47,167 research outputs found
"Building" exact confidence nets
Confidence nets, that is, collections of confidence intervals that fill out
the parameter space and whose exact parameter coverage can be computed, are
familiar in nonparametric statistics. Here, the distributional assumptions are
based on invariance under the action of a finite reflection group. Exact
confidence nets are exhibited for a single parameter, based on the root system
of the group. The main result is a formula for the generating function of the
coverage interval probabilities. The proof makes use of the theory of
"buildings" and the Chevalley factorization theorem for the length distribution
on Cayley graphs of finite reflection groups.Comment: 20 pages. To appear in Bernoull
Transitive factorizations of permutations and geometry
We give an account of our work on transitive factorizations of permutations.
The work has had impact upon other areas of mathematics such as the enumeration
of graph embeddings, random matrices, branched covers, and the moduli spaces of
curves. Aspects of these seemingly unrelated areas are seen to be related in a
unifying view from the perspective of algebraic combinatorics. At several
points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th
birthda
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Order Quasisymmetric Functions Distinguish Rooted Trees
Richard P. Stanley conjectured that finite trees can be distinguished by
their chromatic symmetric functions. In this paper, we prove an analogous
statement for posets: Finite rooted trees can be distinguished by their order
quasisymmetric functions.Comment: 16 pages, 5 figures, referees' suggestions incorporate
Skew Schubert polynomials
We define skew Schubert polynomials to be normal form (polynomial)
representatives of certain classes in the cohomology of a flag manifold. We
show that this definition extends a recent construction of Schubert polynomials
due to Bergeron and Sottile in terms of certain increasing labeled chains in
Bruhat order of the symmetric group. These skew Schubert polynomials expand in
the basis of Schubert polynomials with nonnegative integer coefficients that
are precisely the structure constants of the cohomology of the complex flag
variety with respect to its basis of Schubert classes. We rederive the
construction of Bergeron and Sottile in a purely combinatorial way, relating it
to the construction of Schubert polynomials in terms of rc-graphs.Comment: 10 pages, 7 figure
A two-sided analogue of the Coxeter complex
For any Coxeter system of rank , we introduce an abstract boolean
complex (simplicial poset) of dimension that contains the Coxeter
complex as a relative subcomplex. Faces are indexed by triples , where
and are subsets of the set of simple generators, and is a
minimal length representative for the parabolic double coset . There
is exactly one maximal face for each element of the group . The complex is
shellable and thin, which implies the complex is a sphere for the finite
Coxeter groups. In this case, a natural refinement of the -polynomial is
given by the "two-sided" -Eulerian polynomial, i.e., the generating function
for the joint distribution of left and right descents in .Comment: 26 pages, several large tables and figure
- …