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On $k$-crossings and $k$-nestings of permutations

Abstract : We introduce $k$-crossings and $k$-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of $k$-noncrossing permutations is equal to the number of $k$-nonnesting permutations. We also provide some enumerative results for $k$-noncrossing permutations for some values of $k$.
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Sophie Burrill, Marni Mishna, Jacob Post. On $k$-crossings and $k$-nestings of permutations. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.593-600. ⟨hal-01186302⟩

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