  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 $\#\textbf{P}$-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 $\textit{balanced bias}$.
  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 $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions,
expected $O(n^{9/4})$ time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.
