158 research outputs found
Chromatic polynomials of simplicial complexes
We consider s-chromatic polynomials of simplicial complexes, higher
dimensional analogues of chromatic polynomials for graphs.Comment: 16 page
An ansatz for the asymptotics of hypergeometric multisums
Sequences that are defined by multisums of hypergeometric terms with compact
support occur frequently in enumeration problems of combinatorics, algebraic
geometry and perturbative quantum field theory. The standard recipe to study
the asymptotic expansion of such sequences is to find a recurrence satisfied by
them, convert it into a differential equation satisfied by their generating
series, and analyze the singulatiries in the complex plane. We propose a
shortcut by constructing directly from the structure of the hypergeometric term
a finite set, for which we conjecture (and in some cases prove) that it
contains all the singularities of the generating series. Our construction of
this finite set is given by the solution set of a balanced system of polynomial
equations of a rather special form, reminiscent of the Bethe ansatz. The finite
set can also be identified with the set of critical values of a potential
function, as well as with the evaluation of elements of an additive -theory
group by a regulator function. We give a proof of our conjecture in some
special cases, and we illustrate our results with numerous examples.Comment: 22 pages and 2 figure
Shuffles on Coxeter groups
The random-to-top and the riffle shuffle are two well-studied methods for
shuffling a deck of cards. These correspond to the symmetric group , i.e.,
the Coxeter group of type . In this paper, we give analogous shuffles
for the Coxeter groups of type and . These can be interpreted as
shuffles on a ``signed'' deck of cards. With these examples as motivation, we
abstract the notion of a shuffle algebra which captures the connection between
the algebraic structure of the shuffles and the geometry of the Coxeter groups.
We also briefly discuss the generalisation to buildings which leads to
q-analogues
Piecewise polynomials on polyhedral complexes
For a d-dimensional polyhedral complex P, the dimension of the space of
piecewise polynomial functions (splines) on P of smoothness r and degree k is
given, for k sufficiently large, by a polynomial f(P,r,k) of degree d. When d=2
and P is simplicial, Alfeld and Schumaker determined a formula for all three
coefficients of f. However, in the polyhedral case, no formula is known. Using
localization techniques and specialized dual graphs associated to
codimension--2 linear spaces, we obtain the first three coefficients of
f(P,r,k), giving a complete answer when d=2.Comment: 12 pages, 7 figures v2 removed superfluous Prop 3.
The average simplex cardinality of a finite abstract simplicial complex
We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of a
finite abstract simplicial complex G. The functional is a homomorphism from the
monoid of simplicial complexes to the rationals: the formula Dim^+(G + H) =
Dim^+(G) + Dim^+(H) holds for the join + similarly as for the augmented
inductive dimension dim^+(G) = dim(G)+1 where dim is the inductive dimension
dim(G) = 1+ sum_x dim(S(x))/|G| with unit sphere S(x) (a recent theorem of
Betre and Salinger). In terms of the generating function f(t) = 1+v_0 t + v_1
t^2 + ... +v_d t^(d+1) defined by the f-vector (v_0,v_1, \dots) of G for which
f(-1) is the genus 1-X(G) with Euler characteristic X and f(1)=|G|+1 is the
augmented number of simplices, the average cardinality is the logarithmic
derivative Dim^+(f) = f'(1)/f(1) of f at 1. Beside introducing the average
cardinality and establishing its compatibility with arithmetic, we prove two
results: 1) the inequality dim^+(G)/2 <= Dim^+(G) with equality for complete
complexes. 2) the limit C_d of Dim^+(G_n) for n to infinity is the same for any
initial complex G_0 of maximal dimension d and the constant c_d is explicitly
given in terms of the Perron-Frobenius eigenfunction of the universal
Barycentric refinement operator and is for positive d always a rational number
in the open interval ((d+1)/2,d+1).Comment: 19 page, 8 figure
-Vectors of Barycentric Subdivisions
For a simplicial complex or more generally Boolean cell complex we
study the behavior of the - and -vector under barycentric subdivision. We
show that if has a non-negative -vector then the -polynomial of
its barycentric subdivision has only simple and real zeros. As a consequence
this implies a strong version of the Charney-Davis conjecture for spheres that
are the subdivision of a Boolean cell complex.
For a general -dimensional simplicial complex the
-polynomial of its -th iterated subdivision shows convergent behavior.
More precisely, we show that among the zeros of this -polynomial there is
one converging to infinity and the other converge to a set of real
numbers which only depends on
The Partition Lattice in Many Guises
This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-element
Scheduling Problems
We introduce the notion of a scheduling problem which is a boolean function
over atomic formulas of the form . Considering the as
jobs to be performed, an integer assignment satisfying schedules the jobs
subject to the constraints of the atomic formulas. The scheduling counting
function counts the number of solutions to . We prove that this counting
function is a polynomial in the number of time slots allowed. Scheduling
polynomials include the chromatic polynomial of a graph, the zeta polynomial of
a lattice, the Billera-Jia-Reiner polynomial of a matroid.
To any scheduling problem, we associate not only a counting function for
solutions, but also a quasisymmetric function and a quasisymmetric function in
non-commuting variables. These scheduling functions include the chromatic
symmetric functions of Sagan, Gebhard, and Stanley, and a close variant of
Ehrenborg's quasisymmetric function for posets.
Geometrically, we consider the space of all solutions to a given scheduling
problem. We extend a result of Steingr\'immson by proving that the -vector
of the space of solutions is given by a shift of the scheduling polynomial.
Furthermore, under certain niceness conditions on the defining boolean
function, we prove partitionability of the space of solutions and positivity of
fundamental expansions of the scheduling quasisymmetric functions and of the
-vector of the scheduling polynomial.Comment: Final version, substantially shortened, 18 page
Symmetric decompositions and real-rootedness
In algebraic, topological, and geometric combinatorics inequalities among the
coefficients of combinatorial polynomials are frequently studied. Recently a
notion called the alternatingly increasing property, which is stronger than
unimodality, was introduced. In this paper, we relate the alternatingly
increasing property to real-rootedness of the symmetric decomposition of a
polynomial to develop a systematic approach for proving the alternatingly
increasing property for several classes of polynomials. We apply our results to
strengthen and generalize real-rootedness, unimodality, and alternatingly
increasing results pertaining to colored Eulerian and derangement polynomials,
Ehrhart -polynomials for lattice zonotopes, -polynomials of
barycentric subdivisions of doubly Cohen-Macaulay level simplicial complexes,
and certain local -polynomials for subdivisions of simplices. In particular,
we prove two conjectures of Athanasiadis
Polynomial splitting measures and cohomology of the pure braid group
We study for each a one-parameter family of complex-valued measures on
the symmetric group , which interpolate the probability of a monic, degree
, square-free polynomial in having a given factorization
type. For a fixed factorization type, indexed by a partition of ,
the measure is known to be a Laurent polynomial. We express the coefficients of
this polynomial in terms of characters associated to -subrepresentations
of the cohomology of the pure braid group . We
deduce that the splitting measures for all parameter values
(resp. ), after rescaling, are characters of
-representations (resp. virtual -representations.)Comment: To appear in the Arnold Mathematical Journa
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