Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the combinatorics of tableaux -- Graphical representation of insertion algorithms

Dale R. Worley

TL;DR

The work unifies numerous tableau insertion algorithms within a single growth-diagram framework, using R-correspondences and insertion diagrams to visualize bumping, insertion, and state propagation. It introduces new algorithms (e.g., jitter and double-circle insertion) and extends the framework with biweighting, enabling simultaneous control of multiple edge weights and colors, including shifted and biweighted variants. Dualities (inversion and transpose) are developed to relate different insertion schemes and to relate P and Q tableaux under permutations and transposes. The approach yields a graphic catalog of unshifted and shifted tableau insertions, clarifies relationships among algorithms, and provides software to construct and experiment with growth diagrams, promising applications in combinatorial identities and enumeration. The framework thus offers a versatile, visual, and extensible method for designing, analyzing, and comparing tableau insertion processes across a broad spectrum of cases.

Abstract

Many algorithms for inserting elements into tableaux are known, starting with the Robinson-Schensted algorithm. Much of those processes can be incorporated into the general framework of Fomin's "growth diagrams". Even for single types of tableaux, there are various alternative insertion algorithms and, due to the varying ways they are described, the relationships between the algorithms can be obscure. The distinguishing features of many algorithms can be codified into graphic "insertion diagrams" which make important aspects of the algorithms immediately apparent. We use insertion diagrams to build a graphic catalog or picture book of many of the tableau insertion algorithms in the literature.

On the combinatorics of tableaux -- Graphical representation of insertion algorithms

TL;DR

The work unifies numerous tableau insertion algorithms within a single growth-diagram framework, using R-correspondences and insertion diagrams to visualize bumping, insertion, and state propagation. It introduces new algorithms (e.g., jitter and double-circle insertion) and extends the framework with biweighting, enabling simultaneous control of multiple edge weights and colors, including shifted and biweighted variants. Dualities (inversion and transpose) are developed to relate different insertion schemes and to relate P and Q tableaux under permutations and transposes. The approach yields a graphic catalog of unshifted and shifted tableau insertions, clarifies relationships among algorithms, and provides software to construct and experiment with growth diagrams, promising applications in combinatorial identities and enumeration. The framework thus offers a versatile, visual, and extensible method for designing, analyzing, and comparing tableau insertion processes across a broad spectrum of cases.

Abstract

Many algorithms for inserting elements into tableaux are known, starting with the Robinson-Schensted algorithm. Much of those processes can be incorporated into the general framework of Fomin's "growth diagrams". Even for single types of tableaux, there are various alternative insertion algorithms and, due to the varying ways they are described, the relationships between the algorithms can be obscure. The distinguishing features of many algorithms can be codified into graphic "insertion diagrams" which make important aspects of the algorithms immediately apparent. We use insertion diagrams to build a graphic catalog or picture book of many of the tableau insertion algorithms in the literature.
Paper Structure (33 sections, 1 theorem, 15 equations, 44 figures)

This paper contains 33 sections, 1 theorem, 15 equations, 44 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Basics
  4. The poset of points
  5. The distributive lattice of diagrams
  6. The weight function
  7. The ascending and descending graphs
  8. Tableaux
  9. Growth diagrams
  10. Generalized permutations
  11. R-correspondence
  12. The growth process
  13. Relationship of R-correspondences to insertion algorithms
  14. Duality
  15. Inversion duality
  16. ...and 18 more sections

Key Result

Proposition 2.1

If a poset $P$ has finite covers, the following are equivalent:

Figures (44)

  • Figure 1: The points of the quadrant $\mathbb{Q} = \mathbb{P}^2$
  • Figure 2: The points of the quadrant $\mathbb{Q}$ without coordinates
  • Figure 3: Young diagrams
  • Figure 4: Young's lattice $\mathbb{Y}$ of partitions
  • Figure 5: (a) Hasse diagram of the quadrant $\mathbb{Q}$, (b) with weights $w(p) = 1$
  • ...and 39 more figures

Theorems & Definitions (2)

  • Proposition 2.1
  • proof