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

We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás-Riordan polynomial RG L of a ribbon graph associated with L. This result generalizes the celebrated relation between the classical Kauffman bracket and the Tutte polynomial of planar graphs.

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We present several direct bijections between different combinatorial interpretations of the Littlewood-Richardson coefficients. The bijections are defined by explicit linear maps which have other applications. * This work was done during a sabbatical stay at MIT Mathematics Department. I would like to thank CONACYT and DGAPA-UNAM for financial support.We start with three major combinatorial interpretations of the LR coefficients which we view as integer points in certain cones. We present simple linear maps between the cones which produce explicit bijections for all triples of partitions involved in the LR rule. These bijections are quite natural in this setting and in a certain sense can be shown to be unique. Below we further emphasize the importance of the linear maps.A classical version of the LR rule, in terms of certain Young tableaux, is now well understood, and its proof has been perfected for decades. We refer to [14] for a beautifully written survey of the "classical" approach, with a historical overview and connections to the jeu-de-taquin, Schützenberger involution, etc. Unfortunately, the language of Young tableaux is often too rigid to be able to demonstrate the inherent symmetries of the LR coefficients.A radically different combinatorial interpretation in due to Berenstein and Zelevinsky, in terms of the so called BZ triangles, which makes explicit all but one symmetry of the LR coefficients † . The authors' proof in [6] relies on a series of previous papers [10,4,5], a situation that is hardly satisfactory. A paper [8] establishes a technically involved bijection with the contratableaux associated with certain Yamanouchi words, which gives another combinatorial interpretation of the LR rule. This combinatorial interpretation is in fact different from the one given by LR tableaux, which makes the matter even more confusing.In a subsequent development, Knutson and Tao introduced [13] the so called honeycombs, which are related to BZ triangles by a bijection that they sketch at the end. The paper [11] uses a related construction of "web diagrams" for a different purpose. The appendix in [13] also introduces a different language of hives, which proved to be more flexible to restate the Knutson-Tao proof of saturation conjecture [7].In the appendix to [7], Fulton described in a simple language a bijection with a set of certain contratableaux, similar to that of Carré [8]. As mentioned at the end of the appendix (cf. also [9]), the latter are in a well known bijection with the classical LR tableaux. Unfortunately, this bijection is based on the Schützenberger involution, which is in fact quite involved and goes beyond the scope of this paper. Now, let us return to the linear maps establishing the bijections. First, these maps show that the LR cones have the same combinatorial structure. Despite a visual difference between definitions of LR tableaux, hives, and BZ triangles, these combinatorial objects are essentially the same and should be treated as equivalent. In a sense, this varying nature of these...