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In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable 'temperate zone' in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec… (More)

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions ((Kas61), (TF61)). In earlier work, Stanley (Sta85) proved a reciprocity principle governing the number N(m, n) of dimer coverings of an m by n rectangular… (More)

We show that the set R(w 0) of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, R(w 0) possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the… (More)

We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit simplicial complexes whose f-vectors are… (More)

The napkin problem was first posed by John H. Conway, and written up as a 'toughie' in " Mathematical Puzzles: A Connoisseur's Collection, " by Peter Winkler. To paraphrase Winkler's book, there is a banquet dinner to be served at a mathematics conference. At a particular table, n men are to be seated around a circular table. There are n napkins, exactly… (More)

We define a partial order on colored compositions with many properties analogous to Young's lattice. We show that saturated chains correspond to colored permutations, and that covering relations correspond to a Pieri-type rule for colored quasi-symmetric functions. We also show that the poset is CL-shellable. In the case of a single color, we recover the… (More)

Adin, Brenti, and Roichman introduced the pairs of statistics (ndes, nmaj) and (fdes, fmaj). They showed that these pairs are equidistributed over the hyperoctahedral group B n , and can be considered " Euler-Mahonian " in that they generalize the Carlitz identity. Further, they asked whether there exists a bijective proof of the equidistribution of their… (More)

In the context of generating functions for P-partitions, we revisit three flavors of quasisymmetric functions: Gessel's qua-sisymmetric functions, Chow's type B quasisymmetric functions, and Poirier's signed quasisymmetric functions. In each case we use the inner coproduct to give a combinatorial description (counting pairs of permutations) to the… (More)

For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative g-vector. A sufficient condition for unimodality is having a non-negative γ-vector, though one can have negative entries in the γ-vector and still have a nonnegative g-vector. In this paper we provide combinatorial models for three families of γ-vectors that… (More)