An RSK correspondence for cylindric tableaux

TL;DR

This paper develops a new bijective correspondence for cylindric tableaux, extending the classical Robinson--Schensted correspondence, and explores its combinatorial and enumerative implications for pattern-avoiding permutations.

Contribution

It introduces a Robinson--Schensted type bijection for cylindric tableaux and generalizes it to semistandard and oscillating tableaux, expanding combinatorial understanding.

Findings

Constructed a bijection between pattern-avoiding permutations and cylindric tableaux.

Derived asymptotic formulas for the number of pattern-avoiding permutations.

Extended classical combinatorial correspondences to cylindric and oscillating tableaux.

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 an asymptotic for the number of permutations in $S_n$ that avoid the patterns $d\cdots 1 (d+1)$ and $1\cdots (L+1)$ as $n\to\infty$.