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An RSK correspondence for cylindric tableaux

Alexander Dobner

Abstract

This paper establishes an analogue of the Robinson--Schensted correspondence for cylindric tableaux. In particular, for any pair of positive integers $(d,L)$, we construct a bijection between permutations that avoid the patterns $d\cdots 1 (d+1)$ and $1\cdots (L+1)$ and pairs of $(d,L)$-cylindric standard Young tableaux with a common shape. This arises as a special case of a Knuth-type generalization involving cylindric semistandard tableaux and a further generalization involving oscillating tableaux. Using these results, we construct several other bijections and derive enumerative consequences involving cylindric tableaux and pattern-avoiding permutations. For example, we give asymptotics for the number of permutations in $S_n$ that avoid the patterns $d\cdots 1 (d+1)$ and $1\cdots (L+1)$ as $n\to\infty$.

An RSK correspondence for cylindric tableaux

Abstract

This paper establishes an analogue of the Robinson--Schensted correspondence for cylindric tableaux. In particular, for any pair of positive integers , we construct a bijection between permutations that avoid the patterns and and pairs of -cylindric standard Young tableaux with a common shape. This arises as a special case of a Knuth-type generalization involving cylindric semistandard tableaux and a further generalization involving oscillating tableaux. Using these results, we construct several other bijections and derive enumerative consequences involving cylindric tableaux and pattern-avoiding permutations. For example, we give asymptotics for the number of permutations in that avoid the patterns and as .
Paper Structure (41 sections, 29 theorems, 26 equations, 9 figures)

This paper contains 41 sections, 29 theorems, 26 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline
  4. Acknowledgements
  5. Definitions
  6. Pattern-avoiding permutations
  7. Partitions, Young diagrams, and fillings
  8. Semistandard Young tableaux
  9. Oscillating tableaux
  10. Standard tableaux
  11. Main results
  12. Growth diagrams and local rules
  13. Growth diagram correspondences
  14. Properties of the oscillating RSK and $d$-RSK correspondences
  15. Chains in growth diagrams
  16. ...and 26 more sections

Key Result

Theorem 1.1

For all $n,d,L \in \mathbb{N}$, there is an explicit bijective correspondence between Under this bijection, taking the inverse of the permutation corresponds to interchanging $P$ and $Q$.

Figures (9)

  • Figure 1: In white: a semistandard Young tableau. This tableau is $(3,4)$-cylindric since the entries are strictly increasing along columns even after adding shifted copies. The picture can be drawn on a cylinder by gluing the two dashed lines together.
  • Figure 2: The Young diagram of the partition $(4,3,1)$. The lattice points of the diagram are depicted as black dots. The outer boundary (shown in bold) is encoded by the type sequence ${+}{-}{+}{-}{-}{+}{-}$.
  • Figure 3: A filling with highlighted examples of a NE-chain, a se-chain, and an instance of the pattern $213$. This filling avoids the pattern $3 2 1 4$.
  • Figure 4: (a) A depiction of the semistandard Young tableau $T=$$(\emptyset \prec (1) \prec (3,1) \prec (4,3,1) \prec (4,3,1) \prec (5,3,3) \prec (6,3,3) \prec (6,6,3))$. In this case $\mathrm{wt}(T)=(1,3,4,0,3,1,3)$, and $T$ is a $d$-semistandard Young tableau for any $d\geq 3$. (b) A depiction of $T$ augmented with a shifted copy of the first row above row $d=3$. The value of $\mathrm{MCW}_3(T)$ is the minimum shift needed to ensure that the entries in this augmented picture are strictly increasing along columns. It is also required that there are no gaps between cells in a column, so $\mathrm{MCW}_d(T)=6$ for all $d\geq 4$.
  • Figure 5: A single cell in a growth diagram.
  • ...and 4 more figures

Theorems & Definitions (54)

  • Theorem 1.1: Cylindric RS correspondence
  • Proposition 3.1
  • proof
  • Theorem 3.2: Oscillating RSK correspondence; see krattenthaler2006
  • Theorem 3.3: Oscillating $d$-RSK correspondence
  • Proposition 3.4
  • proof
  • Theorem 3.5: see krattenthaler2006
  • Theorem 3.6
  • Corollary 4.1: Oscillating cylindric RSK correspondence
  • ...and 44 more