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Runs and RSK tableaux of boolean permutations

Emily Gunawan, Jianping Pan, Heather M. Russell, Bridget Eileen Tenner

Abstract

We define and construct the "canonical reduced word" of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the "run" statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.

Runs and RSK tableaux of boolean permutations

Abstract

We define and construct the "canonical reduced word" of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the "run" statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.
Paper Structure (12 sections, 23 theorems, 39 equations, 2 figures)

This paper contains 12 sections, 23 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Fully commutative permutations and boolean permutations
  4. Heaps of a boolean permutation
  5. Robinson--Schensted--Knuth tableaux
  6. Runs and longest increasing subsequences
  7. Runs and RSK partitions
  8. Canonical reduced words and the second row of RSK tableaux
  9. Constructing canonical words
  10. From canonical reduced words to RSK tableaux
  11. Characterizing boolean insertion tableaux
  12. Enumerating uncrowded tableaux

Key Result

Theorem 1

For any permutation $w \in S_n$, we have

Figures (2)

  • Figure 1: The heap diagram for the boolean permutation $314569278$.
  • Figure 2: Heap diagram for the boolean permutation having canonical reduced word $\left[{21\cdot98\cdot567\cdot34}\right]$.

Theorems & Definitions (53)

  • Theorem
  • Proposition 2.1: billey-jockusch-stanley
  • Proposition 2.2: tenner-patt-bru
  • Definition 2.3
  • Proposition 2.4: Ste96 and Sta12
  • Example 2.5
  • Proposition 2.6: Scu63
  • Theorem 2.7
  • Example 2.8
  • Proposition 3.1: MT22
  • ...and 43 more