794 research outputs found
Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all
roots of its characteristic polynomial have the same real part. This property
was conjectured by Postnikov and Stanley for certain families of arrangements
which are defined for any irreducible root system and was proved for the root
system . The proof is based on an explicit formula for the
characteristic polynomial, which is of independent combinatorial significance.
Here our previous derivation of this formula is simplified and extended to
similar formulae for all but the exceptional root systems. The conjecture
follows in these cases
Binomial Eulerian polynomials for colored permutations
Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and
Williams on the face enumeration of generalized permutohedra. They are
-positive (in particular, palindromic and unimodal) polynomials which
can be interpreted as -polynomials of certain flag simplicial polytopes and
which admit interesting Schur -positive symmetric function
generalizations. This paper introduces analogues of these polynomials for
-colored permutations with similar properties and uncovers some new
instances of equivariant -positivity in geometric combinatorics.Comment: Final version; minor change
Some applications of Rees products of posets to equivariant gamma-positivity
The Rees product of partially ordered sets was introduced by Bj\"orner and
Welker. Using the theory of lexicographic shellability, Linusson, Shareshian
and Wachs proved formulas, of significance in the theory of gamma-positivity,
for the dimension of the homology of the Rees product of a graded poset
with a certain -analogue of the chain of the same length as . Equivariant
generalizations of these formulas are proven in this paper, when a group of
automorphisms acts on , and are applied to establish the Schur
gamma-positivity of certain symmetric functions arising in algebraic and
geometric combinatorics.Comment: Final version, with a section on type B Coxeter complexes added; to
appear in Algebraic Combinatoric
A survey of subdivisions and local -vectors
The enumerative theory of simplicial subdivisions (triangulations) of
simplicial complexes was developed by Stanley in order to understand the effect
of such subdivisions on the -vector of a simplicial complex. A key role
there is played by the concept of a local -vector. This paper surveys some
of the highlights of this theory and some recent developments, concerning
subdivisions of flag homology spheres and their -vectors. Several
interesting examples and open problems are discussed.Comment: 13 pages, 3 figures; minor changes and update
Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements
Monotone path polytopes arise as a special case of the construction of fiber
polytopes, introduced by Billera and Sturmfels. A simple example is provided by
the permutahedron, which is a monotone path polytope of the standard unit cube.
The permutahedron is the zonotope polar to the braid arrangement. We show how
the zonotopes polar to the cones of certain deformations of the braid
arrangement can be realized as monotone path polytopes. The construction is an
extension of that of the permutahedron and yields interesting connections
between enumerative combinatorics of hyperplane arrangements and geometry of
monotone path polytopes
Power sum expansion of chromatic quasisymmetric functions
The chromatic quasisymmetric function of a graph was introduced by Shareshian
and Wachs as a refinement of Stanley's chromatic symmetric function. An
explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing
the chromatic quasisymmetric function of the incomparability graph of a natural
unit interval order in terms of power sum symmetric functions, is proven. The
proof uses a formula of Roichman for the irreducible characters of the
symmetric group.Comment: Final version, 9 pages; comments by a referee incorporate
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