. It develops a complete combinatorial framework, including an inverse procedure, independence from insertion order, and a connection to the octahedron recurrence, and uses this to derive hook-length formulas and refined weighted identities for reverse plane partitions. The approach also yields a Greene–Kleitman invariant, establishing the equivalence with classical RSK on matrices and linking to broader topics such as dual RSK and -complete posets. Overall, the toggle formulation provides a transparent, local, and structurally rich viewpoint on RSK with direct combinatorial consequences and avenues for generalization.

Abstract

We explain how to define the Robinson-Schensted-Knuth (RSK) correspondence in terms of local transformations called "toggles." (This note, which is not intended for publication and which is based on presentations of Alex Postnikov, was written in 2014 and has been circulating since then. We are finally posting it to the arXiv for preservation purposes.)
Paper Structure (5 sections, 12 theorems, 61 equations, 1 figure)

This paper contains 5 sections, 12 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The RSK algorithm via toggles
  3. An application: hook-length formulas
  4. RSK = RSK: the Greene-Kleitman invariant and the octahedron recurrence
  5. Future directions: dual RSK; other d-complete posets

Key Result

Proposition 9

The $\mathcal{RSK}$ map defined above is well-defined; that is, the order in which we decompose an $\mathbb{N}$-tableaux $T$ does not matter.

Figures (1)

  • Figure 1: The $\mathbb{N}$-tableau $\widehat{T}$, where position $(i,j)$ has been marked by a star.

Theorems & Definitions (24)

  • Definition 1
  • Example 2
  • Definition 3
  • Remark 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Example 8
  • Proposition 9
  • Proposition 10
  • ...and 14 more