Vertex degrees in grid graphs associated with 213-avoiding permutations

Given a permutation of size $n$, we consider its associated grid graph whose $i$th column has height equal to the $i$th entry, with vertical edges between consecutive levels and horizontal edges between equal levels in adjacent columns. We study global degree statistics of these graphs when the permutation is chosen from the Catalan avoidance class $\mathrm{Av}_n(213)$ (and, by reversal, also from $\mathrm{Av}_n(312)$). We first obtain an explicit closed form for the total number of horizontal edges summed over all permutations in $\mathrm{Av}_n(213)$. We then determine, for each degree $r\in\{1,2,3,4\}$, the total number of degree-$r$ vertices accumulated over the same class, yielding closed expressions in terms of central binomial coefficients and powers of four. The proofs rely on the Catalan decomposition induced by the position of the minimum entry, which leads to gluing identities and algebraic functional equations for ordinary generating functions, completed using global vertex and degree-sum identities. As a consequence, we derive asymptotic degree proportions for a uniform random permutation in $\mathrm{Av}_n(213)$: the distribution concentrates and the proportion of degree-$4$ vertices tends to $1$, with a deficit of order $n^{-1/2}$.