Walks 'n' Blocks: Asymptotic enumeration of weighted reflectable walks in Ad1 and A2 and exploration of balanced splittable Hadamard matrices

We exploit algebraic structure in combinatorial objects in two distinct areas. In Part I we consider weighted walks confined to Weyl chambers whose stepset is invariant when applying the action of the underlying Weyl group. Previous results on such walks focus on a single weighted stepset or a family of unweighted stepsets. We compute new asymptotic formulas for weighted walks in Weyl chambers Ad1 and A2. This involves setting up a generating function equation, solving for critical points of the function, expressing the desired terms using the Cauchy Integral Formula around these critical points, and approximating the integral. The asymptotics fall into different regimes depending on the contributing critical points. We analyze each asymptotic regime and identify the difficulties in each case. In Part II we consider Hadamard matrices with additional structure. A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. We construct new infinite families of balanced splittable Hadamard matrices. A rich source of examples is provided by packings of partial difference sets in elementary abelian 2-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.