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  1. ; (, The Australasian journal of combinatorics)
    Albert, Michael; Billington, Elizabeth J (Ed.)
    In the first partial result toward Steinberg’s now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly fewer than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey’s resolution of Ore’s conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable. 
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  2. ; (, The Electronic Journal of Combinatorics)
    Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream. 
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  3. ; ; ; ; (, Algebraic Combinatorics)
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  4. ; (, European Journal of Combinatorics)
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  5. ; ; (, Advances in Mathematics)
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  6. ; ; (, The Electronic Journal of Combinatorics)
    We study the relationship between two notions of pattern avoidance for involutions in the symmetric group and their restriction to fixed-point-free involutions. The first is classical, while the second appears in the geometry of certain spherical varieties and generalizes the notion of pattern avoidance for perfect matchings studied by Jelínek. The first notion can always be expressed in terms of the second, and we give an effective algorithm to do so. We also give partial results characterizing the families of involutions where the converse holds. As a consequence, we prove two conjectures of McGovern characterizing (rational) smoothness of certain varieties. We also give new enumerative results, and conclude by proposing several lines of inquiry that extend our current work. 
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