  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.
