Linked partitions and permutation tableaux
WYC Chen, LH Liu, CJ Wang�- arXiv preprint arXiv:1305.5357, 2013 - arxiv.org
WYC Chen, LH Liu, CJ Wang
arXiv preprint arXiv:1305.5357, 2013•arxiv.orgLinked 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�…
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�…
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 . Let denote the set of linked partitions of with blocks, let denote the set of permutations of with descents, and let denote the set of permutation tableaux of length with rows. Steingr\'{i}msson and Williams found a bijection between the set of permutation tableaux of length with rows and the set of permutations of with weak excedances. Corteel and Nadeau gave a bijection from the set of permutation tableaux of length with columns to the set of permutations of with descents. In this paper, we establish a bijection between and and a bijection between and . Restricting the latter bijection to noncrossing linked partitions, we find that the corresponding permutation tableaux can be characterized by pattern avoidance.
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