Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often #P-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call balanced bias. A noteworthy application of our method is sampling restricted classes of integer partitions of n. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of n from general restricted classes. Several observations allow us to improve the efficiency of this chain to require O(n1/2 log(n)) space, and for unrestricted integer partitions, expected O(n9/4) time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.

The Schelling segregation model attempts to explain possible causes of racial segregation in cities. Schelling considered residents of two types, where everyone prefers that the majority of his or her neighbors are of the same type. He showed through simulations that even mild preferences of this type can lead to segregation if residents move whenever they are not happy with their local environments. We generalize the Schelling model to include a broad class of bias functions determining individuals happiness or desire to move, called the General Influence Model. We show that for any influence function in this class, the dynamics will be rapidly mixing and cities will be integrated (i.e., there will not be clustering) if the racial bias is sufficiently low. Next we show complementary results for two broad classes of influence functions: Increasing Bias Functions (IBF), where an individual's likelihood of moving increases each time someone of the same color leaves (this does not include Schelling's threshold models), and Threshold Bias Functions (TBF) with the threshold exceeding one half, reminiscent of the model Schelling originally proposed. For both classes (IBF and TBF), we show that when the bias is sufficiently high, the dynamics take exponential time to mix and we will have segregation and a large “ghetto” will form.

We study the mixing time of a Markov chain Mnn on permutations that performs nearest neighbor transpositions in the non-uniform setting, a problem arising in the context of self-organizing lists. We are given “positively biased” probabilities {pij ≥ 1/2} for all i < j and let pj,i = 1 − pj,i. In each step, the chain Mnn chooses two adjacent elements k, and ℓ and exchanges their positions with probability pℓ,k. Here we define two general classes and give the first proofs that the chain is rapidly mixing for both. In the first case we are given constants r1, … rn−1 with 1/2 ≤ ri ≤ 1 for all i and we set pi,j = ri for all i < j. In the second we are given a binary tree with n leaves labeled 1, … n and constants q1, … qn−1 associated with all of the internal vertices, and we let pi,j = qi⁁j for all i < j. Our bounds on the mixing time of Mnn rely on bijections between permutations, inversion tables and asymmetric simple exclusion processes (ASEPs) that allow us to express moves of the chain in the context of these other combinatorial families. We also demonstrate that the chain is not always rapidly mixing by constructing an example requiring exponential time to converge to equilibrium. This proof relies on a reduction to biased lattice paths in ℤ2.