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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?

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