Geodesic
Arc permutations
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 301-334.

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Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, and the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircase shape. Finally, a bijection from non-unimodal arc permutations to Young tableaux of certain shapes, which preserves the descent set, is described and applied to deduce a conjectured character formula of Regev.
Classification : 05E15, 05E10, 05A05
Keywords: arc permutation, unimodal permutation, pattern avoidance, affine Weyl group, descent set, shifted staircase, weak order 
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     author = {Elizalde, Sergi and Roichman, Yuval},
     title = {Arc permutations},
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Elizalde, Sergi; Roichman, Yuval. Arc permutations. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 301-334. http://geodesic.mathdoc.fr/item/JAC_2014__39_2_a6/