# Dyck paths weighted by height profile

We are interested in a question concerning a weight function on Dyck paths that penalizes visits to higher heights.

Let $$\rho$$ be a parameter. Let $$D_k$$ be the set of all nearest neighbor random walk paths that connects $$(0,1)$$ to $$(2k,1)$$ and is strictly greater than $$1$$ away from the endpoints. So each $$\gamma\in D_k$$ is a translate up by one of prime Dyck path. Define the height profile $$h(\gamma) = (h_1,\cdots,h_k)$$ where $$h_i$$ is the number of times $$\gamma$$ is at height $$i$$. Let $$\omega(\gamma) = \prod_{i=1}^k (2 + i \rho)^{-h_i}$$ be the penalizing weight function. Define the values $$p_k = \sum_{\gamma \in D_k} \omega(\gamma)$$ and for any integer $$d\geq 2$$ let $$F(\rho) = \sum_{k=1}^\infty p_k d^k.$$ Define the critical value $$\rho_c = \inf \{ \rho \colon F(\rho) < \infty \}.$$ So $$\rho_c$$ is the balance point" at which $$p_k \approx d^{-k}$$ modulo possibly some polynomial terms. We would like to know if $$F(\rho_c) <\infty.$$ Does this seem possible to prove, or even true? It is unclear to us whether this is a common combinatorial object. Are generating functions for height profiles of Dyck paths like $$h(\gamma)$$ understood at all?