Linked Partitions and Permutation Tableaux

TL;DR

This paper establishes new bijections connecting linked partitions, permutations, and permutation tableaux, revealing combinatorial structures and pattern avoidance properties within these mathematical objects.

Contribution

It introduces bijections between linked partitions and both permutations and permutation tableaux, expanding understanding of their combinatorial relationships.

Findings

Bijection between linked partitions and permutations with one fewer descent.

Bijection between linked partitions and permutation tableaux.

Characterization of pattern-avoiding permutation tableaux for noncrossing linked partitions.

Abstract

Linked partitions are introduced by Dykema in the study of transforms in free probability theory, whereas permutation tableaux are introduced by Steingr\'{i}msson 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. Steingr\'{i}msson 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 from the set of permutation tableaux of length $n$ with $k$ columns to 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, we find that the corresponding permutation tableaux can be characterized by pattern avoidance.