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Bijections for rhombic alternative tableaux

Sylvie Corteel, Jang Soo Kim, Olya Mandelshtam, Philippe Nadeau

Abstract

We generalize well-known bijections between alternative tableaux and permutations to bijections between rhombic alternative tableaux (RAT) and assemblées of permutations. We show how these various bijections are connected. As a consequence, we find a refined enumeration formula for RAT. One of our bijections carries many statistics from RAT to assemblées; notably, it sends the number of free cells to the number of crossings, which answers a question of Mandelshtam and Viennot. We also find an $r!$-to-$1$ map from marked Laguerre histories to assemblées, answering a question of Corteel and Nunge.

Bijections for rhombic alternative tableaux

Abstract

We generalize well-known bijections between alternative tableaux and permutations to bijections between rhombic alternative tableaux (RAT) and assemblées of permutations. We show how these various bijections are connected. As a consequence, we find a refined enumeration formula for RAT. One of our bijections carries many statistics from RAT to assemblées; notably, it sends the number of free cells to the number of crossings, which answers a question of Mandelshtam and Viennot. We also find an -to- map from marked Laguerre histories to assemblées, answering a question of Corteel and Nunge.
Paper Structure (22 sections, 30 theorems, 89 equations, 17 figures, 3 algorithms)

This paper contains 22 sections, 30 theorems, 89 equations, 17 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutations and assemblées
  4. Alternative tableaux
  5. Rhombic alternative tableaux
  6. The structure of RAT
  7. Decomposition of a RAT
  8. Flattening map
  9. Three bijections between RAT and assemblées
  10. The insertion map
  11. The inverse of the insertion map
  12. An application of the insertion map
  13. A zigzag map from RAT to assemblées
  14. The fusion-exchange map from assemblées to RAT
  15. Another zigzag map from RAT to assemblées
  16. ...and 7 more sections

Key Result

Theorem 2.7

Corteel2009 The insertion map $\Phi^{\mathop{\mathrm{AT}}\nolimits}_I:\mathop{\mathrm{AT}}\nolimits^+(n+1)\to S_{n+1}$ is a bijection. Moreover, if $T = \Phi^{\mathop{\mathrm{AT}}\nolimits}_I(\pi)$, then for each $i\in[n+1]$, the integer $i$ is the label of a free row of $T$ if and only if $i$ is an

Figures (17)

  • Figure 1: The parameters of the two-species PASEP.
  • Figure 2: An alternative tableau $T\in \mathop{\mathrm{AT}}\nolimits(12)$ of shape $w=002220202002$ on the left and the corresponding extended alternative tableau on the right.
  • Figure 3: The zigzag path from $1$ ends at $8$, and the zigzag path from $3$ ends at $3$.
  • Figure 4: The N-side $N(\tau)$, S-side $S(\tau)$, E-side $E(\tau)$, and W-side $W(\tau)$ of a tile $\tau$ are indicated with letters N, S, E, and W, respectively.
  • Figure 5: The rhombic diagram $\Gamma_w$ of the word $w=2\,2\,1\,0\,1\,0\,2\,0$.
  • ...and 12 more figures

Theorems & Definitions (100)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Zigzag map
  • Definition 2.6: Insertion map
  • Theorem 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 90 more