1,633 research outputs found
Convex Rank Tests and Semigraphoids
Convex rank tests are partitions of the symmetric group which have desirable
geometric properties. The statistical tests defined by such partitions involve
counting all permutations in the equivalence classes. Each class consists of
the linear extensions of a partially ordered set specified by data. Our methods
refine existing rank tests of non-parametric statistics, such as the sign test
and the runs test, and are useful for exploratory analysis of ordinal data. We
establish a bijection between convex rank tests and probabilistic conditional
independence structures known as semigraphoids. The subclass of submodular rank
tests is derived from faces of the cone of submodular functions, or from
Minkowski summands of the permutohedron. We enumerate all small instances of
such rank tests. Of particular interest are graphical tests, which correspond
to both graphical models and to graph associahedra
Subword complexes in Coxeter groups
Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma
and an element in \Pi determine a {\em subword complex}, as introduced in our
paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword
complexes are demonstrated here to be homeomorphic to balls or spheres, and
their Hilbert series are shown to reflect combinatorial properties of reduced
expressions in Coxeter groups. Two formulae for double Grothendieck
polynomials, one of which is due to Fomin and Kirillov, are recovered in the
context of simplicial topology for subword complexes. Some open questions
related to subword complexes are presented.Comment: 14 pages. Final version, to appear in Advances in Mathematics. This
paper was split off from math.AG/0110058v2, whose version 3 is now shorte
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Lexicographic shellability, matroids and pure order ideals
In 1977 Stanley conjectured that the -vector of a matroid independence
complex is a pure -sequence. In this paper we use lexicographic shellability
for matroids to motivate a combinatorial strengthening of Stanley's conjecture.
This suggests that a pure -sequence can be constructed from combinatorial
data arising from the shelling. We then prove that our conjecture holds for
matroids of rank at most four, settling the rank four case of Stanley's
conjecture. In general, we prove that if our conjecture holds for all rank
matroids on at most elements, then it holds for all matroids
Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals
We prove a theorem unifying three results from combinatorial homological and
commutative algebra, characterizing the Koszul property for incidence algebras
of posets and affine semigroup rings, and characterizing linear resolutions of
squarefree monomial ideals. The characterization in the graded setting is via
the Cohen-Macaulay property of certain posets or simplicial complexes, and in
the more general nongraded setting, via the sequential Cohen-Macaulay property.Comment: 31 pages, 1 figure. Minor changes from previous version. To appear in
Advances in Mathematic
Geometry of rank tests
We study partitions of the symmetric group which have desirable geometric
properties. The statistical tests defined by such partitions involve counting
all permutations in the equivalence classes. These permutations are the linear
extensions of partially ordered sets specified by the data. Our methods refine
rank tests of non-parametric statistics, such as the sign test and the runs
test, and are useful for the exploratory analysis of ordinal data. Convex rank
tests correspond to probabilistic conditional independence structures known as
semi-graphoids. Submodular rank tests are classified by the faces of the cone
of submodular functions, or by Minkowski summands of the permutohedron. We
enumerate all small instances of such rank tests. Graphical tests correspond to
both graphical models and to graph associahedra, and they have excellent
statistical and algorithmic properties.Comment: 8 pages, 4 figures. See also http://bio.math.berkeley.edu/ranktests/.
v2: Expanded proofs, revised after reviewer comment
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