Prescribed Reading: November 2010

Health Care Matters: My opinion counts Our employees are ‘Leading with care’ Patient Satisfaction Scorecard Welcome to our new provider Nine promotional projects recognized The 2010 CCHS annual report is online NuVal comes to Coborn’s stores Clean-out the cabinet results! Welcome to these new employees Recognition for years of service CentraCare Wound Center opens Nov. 8 Surgery open house planned for Nov. 11 Computer kiosk check-in pilot planned Kudos to . . . National Memory Screening Day is Nov. 1

Cambrian Lattices

For an arbitrary finite Coxeter group W we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a "cluster fan." Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.Comment: Revisions in exposition (partly in response to the suggestions of an anonymous referee) including many new figures. Also, Conjecture 1.4 and Theorem 1.5 are replaced by slightly more detailed statements. To appear in Adv. Math. 37 pages, 8 figure

Generic rectangulations

A rectangulation is a tiling of a rectangle by a finite number of rectangles. The rectangulation is called generic if no four of its rectangles share a single corner. We initiate the enumeration of generic rectangulations up to combinatorial equivalence by establishing an explicit bijection between generic rectangulations and a set of permutations defined by a pattern-avoidance condition analogous to the definition of the twisted Baxter permutations.Comment: Final version to appear in Eur. J. Combinatorics. Since v2, I became aware of literature on generic rectangulations under the name rectangular drawings. There are results on asymptotic enumeration and computations counting generic rectangulations with n rectangles for many n. This result answers an open question posed in the rectangular drawings literature. See "Note added in proof.

Universal geometric cluster algebras

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or reals. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal coefficients is equivalent to finding an R-basis for B (a "mutation-linear" analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan F_B, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between F_B and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure

Noncrossing partitions and the shard intersection order

We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new construction of NC(W) yields a new proof that NC(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of the shard intersection order, like Mobius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC(W). There is a natural dimension-preserving bijection between simplices in the order complex of the shard intersection order (i.e. chains in the shard intersection order) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The shard intersection order is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.Comment: 44 pages, 15 figure

Hong Kong – The new offence of fraud

Letter from Hong Kong by John Reading SC (Senior Assistant Director of Public Prosecutions, Commercial Crime Unit, Department of Justice, Hong Kong Special Administrative Region) describing how Hong Kong’s legislature enacted a statutory offence of fraud by inserting a new section (16A) in the Theft Ordinance. Jean reading prosecutes fraud and corruption cases and is a Senior Counsel. Published in the Letter from … section of Amicus Curiae - Journal of the Institute of Advanced Legal Studies and its Society for Advanced Legal Studies. The Journal is produced by the Society for Advanced Legal Studies at the Institute of Advanced Legal Studies, University of London