There are several examples where the mixing time of a Markov chain can be reduced substantially, often to about its square root, by "lifting", i.e., by splitting each state into several states.In several examples of random walks on groups, the lifted chain not only mixes better, but is easier to analyze.We characterize the best mixing time achievable through lifting in terms of multicommodity flows. We show that the reduction to square root is best possible. If the lifted chain is time-reversible, then the gain is smaller, at most a factor of log(l/na), where 110 is the smallest stationary probability of any state. We give an example showing that a gain of a factor of log(l/~o)/log log(l/rro) is possible.

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

Abstract. The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman-Grassl correspondence, respectively.The main new results are two different q-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. We establish explicit bijections between these objects and families of integer arrays with certain nonzero entries, which also proves the second formula.

We prove that the partition function p(n) is log-concave for all n > 25. We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based on Lehmer's estimates on the remainders of the Hardy-Ramanujan and the Rademacher series for p(n).

We present a bijection between 321-and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson et al. (Ann. Combin. 6 (2003) 427), and Elizalde (Proc. FPSAC 2003). We also show that our bijection preserves additional statistics, which extends the previous results. r