A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett-Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszár model, the inversion model, and the Bradley-Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.

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A finite irreducible real reflection group of rank ℓ and Coxeter number h has root system of cardinality h·ℓ. It is shown that the fake degree for the permutation action on its roots is divisible by [h]q = 1+q +q2 + · · ·+q h−1, and that in simply-laced types it equals [h]q · Pℓ i=1 qd∗i where d ∗ i = ei − 1 are the codegrees and ei are the exponents.

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Incomplete rankings on a set of items {1,..., n} are orderings of the form a1 ≺ · · · ≺ ak, with {a1,... ak} ⊂ {1,..., n} and k < n. Though they arise in many modern applications, only a few methods have been introduced to manipulate them, most of them consisting in rep-resenting any incomplete ranking by the set of all its possible linear extensions on {1,..., n}. It is the major purpose of this paper to introduce a completely novel approach, which allows to treat incomplete rankings directly, representing them as injective words over {1,..., n}. Unexpectedly, operations on incomplete rankings have very simple equivalents in this setting and the topological structure of the complex of injective words can be interpretated in a simple fashion from the perspective of ranking. We exploit this connection here and use recent results from algebraic topology to construct a multiresolution analysis and develop a wavelet frame-work for incomplete rankings. Though purely combinatorial, this construction relies on the same ideas underlying multiresolution analysis on a Euclidean space, and permits to localize the information related to rankings on each subset of items. It can be viewed as a crucial step toward nonlinear approximation of distributions of incomplete rankings and paves the way for many statistical applications, including preference data analysis and the design of recommender systems.