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The symmetry of the joint distribution of the numbers of crossings and nestings of length 2 has been observed in many combinatorial structures, including permutations , matchings, set partitions, linked partitions, and certain families of graphs. These results have been unified in the larger context of enumeration of northeast and southeast chains of length… (More)

In this paper, we focus on a " local property " of permutations: value-peak. A permutation σ has a value-peak σ(i) if σ(i − 1) < σ(i) > σ(i + 1) for some i ∈ [2, n − 1]. Define V P (σ) as the set of value-peaks of the permutation σ. For any S ⊆ [3, n], define V P n (S) such that V P (σ) = S. Let P n = {S | V P n (S) = ∅}. we make the set P n into a poset P… (More)

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