Discrete differential equations of order k are of the form R(z,u,F(z,u),Δ F(z,u),…,Δ^kF(z,u)) = 0, where Δ F(z,u) = (F(z,u)-F(z,0))/u and Δ^k F(z,u) = Δ(Δ^{k-1} F(z,u)) for k ≥ 2. Such equations appear most prominently in planar map enumeration but also in several other contexts such as statistical mechanics, lattice path enumeration, pattern avoiding permutations or stack-sortable permutations. Mostly, one is interested in the function F(z,0) that is usually the corresponding counting generating function. In this work, we consider discrete differential equations with an additional parameter x, where the order of the equation is 1 for x = 1 but k > 1 for x ≠ 1. We call such equations singularly perturbed. The solution theory of higher order discrete differential equations is much more involved than for degree 1 and it is a priori not clear that there is a smooth transition from x = 1 to x ≠ 1. The main contribution of this work is to show that there is actually a smooth transition under certain natural assumptions. As an application of this result we consider pattern counts in triangular planar maps and derive a central limit theorem for these counts.

Tree walks are a class of closed walks on a complete graph constrained to span trees. In this work, we focus on a special subclass called $k$-tours, which were recently introduced by Gunnells and are enumerated by the hypergraph Catalan numbers $ c_n^{(k)}$. Gunnells conjectured an asymptotic formula for $c_n^{(k)}$ which we confirm through an alternative approach to their enumeration. As it turns out, the asymptotic growth is governed by the number of $k$-tours on star-like trees.