Some statistics on the hypercubes of Catalan permutations

For a permutation σ of length 3, we define the oriented graph Qn(σ). The graph Qn(σ) is obtained by imposing edge constraints on the classical oriented hypercube Qn, such that each path going from 0^n to 1^n in Qn(σ) bijectively encodes a permutation of size n avoiding the pattern σ. The orientation of the edges in Qn(σ) naturally induces an order relation ≼_σ among its nodes. First, we characterize ≼_σ. Next, we study several enumerative statistics on Qn(σ), including the number of intervals, the number of intervals of fixed length k, and the number of paths (or permutations) intersecting a given node.