Linked partitions and permutation tableaux

William Y.C. Chen¹ ; Lewis H. Liu² ; Carol J. Wang³

¹Tianjin University
²Nankai University
³Beijing Technology and Business University

The electronic journal of combinatorics, Tome 20 (2013) no. 3

Résumé

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.

Détail

DOI : 10.37236/3408

Classification : 05A05, 05A19
Mots-clés : linked partition, permutation, descent, permutation tableau