We provide dual algorithms for sampling the space of abstract simplicial complexes on a fixed number of vertices. We develop a generative and descriptive sampler designed with heuristics to help balance the combinatorial multiplicities of the states and more widely sample across the space of nonisomorphic complexes. We provide a formula for the exact probabilities with which this algorithm will produce a requested labeled state, and compare with an existing benchmark. We also design a highly conductive local ergodic random walk with known transition probabilities. We characterize the autocorrelation of the walk, and numerically test it against our sampler to illustrate its efficacy.