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A bijection between noncrossing and nonnesting partitions of types A and B

Abstract : The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{ n+1} \binom{2n}{n}$ when $\Psi =A_{n-1}$, and the binomial coefficient $\binom{2n}{n}$ when $\Psi =B_n$, and these numbers coincide with the correspondent number of nonnesting partitions. For type $A$, there are several bijective proofs of this equality; in particular, the intuitive map, which locally converts each crossing to a nesting, is one of them. In this paper we present a bijection between nonnesting and noncrossing partitions of types $A$ and $B$ that generalizes the type $A$ bijection that locally converts each crossing to a nesting.
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Ricardo Mamede. A bijection between noncrossing and nonnesting partitions of types A and B. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.597-610. ⟨hal-01185378⟩

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