Linked Partitions and Permutation Tableaux

Abstract

Linked partitions were introduced by Dykema in the study of transforms in free probability theory, whereas permutation tableaux were introduced by Steingrímsson and Williams in the study of totally positive Grassmannian cells. Let $[n]=\{1,2,\ldots,n\}$. Let $L(n,k)$ denote the set of  linked partitions of $[n]$ with $k$ blocks, let $P(n,k)$ denote the set of  permutations of $[n]$ with $k$ descents, and let $T(n,k)$ denote the set of permutation tableaux of length $n$ with $k$ rows. Steingrímsson and Williams found a bijection between the set of permutation tableaux of length $n$ with $k$ rows and the set of permutations of $[n]$ with $k$ weak excedances. Corteel and Nadeau gave a bijection between the set of permutation tableaux of length $n$ with $k$ columns and the set of permutations of $[n]$ with $k$ descents. In this paper, we establish a bijection between $L(n,k)$ and $P(n,k-1)$ and a bijection between $L(n,k)$ and $T(n,k)$. Restricting the latter bijection to  noncrossing linked partitions and nonnesting linked partitions, we find that the corresponding  permutation tableaux can be characterized by pattern avoidance.