Some papers Vic Reiner
 Harmonics and graded Ehrhart theory (with Brendon Rhoades)

The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a qdeformation of the Ehrhart series, basedon the notions of harmonic spaces and Macaulay's inverse systems for coordinate rings of finite point configurations. We conjecture that this qEhrhart series is a rational function, and introduce and study a bigraded algebra whose Hilbert series matches the qEhrhart series. Defining this algebra requires a new result on Macaulay inverse systems for Minkowski sums of point configurations.

(Math arXiv preprint
arxiv:2407.06511
)
 Koszulity, supersolvability and Stirling Representations (with Ayah Almousa and Sheila Sundaram)

Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their
OrlikSolomon algebras and graded VarchenkoGel'fand algebras.
We explore how this interacts with group actions, particularly for the braid arrangement and the action
of the symmetric group, where the Hilbert functions of
the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively.
The corresponding symmetric group representations exhibit
branching rules that interpret Stirling number recurrences,
which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.

(Math arXiv preprint
arXiv:2404.10858
)
 Sandpile groups for cones over trees
(with Dorian Smith)

Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex to a tree. For example, it is shown that the number of generators of the sandpile group is at most one less than the number of leaves in the tree. For trees on a fixed number of
vertices, the paths and stars are shown to provide extreme behavior, not only for the number of generators, but also for the number of spanning trees, and for Tutte polynomial evaluations that count the recurrent sandpile configurations by their numbers of chips.

(Math arXiv preprint
arXiv:2402.15453
)
 Chow rings of matroids as permutation representations
(with Robert Angarone and Anastasia Nathanson)

Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincaré duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.

(Math arXiv preprint
arXiv:2309.14312
)
 Equivariant resolutions over Veronese rings
(with Ayah Almousa, Michael Perlman, Alexandra Pevzner, Keller VandeBogert)

Working in a polynomial ring S=k[x_{1},...,x_{n}] where k is an arbitrary commutative ring with 1, we consider the dth Veronese subalgebras R=S^{(d)},
as well as natural Rsubmodules M=S^{(≥r,d)} inside S.
We develop and use characteristicfree theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple GL_{n}(k)equivariant minimal
free Rresolutions for the quotient ring k=R/R_{+} and
for these modules M. These also lead to elegant descriptions of Tor^{R}_{i}(M,M') for all i and Hom_{R}(M,M') for any pair of these modules M,M'.

(Math arXiv preprint
arXiv:2210.16342,
journal version)
 Invariant theory for the free leftregular band and a qanalogue
(with Sarah Brauner and Patricia Commins))

We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups: the free leftregular band on n letters, acted on by the symmetric group, and one of its qanalogues, acted on by the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and qStirling numbers. We then use results from the theory of random walks and randomtotop shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs.

(Math arXiv preprint
arXiv:2206.11406
)
 Topology of augmented Bergman complexes
(with E. Bullock, A. Kelley, K. Ren, G. Shemy, D. Shen, B. Sun, A. Tao, J. Zhang)

The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two wellstudied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex.
It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings reinterprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.

(Math arXiv preprint
arXiv:2108.13394)
 The "Grothendieck to Lascoux" conjecture
(with A. Yong)

This report formulates a conjectural combinatorial rule that positively expands
Grothendieck polynomials into Lascoux polynomials. It generalizes one such
formula expanding Schubert polynomials into key polynomials, and refines another
one expanding stable Grothendieck polynomials.

(Math arXiv preprint
arXiv:2102.12399)
 A colorful Hochster formula and universal parameters for face rings
(with A. Adams)

This paper has two related parts. The first generalizes Hochster's formula on
resolutions of StanleyReisner rings to a colorful version, applicable to any
proper vertexcoloring of a simplicial complex. The second part examines a
universal system of parameters for StanleyReisner rings of simplicial
complexes, and more generally, face rings of simplicial posets. These
parameters have good properties, including being fixed under symmetries, and
detecting depth of the face ring. Moreover, when resolving the face ring over
these parameters, the shape is predicted, conjecturally, by the colorful
Hochster formula.

(Math arXiv preprint
arXiv:2007.13021,
journal version,)
 Cyclic sieving for cyclic codes
(with A. Mason and S. Sridhar)

Prompted by a question of Jim Propp, this paper examines the cyclic sieving phenomenon (CSP) in certain cyclic codes. For example, it is shown that, among dual Hamming codes over F_{q}, the generating function for codedwords according to the major index statistic (resp. the inversion statistic) gives rise to a CSP when q=2 or q=3 (resp. when q=2). A byproduct is a curious characterization of the irreducible polynomials in F_{2}[x] and F_{3}[x] that are primitive.

(Math arXiv preprint
arXiv:2004.11998)
 Invariant theory for coincidental complex reflection groups
(with A.V. Shepler and E. Sommers)

V.F. Molchanov considered the Hilbert series for the space of invariant
skewsymmetric tensors and dual tensors with polynomial coefficients under the
action of a real reflection group, and speculated that it had a certain product
formula involving the exponents of the group. We show that Molchanov's
speculation is false in general but holds for all coincidental complex
reflection groups when appropriately modified using exponents and coexponents.
These are the irreducible wellgenerated (i.e., duality) reflection groups with
exponents forming an arithmetic progression and include many real reflection
groups and all nonreal Shephard groups, e.g., the ShephardTodd infinite
family G(d,1,n). We highlight consequences for the qNarayana and
qKirkman polynomials, giving simple product formulas for both, and give a
qanalogue of the identity transforming the hvector to the fvector for
the coincidental finite type cluster/Cambrian complexes of FominZelevinsky
and Reading.

(Math arXiv preprint
arXiv:1908.02663)
 Whitney numbers for poset cones
(with G. DorpalenBarry and Jang Soo Kim)

Hyperplane arrangements dissect R^{n} into connected components called chambers, and a wellknown theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes x_{i}=x_{j} inside R^{n} for the symmetric group, thought of as the type A_{n1} reflection group. Here cones correspond to posets, chambers within the cone correspond to linear extensions of the poset, and the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. We interpret this refinement explicitly for two families of posets: width two posets, and disjoint unions of chains. In the latter case, this gives a geometric reinterpretation to Foata's theory of cycle decomposition for multiset permutations, and leads to a simple generating function compiling these Whitney numbers.

(Math arXiv preprint
arXiv:1906.00036)
 Cyclic quasisymmetric functions
(with R. Adin, I.M. Gessel, and Y. Roichman)

The ring of cyclic quasisymmetric functions and its nonEscher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasisymmetric functions; for the nonEscher subring they arise as toric Ppartition enumerators, for toric posets P with a total cyclic order. The associated structure constants are determined by cyclic shuffles of permutations. We then prove the following positivity phenomenon: for every nonhook shape λ, the coefficients in the expansion of the Schur function s_{λ} in terms of fundamental cyclic quasisymmetric functions are nonnegative. The proof relies on the existence of a cyclic descent map on the standard Young tableaux (SYT) of shape λ. The theory has applications to the enumeration of cyclic shuffles and SYT by cyclic descents. We conclude by providing a cyclic analogue of Solomon's descent algebra.

(Math arXiv preprint
arXiv:1811.05440)
 Weak order and descents for monotone triangles
(with Z. Hamaker)

Monotone triangles are a rich extension of permutations that biject with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous properties. It is shown that any linear extension of the weak order gives rise to a shelling order on a poset, recently introduced by Terwilliger, whose maximal chains biject with monotone triangles; among these shellings are a family of ELshellings.
The weak order turns out to encode an action of the 0Hecke monoid of type A on the monotone triangles, generalizing the usual bubblesorting action on permutations. It also leads to a notion of descent set for monotone triangles, having another natural property: the surjective algebra map from the Malvenuto Reutenauer Hopf algebra of permutations into quasisymmetric functions extends in a natural way to an algebra map out of the recentlydefined CheballahGiraudoMaurice algebra of alternating sign matrices.

(Math arXiv preprint
arXiv:1809.10571)
 On configuration spaces and Whitehouse's lifts of the Eulerian representations
(with N. Early)

S. Whitehouse's lifts of the Eulerian representations of the symmetric group S_{n} to S_{n+1} are reinterpreted, topologically and ringtheoretically, with inspiration from A. Ocneanu's theory of permutohedral blades.

(Math arXiv preprint
arXiv:1808.04007)
 On cyclic descents for tableaux
(with R. Adin and Y. Roichman)

The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical.
Cellini introduced a natural notion of cyclic descent set for permutations,
and Rhoades introduced such a notion for SYT  but only for rectangular shapes.
In this work we define cyclic extensions of descent sets in a general context,
and prove existence and essential uniqueness for SYT of almost all shapes.
The proof applies nonnegativity properties of Postnikov's toric Schur polynomials,
providing a new interpretation of certain GromovWitten invariants.

(Math arXiv preprint
arXiv:1710.06664)

The Koszul homology algebra of the second Veronese is generated by the
lowest strand
(with Aldo Conca and Lukas Katthän)

We show that the Koszul homology algebra of the second Veronese subalgebra of
a polynomial ring over a field of characteristic zero is generated, as an
algebra, by the homology classes corresponding to the syzygies of its linear strand.

(Math arXiv preprint
arXiv:1710.04293)

A refined count of Coxeter element factorizations
(with Elise delMas and Thomas Hameister)

For wellgenerated complex reflection groups, Chapuy and Stump gave a simple
product for a generating function counting reflection factorizations of a
Coxeter element by their length. This is refined here to record the number of
reflections used from each orbit of hyperplanes. The proof is casebycase via
the classification of wellgenerated groups. It implies a new expression for
the Coxeter number, expressed via data coming from a hyperplane orbit; a
casefree proof of this due to J. Michel is included.

(Math arXiv preprint
arXiv:1708.06292)

Critical groups for Hopf algebra modules
(with Darij Grinberg and Jia Huang)

This paper considers an invariant of modules over a finitedimensional Hopf
algebra, called the critical group. This generalizes the critical groups of
complex finite group representations studied by Benkart, Klivans, Reiner and
Gaetz. A formula is given for the cardinality of the critical group generally,
and the critical group for the regular representation is described completely.
A key role in the formulas is played by the greatest common divisor of the
dimensions of the indecomposable projective representations.

(Math arXiv preprint
arXiv:1704.03778,
journal version.
)

Invariant derivations and differential forms for reflection groups
(with Anne V. Shepler)

Classical invariant theory of a complex reflection group W highlights three beautiful structures:
 the Winvariant polynomials constitute a polynomial algebra, over which
 the Winvariant differential forms with polynomial coefficients constitute an exterior algebra, and
 the relative invariants of any Wrepresentation constitute a free module.
When W is a duality (or wellgenerated) group, we give an explicit description of the isotypic component within the differential forms of the irreducible reflection representation. This resolves a conjecture of Armstrong, Rhoades and the first author, and relates to Lietheoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and also Deconcini, Papi, and Procesi. We establish this result by examining the space of Winvariant differential derivations; these are derivations whose coefficients are not just polynomials, but differential forms with polynomial coefficients.
For every complex reflection group W, we show that the space of invariant differential derivations is finitely generated as a module over the invariant differential forms by the basic derivations together with their exterior derivatives. When W is a duality group, we show that the space of invariant differential derivations is free as a module over the exterior subalgebra of Winvariant forms generated by all but the topdegree exterior generator. (The basic invariant of highest degree is omitted.)
Our arguments for duality groups are casefree, i.e., they do not rely on any reflection group classification.

(Math arXiv preprint
arXiv:1612.01031)

Weyl group qKreweras numbers and cyclic sieving
(with Eric Sommers)

The paper concerns a definition for qKreweras numbers
for finite Weyl groups W, refining the qCatalan numbers for W, and arising
from work of the second author. We give explicit formulas
in all types for the qKreweras numbers. In the classical types A, B, C,
we also record formulas for the qNarayana numbers and in the process show that the formulas
depend only on the Weyl group (that is, they coincide in types B and C).
In addition we verify that in the classical types
A,B,C,D that the qKreweras numbers
obey the expected cyclic sieving phenomena when evaluated at
appropriate roots of unity.

(Math arXiv preprint
arXiv:1605.09172,
journal version).

Poset edge densities, nearly reduced words, and barely setvalued tableaux
(with Bridget Eileen Tenner and Alexander Yong)

In certain finite posets, the expected downdegree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young's lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0Hecke algebras; that is, factorizations that are one letter longer than the Coxeter group length.

(Math arXiv preprint
arXiv:1603.09589,
journal link)

Circuits and Hurwitz action in finite root systems
(with Joel Brewster Lewis)

In a finite real reflection group, two factorizations of a Coxeter element
into an arbitrary number of reflections are shown to lie in the same
orbit under the Hurwitz
action if and only if they use the same multiset of reflection
conjugacy classes.
The proof uses a surprising lemma, derived from a classification of
the minimal linear dependences (matroid circuits) in finite root systems:
any set of roots forming a minimal linear dependence with
positive coefficients always has a disconnected graph of
pairwise acuteness.

(Math arXiv preprint
arXiv:1603.05969,
journal version)

Chip firing on Dynkin diagrams and McKay quivers
(with Georgia Benkart and Caroline Klivans)

Two classes of avalanchefinite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chipfiring/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKayCartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKayCartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.

(Math arXiv preprint
arXiv:1601.06849,
journal version)

Absolute order in general linear groups
(with Jia Huang and Joel Brewster Lewis)

This paper studies a partial order on the general linear group GL(V) called the absolute order, derived from viewing GL(V) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including selfduality of its intervals.
Working over a finite field F_{q}, it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL_{n}(F_{q}) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.

(Math arXiv preprint
arXiv:1506.03332,
journal version)

Representation stability for cohomology of configuration spaces in R^{d}
(with Patricia Hersh)

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group S_{n} on the cohomology of the configuration space of n ordered points in R^{d}. This cohomology is known to vanish outside of dimensions divisible by d1; it is shown here that the S_{n}representation on the i(d1)st cohomology stabilizes sharply at n=3i (resp. n=3i+1) when d is odd (resp. even).
The result comes from analyzing S_{n}representations known to control the cohomology: the Whitney homology of set partition lattices for d even, and the higher Lie representations for d odd. A similar analysis shows that the homology of any rankselected subposet in the partition lattice stabilizes by n ≥ 4i, where i is the maximum rank selected.
Further properties of the Whitney homology and more refined stability statements for S_{n}isotypic components are also proven, including conjectures of J. WiltshireGordon.

(Math arXiv preprint
arXiv:1505.04196,
PDF file)

On nonconjugate Coxeter elements in wellgenerated reflection groups
(with Vivien Ripoll and Christian Stump)

ABSTRACT:
Given an irreducible wellgenerated complex reflection group W with Coxeter
number h, we show that the class of regular elements of order h form a single
orbit in W under the action of reflection automorphisms. For Coxeter and
Shephard groups, this implies that an element c is hregular if and only if
there exists a simple system S of reflections such that c is the product of the
generators in S. We moreover deduce multiple further implications of this
property. In particular, we obtain that all noncrossing partition lattices of W
associated to different regular elements of order h are isomorphic. We also
prove that there is a simply transitive action of the Galois group of the field
of definition of W on the conjugacy classes of hregular elements. Finally, we
extend several of these properties to regular elements of arbitrary order. We
show that the action of reflection automorphisms also preserves, and is
transitive on, the set of regular elements of a given order d, and we study the
action of the Galois group on conjugacy classes of dregular elements.

(Math arXiv preprint
arXiv:1404.5522)
 Invariants of GL_{n}(F_{q}) mod Frobenius powers
(with Joel Lewis and Dennis Stanton)

ABSTRACT:
Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups, one for invariants mod Frobenius powers of the irrelevant ideal, one for cofixed spaces of polynomials.

(Math arXiv preprint
arXiv:1403.6521,
journal version (Proc. Roy. Soc. Edinburgh) )
 Pseudodeterminants and perfect square spanning tree counts
(with Jeremy Martin, Molly Maxwell, and Scott O. Wilson)

ABSTRACT:
The pseudodeterminant pdet(M) of a square matrix is the last
nonzero coefficient in its characteristic polynomial;
for a nonsingular matrix, this is just the determinant.
If A is a symmetric or skewsymmetric
matrix then pdet(A A^{t})=pdet(A)^{2}.
Whenever A is the k^{th} boundary map of a selfdual CWcomplex X,
this linearalgebraic identity implies that the torsionweighted generating
function for cellular ktrees in X is a perfect square. In the case that
X is an antipodally selfdual CWsphere of odd dimension 2k1, the
pseudodeterminant of its k^{th}cellular boundary map can be interpreted
directly as a torsionweighted generating function both for ktrees and for
(k1)trees, complementing the analogous result for evendimensional spheres
given by the second author. The argument relies on the topological fact that
any selfdual evendimensional CWball can be oriented so that its middle
boundary map is skewsymmetric.

(PDF file)
 What is ... cyclic sieving?
(with Dennis Stanton and Dennis White)

ABSTRACT:
This is a short article for the Notices of the AMS on what we call the cyclic sieving phenomenon.

(PDF file)
 Reflection factorizations of Singer cycles
(with Joel Lewis and Dennis Stanton)

ABSTRACT:
The number of shortest factorizations into reflections for a Singer cycle in
GL(n,F_q) is shown to
be (q^{n} −1)^{n−1}. Formulas counting factorizations of any length, and counting those with reflections of fixed
conjugacy classes are also given.

(Math arXiv preprint
arXiv:1308.1468,
PDF file)
 Critical groups of covering, voltage and signed graphs
(with Dennis Tseng)

ABSTRACT:
Graph coverings are known to induce
surjections of their critical groups. Here we describe
the kernels of these morphisms in terms of data parametrizing the covering.
Regular coverings are parametrized by voltage graphs, and the above kernel can
be identified with a naturally defined voltage graph critical group.
For double covers, the voltage graph is a signed graph, and the theory takes a
particularly pleasant form, leading also to a theory of double covers of signed
graphs.

(Math arXiv preprint
arXiv:1301.2977)
 Toric partial orders
(with Mike Develin and Matthew Macauley)

ABSTRACT:
We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders.

(Math arXiv preprint
arXiv:1211.4247,
journal offprint)
 A universal coefficient theorem for Gauss's Lemma
(with William Messing)

ABSTRACT:
We prove a version of Gauss’s Lemma that recursively constructs polynomials
{c_{k}} for k=0,...,m+n in Z[a_{i},A_{i},b_{j},B_{j}] for i=0,1,...,m,j=0,1,...,n, of degree at most m+n, such that whenever
Σ_{k} C_{k} X^{k} =
(Σ _{i} A_{i} X^{i})
(Σ_{j} B_{j} X^{j})
and 1=Σ_{i} a_{i} A_{i}
=Σ_{j} b_{j} B_{j},
one has
1=Σ_{k} c_{k} C_{i}.

(Math arXiv preprint
arXiv:1209.6307)
 Parking spaces
(with Drew Armstrong and Brendon Rhoades)

ABSTRACT:
Let W be a Weyl group with root lattice Q and Coxeter number h.
The elements of the finite torus Q=(h+1)Q are called the Wparking functions,
and we call the permutation representation of W on the set of Wparking
functions the (standard) Wparking space. Parking spaces have interesting
connections to enumerative combinatorics, diagonal harmonics, and rational
Cherednik algebras. In this paper we define two new Wparking spaces, called
the noncrossing parking space and the algebraic parking space, with the following
features:
 They are defined more generally for real reflection groups.
 They carry not just Wactions, but W x Cactions, where C is the cyclic
subgroup of W generated by a Coxeter element.
 In the crystallographic case, both are isomorphic to the standard Wparking space.
Our Main Conjecture is that the two new parking spaces are isomorphic to each
other as permutation representations of W x C. This conjecture ties together
several threads in the Catalan combinatorics of finite reflection groups. We
provide evidence for the conjecture, proofs of some special cases, and suggest
further directions for the theory.

(Math ArXiv preprint
arXiv:1204.1760)
 A survey of the higher StasheffTamari orders
(with Joerg Rambau)

ABSTRACT:The Tamari lattice, thought as a poset on the set of triangulations of a convex polygon
with n vertices, generalizes to the higher StasheffTamari orders on the set of triangulations of a cyclic ddimensional polytope having n vertices. This survey discusses what is known about these orders, and what one would like to know about them.

(PDF file of the survey appearing in
Progress in Mathematics, Vol. 299, Birkhauser 2012)
 Fake degrees for reflection actions on roots
(with Zhiwei Yun)

ABSTRACT: 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+q^{2}+...+q^{h1}, and that in simplylaced types, it equals [h]_{q} times the summation of q^{ei1} where e_{i} runs through the exponents, so that e_{i}1 are the codegrees.

(Math ArXiv preprint
arXiv:1201.0032)
 The negative qbinomial
(with S. Fu, D. Stanton, and N. Thiem)

ABSTRACT:
Interpretations for the qbinomial coefficient evaluated at q are discussed.
A (q,t)version is established, including an instance of a cyclic sieving phenomenon
involving unitary spaces.

(Math ArXiv preprint
arxiv:1108.4702)
 A multivariate "inv" hook formula for forests
(with F. Hivert)

ABSTRACT: Björner and Wachs provided two qgeneralizations of
Knuth’s hook formula counting linear extensions of forests:
one involving the major index statistic, and one involving the
inversion number statistic. We prove a multivariate generalization
of their inversion number result, motivated by specializations
related to the modular invariant theory of finite general linear groups.

(Math ArXiv preprint
arXiv:1107.3508)
 Ppartitions revisited
(with V. Féray)

ABSTRACT: We compare a traditional and nontraditional view
on the subject of Ppartitions, leading to formulas counting
linear extensions of certain posets.

(Math ArXiv preprint
arXiv:1106.6235)
 Spectra of symmetrized shuffling operators
(with F. Saliola and V. Welker)

ABSTRACT:
For a finite real reflection group W and a
a conjugacy class of its parabolic subgroups,
we introduce a statistic on elements of W. We study
the operator of rightmultiplication within the group algebra of W by the
element whose coefficients are given by this statistic.
We interpret this geometrically in terms of the
reflection arrangement for W, and
show that these operators are selfadjoint and positive
semidefinite, via two explicit factorizations into a symmetrized form A^t A.
In one of these factorizations, A comes from the
the BidigareHanlonRockmore random walks on the chambers of an arrangement.

We use representation theory to show that for rank one parabolics in W,
the corresponding operator has integer spectrum, via
a new family of twisted Gelfand pairs for W.

For the conjugacy classes of Young subgroups of type (k,1^{nk}).
We show these operators pairwise commute, and further
conjecture that they have integer spectrum, generalizing a conjecture of
UyemuraReyes for the case k=n1, where they are the randomtorandom shuffling operators.

For the conjugacy classes of Young subgroups of type
(2^{k},1,^{n2k}), a Gelfand model for the
symmetric group is used to show that these operators pairwise commute and
have integer spectrum.

(Math ArXiv preprint
arXiv:1102.2460,
PDF file)
 The cyclotomic polynomial topologically
(with G. Musiker)

ABSTRACT:
We interpret the coefficients of the cyclotomic polynomial
in terms of simplicial homology.

(Math ArXiv preprint
arXiv:1012.1844)
 Linear extension sums as valuations of cones
(with A. Boussicault, V. Feray and A. Lascoux)

ABSTRACT:
The geometric and algebraic theory of valuations on cones is applied to understand identities involving summing certain rational functions over the set of linear extensions of a poset.

(Math ArXiv preprint
arXiv:1008.3278)
 Constructions for cyclic sieving phenomena
(with A. Berget and S.P. Eu)

ABSTRACT:
We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs.

(Math ArXiv preprint
arXiv:1004.0747)
 Diameter of reduced words
(with Y. Roichman)

ABSTRACT:
For finite reflection groups of types A and B, we determine the diameter of the
graph whose vertices are reduced words for the longest element and whose edges
are braid relations. This is deduced from a more general theorem that
applies to supersolvable hyperplane arrangements.

(Math ArXiv preprint
arXiv:0906.4768)
 Koszul incidence algebras, affine semigroups,
and StanleyReisner ideals
(with D. Stamate)

ABSTRACT:
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
CohenMacaulay property of certain posets or simplicial complexes,
and in the more general nongraded setting, via the
sequential CohenMacaulay property.

(Math ArXiv preprint
arXiv:0904.1683)
 The critical group of a line graph
(with Andrew Berget, Andrew Manion, Molly Maxwell, Aaron Potechin)

ABSTRACT:
The critical group of a graph is a finite abelian group whose order is the
number of spanning forests of the graph. This paper provides three basic
structural results on the critical group of a line graph.
The first deals with connected graphs containing no cutedge. Here the number
of independent cycles in the graph, which is known to bound the number of
generators for the critical group of the graph, is shown also to bound the
number of generators for the critical group of its line graph.
The second gives, for each prime p, a constraint on the pprimary structure
of the critical group, based on the largest power of p dividing all sums of
degrees of two adjacent vertices.
The third deals with connected graphs whose line graph is regular. Here known
results relating the number of spanning trees of the graph and of its line
graph are sharpened to exact sequences which relate their critical groups.
The first two results interact extremely well with the third. For example,
they imply that in a regular nonbipartite graph, the critical group of the
graph and that of its line graph determine each other uniquely in a simple
fashion.

(Math ArXiv preprint
arXiv:0904.1246)
 Differential posets and Smith normal forms
(with A. Miller)

ABSTRACT:
We conjecture a strong property for the up and down maps U and D in an
rdifferential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In
particular, this would determine the integral structure of the maps U, D, UD,
DU, including their ranks in any characteristic.
As evidence, we prove the conjecture for the YoungFibonacci lattice YF
studied by Okada and its rdifferential generalizations Z(r), as well as
verifying many of its consequences for Young's lattice Y and the rdifferential
Cartesian products Y^r.

(Math ArXiv preprint
arXiv:0811.1983)
 Presenting the cohomology of a Schubert variety
(with A. Woo and A. Yong)

ABSTRACT:
We extend the short presentation due to [Borel '53] of the cohomology ring
of a generalized flag manifold to a relatively short presentation
of the cohomology of any of its Schubert varieties. Our result is stated
in a rootsystem uniform manner by introducing the essential set of a
Coxeter group element, generalizing and giving a new characterization
of [Fulton '92]'s definition for permutations. Further refinements are
obtained in type A.

(Math ArXiv preprint
arXiv:0809.2981)
 Extending the Coinvariant Theorems of Chevalley, ShephardTodd, Mitchell and Springer
(with A.Broer, L. Smith and P. Webb)

ABSTRACT:
We extend in several directions invariant theory results of Chevalley, Shephard and Todd,
Mitchell and Springer. Their results compare the group algebra for a finite reflection
group with its coinvariant algebra, and compare a group representation with its module of
relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.

(Math ArXiv preprint
arXiv:0805.3694)
 (q,t)analogues and GL_{n}(F_{q})
(with D. Stanton)

ABSTRACT:
We start with a (q,t)generalization of a binomial coefficient.
It can be viewed as a polynomial in t that depends upon an integer
q, with combinatorial interpretations when q is a positive integer, and
algebraic interpretations when q is the order of a finite field.
These (q,t)binomial coefficients and their interpretations generalize further in two directions, one
relating to columnstrict tableaux and Macdonald's ``7^{th} variation'' of
Schur functions, the other relating to permutation statistics and
Hilbert series from the invariant theory of GL_{n}(F_{q})

(Math ArXiv preprint
arXiv:0804.3074)
 Betti numbers of monomial ideals and shifted skew shapes
(with U. Nagel)

ABSTRACT:
We present two new problems on lower bounds for resolution Betti numbers of
monomial ideals generated in a fixed degree. The first concerns any such ideal
and bounds the total Betti numbers, while the second concerns ideals that are
quadratic and bihomogeneous with respect to two variable sets, but gives a more
finely graded lower bound.
These problems are solved for certain classes of ideals that generalize (in
two different directions) the edge ideals of threshold graphs and Ferrers
graphs. In the process, we produce particularly simple cellular linear
resolutions for strongly stable and squarefree strongly stable ideals generated
in a fixed degree, and combinatorial interpretations for the Betti numbers of
other classes of ideals, all of which are independent of the coefficient field.

(Math ArXiv preprint
arXiv:0712.2537)
 Bimahonian distributions
(with H. Barcelo and D. Stanton)

ABSTRACT: Motivated by permutation statistics, we define for any complex reflection group W
a family of bivariate generating functions. They are defined either in terms of Hilbert series
for Winvariant polynomials when W acts diagonally on two sets of variables, or equivalently,
as sums involving the fake degrees of irreducible representations for W. It is also shown that
they satisfy a ``bicyclic sieving phenomenon'', which combinatorially interprets their values
when the two variables are set equal to certain roots of unity.

(Math ArXiv preprint
math.CO/0703479)
 Alternating subgroups of Coxeter groups
(with F. Brenti and Y. Roichman)

ABSTRACT: We study combinatorial properties of the alternating subgroup of a Coxeter group,
using a presentation of it due to Bourbaki.

(Math ArXiv preprint
math.CO/0702177)
 Cyclic sieving of noncrossing partitions for complex reflection groups
(with D. Bessis)

ABSTRACT: We prove an instance of the cyclic sieving phenomenon,
occurring in the context of noncrossing parititions for wellgenerated complex reflection groups.

(Math ArXiv preprint
math.CO/0701792)
 Shifted set families, degree sequences, and plethysm
(with C. Klivans)

ABSTRACT:
We study, in three parts, degree sequences of kfamilies (or kuniform hypergraphs) and shifted kfamilies.
The first part collects for the first time in one place, various implications such as:
Threshold implies Uniquely Realizable implies DegreeMaximal implies Shifted, which are equivalent concepts
for 2families (=simple graphs), but strict implications for kfamilies with k > 2.
The implication that uniquely realizable implies degreemaximal seems to be new.
The second part recalls Merris and Roby's reformulation of the characterization due to Ruch and Gutman
for graphical degree sequences and shifted 2families. It then introduces two generalizations which are
characterizations of shifted kfamilies. The third part recalls the connection between degree sequences of
kfamilies of size m and the plethysm of elementary symmetric functions e_m[e_k].
It then uses highest weight theory to explain how shifted kfamilies provide the ``top part'' of these
plethysm expansions, along with offering a conjecture about a further relation.

(Math ArXiv preprint
math.CO/0610787)
 Faces of Generalized Permutohedra
(with A. Postnikov and L. Williams)

ABSTRACT:
The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and gammavectors. These polytopes include permutohedra, associahedra, graphassociahedra, graphical zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas involving descent statistics and calculate generating functions. Additionally, we discuss the relationship with Simon Newcomb's problem and express hvectors for pathlike graphassociahedra in terms of the Narayana numbers. We give a combinatorial interpretation for gammavectors of treeassociahedra, confirming Gal's conjectural nonnegativity of gammavectors in this case. Included is an Appendix on deformations of simple polytopes.

(Math ArXiv preprint
math.CO/0609184)
 A quasisymmetric function for matroids
(with L.J. Billera and N. Jia)

ABSTRACT:
A new isomorphism invariant of matroids is introduced, in the form of a quasisym
metric function.
This invariant
 defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric
functions, which is surjective if one uses rational coefficients,
 is a multivariate generating function for integer weight vectors
that give minimum total weight to a unique base of the matroid,
 is equivalent, via the Hopf antipode, to a generating function for integer weight vectors
which keeps track of how many bases minimize the total weight,
 behaves simply under matroid duality,
 has a simple expansion in terms of $P$partition enumerators, and
 is a valuation on decompositions of matroid base polytopes.
This last property leads to an interesting application:
it can sometimes be used to prove that a matroid
base polytope has no decompositions into smaller matroid base polytopes. Existence of
such decompositions is a subtle issue arising in work of Lafforgue,
where lack of such a decomposition implies the matroid has only a finite number of realizations
up to scalings of vectors and overall changeofbasis.

(Math ArXiv preprint
math.CO/0606646)
 Acyclic sets of linear orders via the Bruhat orders
(with A. Galambos)

ABSTRACT:
We describe Abello's acyclic sets of linear orders as the permutations visited by
commuting equivalence classes of maximal reduced decompositions.
This allows us to strengthen Abello's structural result: we show that acyclic sets arising
from this construction are distributive sublattices of the weak Bruhat order. This, in turn, shows
that Abello's acyclic sets are, in fact, the same as ChameniNembua's "distributive covering sublattices".
Fishburn's "alternating scheme" is shown to be a special case of the
Abello/ChameniNembua acyclic sets. Any acyclic set that arises in this way can
be represented by an arrangement of pseudolines, and we use this representation to derive a
simple closed form for the cardinality of the alternating scheme. The higher Bruhat orders
prove to be a natural mathematical framework for this approach to the acyclic sets problem.

(PostScript file,
gzipped PostScript file,
PDF file)
 Coincidences among skew Schur functions
(with K. Shaw and S. van Willigenburg)

ABSTRACT:
New sufficient conditions and necessary conditions are developed for two skew diagrams to
give rise to the same skew Schur function.
The sufficient conditions come from a variety of new operations related to ribbons
(also known as border strips or rim hooks).
The necessary conditions relate to the extent of overlap among the rows or among the
columns of the skew diagram.

(Math ArXiv preprint
math.CO/0602634)
 Bergman complexes, Coxeter arrangements, and graph associahedra
(with F. Ardila and L. Williams)

ABSTRACT:
Tropical varieties play an important role in algebraic geometry.
The Bergman complex B(M) and the positive Bergman complex B+(M) of an
oriented matroid M generalize to matroids the notions of the tropical variety and
positive tropical variety associated to a linear ideal. Our main result is
that if A is a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi,
then B+(M_Phi) is dual to the graph associahedron of type Phi,
and B(M_Phi) equals the nested set complex of A.
In addition, we prove that for any orientable matroid M, one can find mu(M)
different reorientations of M such that the corresponding positive Bergman
complexes cover B(M), where mu(M) denotes the Mobius function of the lattice of flats of M.

(Math ArXiv preprint
math.CO/0508240,
Seminaire Lotharingien de Combinatoire,
Vol. B54Aj (2006), 25 pp )
 Rigidity theory for matroids
(with M. Develin and J. Martin)

ABSTRACT:
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of
barjoint frameworks in R^d in terms of the structure of the underlying
graph G. The goal of this article is to broaden the foundations of
combinatorial rigidity theory by replacing G with an arbitrary representable
matroid M. The ideas of rigidity independence and parallel
independence, as well as Laman's and Recski's combinatorial characterizations
of 2dimensional rigidity for graphs, can naturally be extended to this wider setting.
As we explain, many of these fundamental concepts really
depend only on the matroid associated with G (or its Tutte polynomial), and
have little to do with the special nature of graphic matroids or the real field.
Our main result is a ``nesting theorem'' relating the various
kinds of independence.
Immediate corollaries include generalizations of Laman's Theorem, as well as
the equality of 2rigidity and 2parallel independence.
A key tool in our study is the space of photos of M,
a natural algebraic variety whose irreducibility
is closely related to the notions of rigidity
independence and parallel independence. The number of points on this variety,
when working over a finite field,
turns out be an interesting Tutte polynomial evaluation.

(arXiv:0503050,
PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Springer's regular elements over arbitrary fields
(with D. Stanton and P. Webb)

ABSTRACT:
Springer's theory of regular elements in complex reflection groups
is generalized to arbitrary fields. Consequences for the
cyclic sieving phenomenon in combinatorics are discussed.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Stanley's simplicial poset conjecture, after M. Masuda
(with E. Miller)

ABSTRACT:
M. Masuda recently provided the missing piece proving a conjecture
of R.P. Stanley on the characterization of fvectors for Gorenstein*
simplicial posets. We propose a slight simplification of Masuda's proof.

(PDF file from journal, or
PS file,
DVI file,
PDF file of preprint)
 Finer rook equivalence for Ferrers boards: classification of
Ding's partition Schubert varieties
(with M. Develin and J. Martin)

ABSTRACT:
K. Ding studied a class of Schubert varieties
in type A partial
flag manifolds, indexed by
integer partitions and in bijection
with dominant permutations. He observed that the
Schubert cell structure of such a variety is indexed by maximal rook
placements on the Ferrers board, and that the
integral cohomology groups of two such varieties are
additively isomorphic exactly when the Ferrers boards
satisfy the combinatorial condition of rookequivalence.
We classify these varieties up to isomorphism, distinguishing them
by their graded cohomology rings with integer coefficients. The crux of our approach
is studying the nilpotence orders of linear forms in
the cohomology ring.

(PostScript file,
gzipped PostScript file,
DVI file,
PDF file)
 Reciprocal domains and CohenMacaulay dcomplexes in R^{d}
(with E. Miller)

ABSTRACT:
We extend a reciprocity theorem of Stanley about enumeration of
integer points in polyhedral cones when one exchanges strict and weak
inequalities. The proof highlights the roles played by
CohenMacaulayness and canonical modules. The extension raises the
issue of whether a CohenMacaulay complex of dimension d embedded
piecewiselinearly in dspace is necessarily a dball. This is
observed to be true for d at most 3, but false for d=4.

(PostScript file,
gzipped PostScript file,
DVI file,
PDF file)
 Cyclotomic and simplicial matroids
(with J. Martin)

ABSTRACT:
Two naturally occurring matroids representable over Q are
shown to be dual: the cyclotomic matroid represented
by the nth roots of unity inside a cyclotomic extension,
and a direct sum of copies of a certain simplicial
matroid, considered originally by Bolker in the context
of transportation polytopes.
A result of Adin leads to an upper bound for the number of
Qbases for the cyclotomic extension among the nth roots of unity, which
is tight if and only if n has at most two odd prime factors.
In addition, we study the Tutte polynomial in the case that n
has two prime factors.

(PostScript file,
gzipped PostScript file,
PDF file)
 The cyclic sieving phenomenon
(with D. Stanton and D. White)

ABSTRACT:
The cyclic sieving phenomenon is defined for
generating functions of a set affording a cyclic
group action, generalizing Stembridge's q=1 phenomenon.
The phenomenon is shown to appear in various
situations, involving qbinomial coefficients,
Polya theory, polygon dissections, noncrossing
partitions, finite reflection groups, and some finite
field qanalogues.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Noncrossing partitions for the group D_{n}
(with C. A. Athanasiadis)

ABSTRACT:
The poset of noncrossing partitions can be naturally defined for any finite
Coxeter group W. It is a selfdual, graded lattice which reduces to the
classical lattice of noncrossing partitions of {1, 2, ... ,n} defined by
Kreweras (1970) when W is the symmetric group S_n, and to its type B
analogue defined by the second author (1997) when W is the hyperoctahedral
group. We
give a combinatorial description of this lattice in terms of noncrossing
planar graphs in the case of the Coxeter group of type D_n, thus
answering a question of Bessis. Using this description, we compute a number
of fundamental enumerative invariants of this lattice, such as the rank
sizes, number of maximal chains and Moebius function.
We also extend to the
type D case the statement that noncrossing partitions are equidistributed
to nonnesting partitions by block sizes, previously known for types A, B
and C. This leads to a (casebycase) proof of a theorem valid for all
root systems: the noncrossing and nonnesting subspaces within the
intersection lattice of the Coxeter hyperplane arrangement have the same
distribution according to Worbits.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Note on the expected number of YangBaxter moves applicable
to reduced decompositions

ABSTRACT:
It is observed that the expected number of YangBaxter moves applicable
to reduced decompositions of the longest element in the symmetric group
is always 1.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Factorizations of some weighted spanning tree
enumerators
(with J. Martin)

ABSTRACT:
For two classes of graphs, threshold graphs and Cartesian products of complete
graphs, full or partial factorizations are given for spanning tree enumerators
that keep track of fine weights related to degree sequences
and edge directions.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 On the CharneyDavis and NeggersStanley Conjectures
(with V. Welker)

ABSTRACT:
For a graded naturally labelled poset P, it is shown that the
the PEulerian polynomial which counts linear extensions of P
by their number of descents has (symmetric and) unimodal coefficients.
This is deduced from McMullen's gTheorem, by exhibiting a
simplicial polytopal sphere whose hpolynomial coincides with this
PEulerian polynomial.
This simplicial sphere turns out to be flag, that is, its minimal
nonfaces all have cardinality two. As a consequence,
the NeggersStanley Conjecture on real zeroes for the PEulerian polynomial
is shown to imply the CharneyDavis Conjecture for this flag simplicial sphere.
It is speculated that the proper context in which to view both
of these conjectures may be the theory of Koszul algebras, and evidence for
this is presented.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 The CharneyDavis quantity for certain graded posets
(with D. Stanton and V. Welker)

ABSTRACT:
Given a naturally labelled graded poset P with r ranks,
the sum over its linear extensions of (1) to
the number of descents is an instance of a quantity occurring in the
CharneyDavis Conjecture on flag simplicial
spheres. When Pr is odd this quantity vanishes.
When Pr is even and P satisfies the NeggersStanley Conjecture,
it has sign (1)^{(Pr)/2}.
We interpret this quantity combinatorially for several classes
of graded posets P, including certain disjoint unions of chains and products of chains.
These interpretations involve alternating multiset permutations, Baxter
permutations, Catalan numbers, and Franel numbers.

(LaTeX file,
PostScript file,
gzipped PostScript file,
PDF file,
DVI file)
 Geochemical phase diagrams and Gale diagrams
(with P.H. Edelman, S.W. Peterson, and J.H. Stout)

ABSTRACT:
The problem of predicting the possible topologies of a geochemical
phase diagram, based on the chemical formula of the phases involved,
is shown to be intimately connected with and aided by wellstudied notions in
discrete geometry: Gale diagrams, triangulations,
secondary fans, and oriented matroids.

(PostScript file,
gzipped PostScript file,
PDF file)
 Coxeterlike complexes
(with E. Babson)

ABSTRACT:
Motivated by the Coxeter complex associated to a Coxeter system (W,S),
we introduce a simplicial regular cell complex
with a Gaction associated to any pair (G,S) where G is a group and S is a finite
set of generators for G which is minimal with respect to inclusion.
We examine the topology of this complex (G,S), and in particular the
representations of G on its homology groups.
We look closely at the case of the
symmetric group on n letters along with a choice of a minimal set of
generating transpositions. This corresponds to a choice of a spanning
tree on vertex set {1,2,...,n}.
This naturally leads to the study of a slightly larger class of
simplicial complexes, including not only the Coxeter complexes
of type A and all of their typeselected subcomplexes, but also the
wellstudied chessboard complexes.

(
Journal page in DMTCS,
PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Equivariant fiber polytopes

ABSTRACT:
The equivariant generalization of Billera and Sturmfels'
fiber polytope construction is described.
This gives a new relation between the associahedron
and cyclohedron, a different natural construction for
the type B permutohedron, and leads to a family of
orderpreserving maps between the face lattice of the type B
permutohedron and that of the cyclohedron.

(Journal page in Doc. Math.,
PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 The combinatorics of the bar resolution in group cohomology
(with P. Webb)

ABSTRACT:
We study a combinatoriallydefined double complex structure on the ordered chains of
any simplicial complex. Its columns turn out to be related to the cell complex
K_{n} whose face poset is isomorphic to the subword ordering on words without
repetition from an alphabet of size n. This complex is known to be shellable and
we provide two applications of this fact.
First, the action of the symmetric group on the homology of K_{n} gives
a representation theoretic interpretation for derangement
numbers and a related symmetric function considered by
Desarmenien and Wachs.
Second, the vanishing of homology below the top dimension for K_{n} and the
links of its faces provides a crucial step in
understanding one of the two spectral sequences associated to the double complex.
We analyze also the other spectral sequence arising from the double complex in the
case of the bar resolution for a group. This
spectral sequence converges to the cohomology of the group and provides a
method for computing group cohomology in terms of the cohomology of subgroups. Its
behavior is influenced by the complex of oriented chains of the simplicial complex
of finite subsets of the group, and we examine the Ext class of this complex.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Local cohomology modules of StanleyReisner rings
with supports in general monomial ideals
(with V. Welker and K. Yanagawa)

ABSTRACT:
We study the local cohomology modules of a StanleyReisner ring
associated to a simplicial complex with support in the ideal
corresponding to a subcomplex. We give a combinatorial
topological formula for the multigraded Hilbert series, and in the
case where the ambient complex is Gorenstein, compare this with a second
combinatorial formula that generalizes results of Mustata and Terai.
The agreement between these two formulae is seen to be a disguised
form of Alexander duality. Other results include a comparison of
the local cohomology with certain Ext modules, results about when it
it is concentrated in a single homological degree, and combinatorial
topological interpretations of some vanishing theorems.

(PostScript file,
DVI file,
LaTeX file)
 The sign representation for Shephard groups
(with A.V. Shepler and P. Orlik)

ABSTRACT:
Shephard groups are unitary reflection
groups arising as the symmetries of regular complex
polytopes. For a Shephard group, we identify the representation carried by
the principal ideal in the coinvariant algebra
generated by the image of the product of all linear forms
defining reflecting hyperplanes. This representation
turns out to have many equivalent guises making it
analogous to the sign representation of a finite Coxeter group.
One of these guises is (up to a twist) the
homology of the Milnor fiber for the isolated singularity at $0$
in the hypersurface defined by any homogeneous invariant of minimal degree.

(PostScript file,
DVI file,
LaTeX file)
 Convex, acyclic, and free sets of an oriented matroid
(with P.H. Edelman and V. Welker)

ABSTRACT:
We study the global and local topology of
three objects associated to an oriented matroid: the
lattice of convex sets, the simplicial complex of acyclic sets,
and the simplicial complex of free sets. Special cases of
these objects and their homotopy types have appeared in several
places in the literature.
The global homotopy types of all three are shown to coincide, and
are either spherical or contractible depending on whether the
oriented matroid is totally cyclic.
Analysis of the homotopy type of links of vertices in the complex
of free sets yields a generalization and more conceptual proof of a
recent result counting the interior points of a point configuration.

(PostScript file,
DVI file,
LaTeX file,
two necessary encapsulated PostScript figures:
Figure 1,
Figure 2
)
 Cohomology of smooth Schubert varieties in partial flag manifolds
(with V. Gasharov)

ABSTRACT:
We use the fact that smooth Schubert varieties in partial flag manifolds
are iterated fiber bundles over Grassmannians to give
a simple presentation for their integral cohomology ring,
generalizing Borel's presentation for the cohomology of the
partial flag manifold itself. More generally, such a presentation is shown
to hold for a larger class of subvarieties of the partial flag manifolds
(which we call subvarieties defined by inclusions). The Schubert
varieties which lie within this larger class are characterized
combinatorially by a pattern avoidance condition.

(LaTeX file,
DVI file
PostScript file)
 Note on a theorem of Eng

ABSTRACT:
We reprove a recent theorem of O. Eng that gives an instance
of Stembridge's "q=1 phenomenon" occurring in finite Coxeter groups.
Eng's proof relied on the classification of irreducible finite
Coxeter groups, whereas our proof is uniform for Weyl groups.
We apply a fact from Hodge theory to the cohomology of the homogeneous
spaces G/P, where G is a semisimple algebraic group and P a parabolic
subgroup.

(LaTeX file,
DVI file
PostScript file)
 Shifted simplicial complexes are Laplacian integral
(with A. Duval)

ABSTRACT:
We show that the combinatorial Laplace operators associated
to the boundary maps in a shifted simplicial complex have
all integer spectra. We give a simple combinatorial interpretation
for the spectra in terms of vertex degree sequences,
generalizing a theorem of Merris for graphs.
We also conjecture a majorization inequality for the spectra
of these Laplace operators in an arbitrary simplicial complex,
with equality achieved if and only if the complex is shifted.
This generalizes a conjecture of Grone and Merris for graphs.

(LaTeX file,
DVI file
PostScript file,
journal version in Trans AMS 354 (2002))
 The signature of a toric variety
(with N.C. Leung)

ABSTRACT: We identify a combinatorial quantity
(the alternating sum of the hvector) defined for any
simple polytope as the signature of a toric variety.
This quantity was introduced by Charney and Davis in
their work, which in particular showed that its nonnegativity
is closely related to a conjecture of Hopf on the Euler
characteristic of a nonpositively curved manifold.
We prove positive (or nonnegative) lower bounds for this
quantity under geometric hypotheses on the polytope.
These hypotheses lead to ampleness (or weaker conditions)
for certain line bundles on toric divisors, and then the
lower bounds follow from calculations using the Hirzebruch
Signature Formula.
Moreoever, we show that under these hypotheses on the polytope,
the ith Lclass of the corresponding toric variety is (1)^i times
an effective class for any i.

(LaTeX file,
DVI file
PostScript file)
 Counting the interior points of a point configuration
(with P. H. Edelman)

ABSTRACT: We prove a formula conjectured by Ahrens, Gordon,
and McMahon for the number of interior points for a point configuration
in affine space. Our method is to show that the formula can be interpreted
as a sum of Euler characteristics of certain complexes associated with
the point configuration, and then compute the homology of these complexes.
This method extends to other examples of convex geometries. We sketch these applications, replicating an earlier result of Gordon, and proving a new
result related to ordered sets.

(LaTeX file,
DVI file,
PostScript file)
 On the linear syzygies of a StanleyReisner ideal
 (with Volkmar Welker)
ABSTRACT: We give an elementary description of the maps in the linear strand
of the minimal free resolution of a squarefree monomial ideal, that is,
the StanleyReisner ideal associated to a
simplicial complex K. The description is in terms of
the homology of the canonical Alexander dual complex K*.

(LaTeX file,
DVI file,
PostScript file)
 A homological lower bound for order dimension of lattices
(with Volkmar Welker).

ABSTRACT: We prove that the proper part of a finite lattice of
order dimension d has vanishing homology in dimensions d1 and higher
with any coefficients.

(PostScript file,
DVI file,
PlainTeX file )
 The distribution of descents and length in a Coxeter group

ABSTRACT: We give a method for computing the "qEulerian distribution" W(t,q) for a Coxeter
system (W,S) as a rational function in t and q, where W(t,q) counts the elements of W by their
length and their number of descents (= number of elements of S which shorten them). Using this
we compute generating functions encompassing these W(t,q) of the classical infinite families of
finite and affine Weyl groups.

(Elec. J. Combinatorics
Volume 2(1), 1995, article R25.)
Some even older ones...
Signed permutation statistics
Signed permutation statistics and cycle type
Signed permutation statistics and upper binomial posets
Signed posets
My PhD thesis (scanned, with help from D. Grinberg. Lost the TeX file, sorry!)
Papers with undergraduate (summer REU) students
 The critical group of a threshold graph
(with H. Christianson, Summer 2001 REU)

ABSTRACT:
The critical group of a connected graph is a finite abelian group, whose order is
the number of spanning trees in the graph. The structure of this group
is a subtle isomorphism invariant that has received much attention recently,
partly due to its relation to the graph Laplacian and chipfiring games.
However, the group structure has been determined for relatively
few classes of graphs.
We conjecture a relation between the group structure and the Laplacian
spectrum for a large class of graphs having integer spectra (the
decomposable graphs of Kelmans). Based on computer evidence,
we conjecture the exact group structure for the wellstudied subclass of
threshold graphs, and prove this conjecture for the subclass which
we call generic threshold graphs.

(PostScript file,
gzipped PostScript file,
DVI file)
 Note on the Pfaffian MatrixTree Theorem
(with S. Hirschman, Summer 2002 REU)

ABSTRACT:
The Pfaffian MatrixTree Theorem of Masbaum and Vaintrob gives a generating
function for 3trees in a 3uniform hypergraph, analogous to Kirchoff's MatrixTree
Theorem counting trees in graphs. They prove their result via the analogue of
deletioncontraction induction. This paper gives a proof via a signreversing
involution, analogous to the proof of Kirchoff's Theorem due to Chaiken.

(PostScript file,
gzipped PostScript file,
PDF file)
 Critical groups for complete multipartite
graphs and Cartesian products of complete graphs
(with B. Jacobson and A. Niedermaier, Summer 2002 REU)

ABSTRACT:
The critical group of a connected graph is a finite abelian group, whose order is
the number of spanning trees in the graph, and which is closely
related to the graph Laplacian.
Its group structure has been determined for relatively
few classes of graphs, e.g. complete graphs, and complete bipartite graphs.
For complete multipartite graphs, we describe the
critical group structure completely. For Cartesian products of complete
graphs, we generalize results of H. Bai on the kdimensional cube, by bounding
the number of invariant factors in the critical group, and describing completely
its pprimary structure for all primes p that divide none of
the sizes of the complete graph factors.

(PostScript file,
gzipped PostScript file,
PDF file)
 Note on 1crossing partitions
(with M. Bergerson, A. Miller, A. Pliml, P. Shearer,
D. Stanton, and N. Switala, from Summer 2006 REU)

ABSTRACT:
It is shown that there are (2nr1 choose nr)
noncrossing partitions of an nset together with a distinguished block of size r,
and (n choose k1)(nr1 choose k2) of these have k blocks,
generalizing a result of Bona on partitions with one crossing.
Furthermore, when one evaluates natural qanalogues of these formulae
for q an nth root of unity of order d, one obtains the number of such objects having
dfold rotational symmetry.

(PDF file)
 Filtering cohomology of ordinary and Lagrangian Grassmannians
(with the 2020 Polymath Jr. REU "qbinomials and the Grassmannian group")

ABSTRACT:
This paper studies, for a positive integer m, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most m. We build in two ways upon a conjecture for the Hilbert series of this subalgebra due to Reiner and Tudose. The first reinterprets it in terms of the operation of kconjugation, suggesting two conjectural bases for the subalgebras that would imply their conjecture. The second introduces an analogous conjecture for the cohomology of Lagrangian Grassmannians.

(arXiv version, appeared in Involve 2022.)
Unpublished manuscripts and notes for the fun of it
 Characters and inversions in the symmetric group.
(with A. de Medicis and M. Shimozono)

ABSTRACT:
We consider sums over permutations in the symmetric group
of the value of a skew character
times q^inversions. Our main result gives a lower bound on the number of
factors of 1+q and 1q which divide the sum, and is shown to be sharp when
the skew shape is a hook shape.
(This appeared as an extended abstract in the proceedings of the 6th Formal
Power Series and Algebraic Combinatorics conference, at DIMACS in May 1994.)

(PostScript file,
Gzipped PostScript file,
PDF file,
DVI file)
 On some instances of the generalized Baues problem.

ABSTRACT: We present an approach applicable to certain instances of
the generalized Baues problem of Billera, Kapranov, and Sturmfels.
This approach involves two applications of Alexander/SpanierWhitehead
duality. We use this to show that the generalized Baues problem has
a positive answer for the surjective map of cyclic polytopes
C(n,d) > C(n,2) if n < 2d and d is at least 10.
(These results were later superseded by results of Athanasiadis, Rambau,
and Santos which verified the Baues problem positively for all of the
maps C(n,d) > C(n,d') between cyclic polytopes.)

(LaTeX file,
DVI file,
PostScript file)
 Bernstein's Theorem over fields with discrete valuation
(with S. Sperber; appendix by W. Messing)

ABSTRACT:
For fields complete with respect to a discrete valuation, we
prove a refinement of Bernstein's theorem counting the
generic number of solutions to a system of n polynomial equations
in n unknowns.
The refinement predicts the number of solutions whose coordinates have
given valuations, generalizing to several variables the classical use of
Newton polygons for determining the valuations of the roots of a
polynomial in one variable.
( We later discovered that this result was independently found by
Smirnov. )

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Conjectures on the cohomology of the Grassmannian
(with G. Tudose)

ABSTRACT:
We give a series of successively weaker conjectures on the cohomology ring of the Grassmannian,
starting with the Hilbert series of a certain natural filtration

ArXiv link
 Springer's theorem for modular coinvariants
of GL_{n}(F_{q})
(with D. Stanton and P. Webb)

ABSTRACT:
Two related results are proven in the modular invariant theory of
GL_{n}(F_{q}).
The first is a finite field analogue of a result of Springer on
coinvariants of the symmetric group in characteristic zero.
The second result is a related statement about parabolic invariants and
coinvariants.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 The Tutte polynomial of a finite projective space
(with M. Barany)

ABSTRACT:
We give a generating function for the Tutte polynomials for the arrangements
of all hyperplanes in a finite projective space.

(PostScript file,
PDF file)
 Notes on Poincare series of finite and affine Coxeter groups

ABSTRACT:
There are two famous formulae relating the Poincare series of a
a finite/affine Weyl groups to the degrees of fundamental invariants
for the finite Weyl group. We review the classical proof of the finite
formula that uses the Coxeter complex, and sketch Steinberg's analogous proof
of the affine (Bott) formula using the ``toroidal'' Coxeter complex.

(PDF file, and
L. Grau's Masters Thesis from Karlsruhe, explicating this topic further)
 An old, but cute, proof of the Catalan formula

ABSTRACT:
A few years ago, my colleague Bill Messing suggested to me a cute proof that I'd never seen,
and rather liked, of the formula for the Catalan number. It turns out to be an old proof,
but it's worth knowing.

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
 Some notes on Pólya's Theorem, Kostka numbers and the RSK correspondence
(with D. White)

ABSTRACT: These are notes that followed up on the "Pólya vs. Schensted" problems raised in Dennis White's last U. Minn. combinatorics seminar talk on Nov. 30, 2012, titled "Unfinished business".

(PDF file)
 Hopf algebras in Combinatorics
(with D. Grinberg)

ABSTRACT: Notes on Hopf algebras in combinatorics originating from a onesemester topics class at the University of Minnesota.

(arXiv link)
Book reviews
 Single and multivariable calculus ,
an opensource text by David Guichard
(coreviewed with Albert Schueller)

(MAA review)
 Combinatorics of minuscule representations,
Cambridge Tracts in Mathematics 199, by Richard M. Green

(review in PDF,
Bull. Lond. Math. Soc. web page)
Semihumorous songs honoring the mathematical legacy of one's advisor
 Countin' like the wind
(with C. Chan, I. Gessel, J. Propp, L. Rose, and B. Sagan)

ABSTRACT:
This work honors one of the greats of enumeration (and was sung at the 60th birthday
Fest for Richard P. Stanley).

(PostScript file,
gzipped PostScript file,
PDF file
DVI file)
Humorous songs written by others honoring the mathematical legacy of one's advisor
 EC1
(lyrics by Jim Propp,
music by Frank Loesser from "Guys and Dolls" arranged by Noam Elkies,
performed by S. Billey, C. Chan, K. Edwards, T. Roby,
L. Rose, M. Skandera, L. Williams, P. Winkler
)

ABSTRACT:
(This transpired at the Stanley@70 conference banquet on June 23, 2014.)

(Lyrics in PowerPoint,
Lyrics in PDF)
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