We define Pn,c to be the set of all diagrams consisting of two rows of n vertices with edges, each colored with an element in a set of c possible colors, connecting vertices in different rows. Each vertex can have at most one edge incident to it, and no edges of the same color can cross. In this paper, we find a complete set of irreducible representations of CPn,c. We show that the Bratteli diagram of CP0,c ⊆ CP1,c ⊆ CP2,c ⊆ • • • is Pascal's (c + 1)-simplex, and use this to provide an alternative proof of the well-known recursive formula for multinomial coefficients.

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We provide negative answers to questions posed by Durham, Hagen, and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from rightangled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups. arXiv:1610.07691v1 [math.GT]

We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a 2-stable, quasi-möbius homeomorphism between their Morse boundaries. Our result extends a recent result of Charney-Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental groups of 3-manifolds without Nil or Sol components.

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