Robinson-Schensted shapes arising from cycle decompositions

Abstract

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