Robinson-Schensted shapes arising from cycle decompositions

TL;DR

This paper characterizes the possible Robinson-Schensted shapes arising from permutations with a given cycle structure, focusing on the case of two cycles, and introduces a coloring method to construct such permutations.

Contribution

It explicitly describes the set of RS shapes for permutations with two cycles and introduces the concept of alpha-coloring to construct permutations with specified cycle types and shapes.

Findings

Identifies the set of RS shapes for permutations with two cycles.

Introduces alpha-coloring as a method for construction.

Provides a combinatorial framework linking cycle types and RS shapes.

Abstract

In the symmetric group $S_n$, each element $\sigma$ has an associated cycle type $\alpha$, a partition of $n$ that identifies the conjugacy class of $\sigma$. The Robinson-Schensted (RS) correspondence links each $\sigma$ to another partition $\lambda$ of $n$, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to $\sigma$. Surprisingly, the relationship between these two partitions, namely the cycle type $\alpha$ and the RS shape $\lambda$, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes $\lambda$ that can arise from elements of each cycle type $\alpha$ in cases where $\alpha$ consists of two cycles. To do this, we introduce the notion of an $\alpha$-coloring, where one colors the entries in a certain tableau of shape $\lambda$, in such a way as to construct a permutation $\sigma$ with cycle type $\alpha$ and RS shape $\lambda$.