Preprints:
Publications:
 31. (with
Cristian Lenart)
From the weak Bruhat order to crystal posets; also available at arXiv:1510.05636. (30 pages  version of September 14, 2016).
Mathematische Zeitschrift, 286 (2017), 14351464.
Studies crystal graphs associated to integrable highest weight representations of symmetrizable KacMoody algebras, by interpreting the crystal graphs as partially ordered sets. For lower intervals, we determine poset topology and Moebius functions, and we prove a crystaloperator analogue of Matsumoto's Theorem, namely for the statement that any two reduced expressions for the same Coxeter group element are connected by a series of braid moves. For arbitrary intervals, we give negative examples showing that the corresponding properties fail arbitrarily badly even in type A. Finally, for crystals coming from highest weight representations in the simply laced case we use the method of SBlabeling to show that any interval whose Mobius function is something other than 0,1 or 1 must include a relation amongst crystal operators not implied by Stembridge's local relations.
 30. (with
Karola Mészáros)
SBlabelings and posets with each interval homotopy equivalent to a sphere or a ball; also available at
arXiv:1407.5311. (16 pages  version of April 30, 2017).
Journal of Combinatorial Theory, Series A, 152 (2017), 104120.
Introduces a new type of poset edge labeling which implies for a finite lattice that each open interval of the order complex (nerve) is homotopy equivalent to a sphere or a ball. Such labelings are constructed for weak Bruhat order of a finite Coxeter group, for the Tamari lattice, and for finite distributive lattices.
 29. (with
Victor Reiner and with an appendix coauthored with Vic Reiner and Steven Sam)
Representation stability for cohomology of configuration spaces in R^d; also available at
arXiv:1505.04196. (42 pages  version of December 10, 2015).
IMRN, 2017 (2017), no. 4, 14331486.
Proves a new, sharp stability bound (in the sense of Thomas Church, Benson Farb, et al) for the ith cohomology of the configuration space of n distinct points in the plane by relating this to Whitney homology of the partition lattice and using symmetric functions to analyze that. Also develops other properties of this Whitney homology as a symmetric group module.
 28.
Regular cell complexes
in total positivity; also available
at
arXiv:0711.1348.
Inventiones Mathematicae, 197 (2014), no. 1, 57114.
Proves a conjecture of Fomin and Shapiro that
a collection of
stratified spaces from total positivity theory (arising out of work of Lusztig related to canonical bases) having the Bruhat intervals of a
finite Weyl group as their
closure posets are regular CW complexes homeomorphic to closed balls. This includes the space of totally nonnegative, upper triangular matrices with 1's on the diagonal and entries just above the diagonal summing to a positive constant, stratified according to which minors are positive and which are 0. A key ingredient is a new criterion for determining
whether a finite CW complex is regular with respect to a choice
of characteristic maps. It involves an interplay of combinatorics of posets of closure relations with the topology of the codimension one cell
incidences.
 27. (with
John Shareshian and
Dennis Stanton)
The q=1 phenomenon via homology concentration; also available at
arXiv:1310.7066.
Journal of Combinatorics, 5 (2014), no. 2, 167194.
Provides a topological (Euler characteristic) proof of Stembridge's
q=1 phenomenon in the cases of partitions in a rectangle and solid
partitions in a box by proving homology concentration in even degrees.
 26. (with
Ruth Davidson)
A lexicographic shellability
characterization of geometric lattices; also available at
arXiv:1108.2056.
J. Combinatorial Theory, Ser. A, 123 (2014), no. 1, 813.
Gives a new characterization of geometric lattices as those finite atomic lattices such that every atom order induces an ELlabeling.
 25. (with
Alex Engström and
Bernd Sturmfels)
Toric cubes; also available at
arXiv:1202.4333.
Rendiconti del Circolo Matematico di Palermo (2), 62 (2013), no. 1, 6778.
Studies images of monomial maps on unit cubes (i.e. on probability spaces), providing a CW decomposition for the image, generalizing known results on the edgeproduct space of phylogenetic trees. Results of Basu, Gabrielov and Vorobjov imply our CW decompositions are regular.
 24. (with
Anne Schilling)
Symmetric chain
decomposition for cyclic quotients of Boolean algebras and relation
to cyclic crystals; also available at
arXiv:1107.4073.
IMRN, 2013, (2013), 463473.
Gives a completely explicit symmetric chain decomposition of the necklace poset via a map downward through the symmetric chains that is a cyclic analogue of the GreeneKleitman SCD map and of Kashiwara's type A lowering operator from the theory of crystal bases.
 23.
(with
Drew Armstrong)
Sorting orders, subword complexes, Bruhat
order and total positivity; also available at
arXiv:0909.2828.
Advances in Applied Mathematics (special volume in honor of Dennis Stanton's 60th birthday), 46 (2011), 4653.
Provides a poset map from a Boolean algebra to Bruhat order with subword complexes as fibers and the sorting order as the suborder on distinguished elements of the fibers. This gives a new proof by the Quillen fiber lemma that each open interval in Bruhat order is homotopy equivalent to a sphere and a geometric interpretation for sorting order. The paper concludes by showing that the union of all sorting orders is Bruhat order while the intersection of all sorting orders is weak order.
 22.
(with
Cristian Lenart)
Combinatorial constructions of weight bases: the GelfandTsetlin basis (linked
to electronic journal); also available at
arXiv:0804.4719.
Electronic J. Combinatorics, 17 (2010), no. 1, R33, 14 pp.
In an effort to explicitly construct the irreducible representations of
semisimple Lie algebras in type A, Molev found rational functions
for the coefficients which arise when one applies a raising or lowering
operator to a GelfandTsetlin pattern so as to obtain a linear
combination of other GelfandTsetlin patterns. We give an explanation
for these rational functions using generalized hooklengths, and then
also give an algorithm for recursively determining these coefficients by moving across the poset.
 21. Shelling Coxeterlike complexes and sorting on trees; also available at
arXiv:0809.2414.
Advances in Mathematics, 221 (2009), no. 3, 812829.
Any finite group together with
any minimal generating set gives rise to a Coxeterlike complex, as defined
by Babson and Reiner.
We introduce notions of inversions and weak order on the labellings
of a tree, and use these to prove
a connectivity lower bound conjectured by Babson and
Reiner for Coxeterlike complexes for the symmetric group
with (not necessarily adjacent)
transpositions as the chosen generators.
The paper proves that the existence of an inversion function on
a chosen tree implies shellability of an associated Coxeterlike complex,
and also that a tree admits an inversion function iff the
tree admits greedy sorting of
its labels by swapping neighboring labels that form
inversions. Such an inversion function is constructed for trees in which
each node has ``capacity'' at least its degree minus one, and this leads
to the aforementioned connectivity bound.
 20. (with Sam Hsiao)
Random walks on quasisymmetric functions; also available at
arXiv:0709.1477.
Advances in Mathematics, 222 (2009), no. 3, 782808.
Gives conditions under which an endomorphism on QSYM gives rise to
a random walk on the descent algebra which is a lumping of a random walk
on permutations. Several important random walks are shown to fit into
this context.
 19. (with
John Shareshian and
Dennis Stanton)
The q=1 phenomenon for bounded (plane) partitions via homology concentration.
Discrete Mathematics and Theoretical Computer Science (2009), 471484. (special volume for FPSAC)
Provides a topological (Euler characteristic) proof of Stembridge's
q=1 phenomenon in the cases of partitions in a rectangle and solid
partitions in a box by proving homology concentration in even degrees. We plan to add more details in a future full paper on this project, expanding this FPSAC conference extended abstract.
 18. (with Robert Kleinberg)
A multiplicative
deformation of the Moebius function for the poset of partitions of a
multiset
Contemporary Mathematics (special volume in honor of Joe Gallian's
65th birthday), 479 (2009), 113119.
While the Moebius function of the poset of partitions of a multiset is
not well behaved, this note introduces a variant which does satisfy
very pleasant formulas. This note
is partly expository, not assuming much background in algebraic
combinatorics and touching upon a number of open questions. The target
audience is bright undergraduates, i.e. it's meant to be at an
appropriate level e.g. for students in Joe Gallian's REU.
 17. (with
Stephen Fienberg,
Alessandro Rinaldo and
Yi Zhou)
On maximum likelihood estimation in latent class models for contingency table data; also at
arXiv:0709.3535).
Book chapter in the
book ``Algebraic and geometric methods in statistics'', Cambridge University
Press, 2009. ISBN13: 9780521896191
One of the themes of this paper (i.e. the part in which I played a role) is
to study how maxima of the likelihood function with rank constraints
imposed still seem to possess the same symmetries as the unconstrained
maxima of the likelihood function; our approach uses a mixture of
combinatorics and calculus.
 16. (with Ed Swartz)
Coloring complexes and arrangements; also available at
arXiv:0706.3657
J. Algebraic Combinatorics, 27 (2008), no. 2, 205214.
Provides a convex ear decomposition for the coloring complex of any finite graph. By results of Chari, Steingrimsson, and Swartz, this imposes new restrictions on the chromatic polynomials of all finite graphs.
 15. (with
Alexander Berglund and
Jonah Blasiak)
Combinatorics of multigraded Poincaré series
for monomial rings
J. Algebra, 308 (2007), no. 1, 7390.
Expresses the denominator for Poincaré series when one resolves
a residue field over a polynomial ring modulo a monomial ideal
in terms of ranks of homology groups for intersection lattice lower
intervals for graphic arrangements.
Simplifies the denominator for several classes
of monomial ideals and also proves Golodness in new cases.
 14. (with John Shareshian) Chains of modular elements and lattice connectivity
Order, 23 (2006), no. 4, 339342.
Proves that any finite lattice with a (not necessarily maximal) chain
0 < m_1 < ... < m_r < 1 of modular elements has truncated
order complex which is at least
(r2)connected.
Update: Russ Woodroofe has proven my conjecture that the (r1)skeleton is shellable.
 13. On
optimizing discrete Morse functions ; also at
arXiv:0311270.
Advances in Appl. Math., 35 (2005), 294322.
Gives tools for improving discrete Morse functions by gradient path
reversal: (1) conditions under which several gradient paths may
simultaneously
be reversed and (2) conditions under which a single gradient path in a
lexicographic discrete Morse function is reversible. Also gives new
characterization for lexicographically first reduced words and proves
CohenMacaulayness of a new partial order on permutations which was
introduced by Jeff Remmel.
 12. (with Volkmar
Welker) Gröbner basis degree bounds on Tor^{k[\Lambda]}(k,k) and
discrete Morse theory for posets ; also at
arXiv:0312505.
Integer Points in Polyhedrageometry, number theory, algebra, optimization, 101138, Contemp. Math., 374, Amer. Math. Soc.,
Providence, RI, 2005
Constructs discrete Morse functions for monoid posets that combinatorially
relate the degree of a Groebner basis for a toric ideal of syzygies to
bounds on Betti numbers when one resolves a field over a semigroup ring,
i.e. over the coordinate ring of a toric variety. Expresses the critical
cells in such a Morse function as the words generated by a finite state
automaton, implying that the language of critical cells is a regular language,
hence the generating function for Morse numbers is rational.
Extends the lexicographic discrete Morse function construction
(as introduced in the joint paper with Babson below) in an easy, but
seemingly useful way; this extension is what allows us
to construct discrete Morse functions on monoid posets that are
consistent with an arbitrary (i.e. not necessarily lexicographic) monomial
term order.
 11. (with Eric
Babson)
Discrete Morse functions from lexicographic orders
.
Trans. Amer. Math. Soc., 357 (2005), 509534.
Note: the version posted here on my web site corrects a small error in the published version.
Gives a way of constructing a discrete Morse function with a relatively small
number of critical cells for the order complex of any finite poset.
Applies this to the poset of partitions of a multiset to show under
certain conditions, that its proper part
is homotopy equivalent to a wedge of spheres of
top dimension. It remains open whether this is true for all
multisets. (In his first paper,
Ziegler found examples which were not CohenMacaulay,
but which were contractible).
 10.
Connectivity
of hcomplexes; also at arXiv:math.CO/0311271.
J. Combinatorial Theory, Ser. A., 105 (2004),
no. 1, 111126.
Proves a conjecture of Edelman and Reiner that basically says that
the homology groups for the hcomplex of a Boolean
algebra vanish except in the middle 1/3 of possible dimensions.
Also suggests some possible generalizations.
 9. (with Phil Hanlon)
A Hodge decomposition for the complex of injective
words; also at
arXiv:math.CO/0311268.
Pacific J. Math., 214 (2004), no. 1, 109125.
Proves two main results: (1) that the Laplacian for the complex of injective
words has integral spectrum iff the randomtorandom shuffle operator does,
and (2) provides a Hodgetype
decomposition for the S_nmodule structure for
the (top) homology for the complex of injective words. Along the way, shows
that Eulerian idempotents intertwine with the simplicial boundary map on
the complex of injective words, much like they do with the boundary map for
Hochschild homology. UyemuraReyes previously conjectured
that the randomtorandom shuffle operator has integral spectrum, but as
far as I know this is still open.
 8. (with Phil Hanlon)
Multiplicity of
the trivial representation in
rankselected homology of the partition lattice; also at
math.CO/0311264.
J. Algebra, 266 (2003), no. 2, 521538.
Gives conditions under which this multiplicity is nonzero and under which it
is zero, including confirmation of two conjectures of Sundaram and short
combinatorial proofs of several results of Sundaram,
of Stanley and of Hanlon. Upper bounds on multiplicities result from
a spectral sequence argument using a filtered complex, while lower bounds come from the partitioning
for the quotient of the order complex of the
partition lattice by the action of the symmetric group (as developed in
the paper ``Lexicographic shellability for balanced complexes'' listed below). Interestingly, these two very different techniques end up both being sensitive to the same structure, allowing us to get the upper and lower bounds to coincide in many cases.
 7.
A
partitioning and related properties for
the quotient complex Delta(B_{lm})/S_l \wr S_m
(with an appendix by Vic
Reiner); also at
math.CO/0311266.
J. Pure and
Appl. Algebra, 178 (2003), no. 3, 255272.
Deduces various properties for this quotient complex: a partitioning,
collapsibility and characteristicdependent
CohenMacaulayness results. Exhibits 2torsion for each pair (l,m)
with l>2, m>1. These translate to results about the
ring k[x_1,...,x_{lm}]^{S_l \wr S_m}. The appendix proves
a previously unpublished result of Vic Reiner's from several years ago
that the paper uses, regarding
when k[x_1,...,x_n]^G is CohenMacaulay.
 6.
Lexicographic
shellability for balanced complexes; also at
math.CO/0311262.
J. Algebraic Combinatorics,
17 (2003), no. 3, 225254.
Extends lexicographic shellability from poset order complexes
to balanced simplicial complexes and
balanced boolean cell complexes (most notably to quotient complexes under a group action on the poset). Gives
a partitioning for the quotient of the partition lattice by the symmetric
group. This yields a combinatorial interpretation for the multiplicity of the
trivial representation in the rankselected homology of the
partition lattice, answering a question of Stanley.
Also provides a lexicographic shelling for Delta(B_{2n})/S_2 wr S_n,
implying the ring of invariants k[x_1,...,x_{2n}]^{S_2 wr S_n} is
CohenMacaulay, independent of field characteristic.
 5.
Chain decomposition and the flag fvector.
J. Combinatorial
Theory, Ser. A, 103 (2003), no. 1, 2752.
Provides flag fvector formulas in terms of quasisymmetric functions
for several classes of posets. These turn out to be
symmetric functions in the posets we consider, and in some cases we prove
Schurpositivity. Also obtains from "chain decompositions" some
further combinatorial formulas and other poset properties (supersolvability,
symmetric chain decompositions, etc.).
 4. (with Isabella
Novik)
A short simplicial hvector and the Upper Bound
Theorem; also at
math.CO/0111302.
Discrete and Computational Geometry, 28 (2002), no. 3,
283289.
Extends the Upper Bound Theorem
to odddimensional Eulerian complexes where the links of
all vertices satisfy the conditions of Novik's previous generalization
of the Upper Bound Theorem. Most notably, this verifies the UBC
for odddimensional Eulerian
complexes with isolated singularities (hence for
triangulations of odddimensional
pseudomanifolds with isolated singularities). Klee
conjectured the UBC for all Eulerian simplicial complexes in 1964,
but this still remains open.
 3. Two generalizations of posets of shuffles.
J. Combinatorial
Theory, Ser. A, 97 (2002), no. 1, 126.
Generalizes posets of shuffles to posets for shuffling words with repetition
of variables and for shuffling k words.
 2. Deformation of chains via a local symmetric group action.
Elec.
J. Combinatorics, R27 (1999), 18 pages. Linked to its EJC
location.
Shows that orbits of
local symmetric group actions on lattices give rise to
product of chains subposets, answering a question of Stanley.
Also provides what Stanley calls an R^*Slabelling for the
noncrossing partition lattices for classical
reflection groups.
 1. On exact nstep domination.
Discrete Mathematics, 205 (1999), 235239. Written as an undergrad in Joe Gallian's REU in Duluth, Minnesota.
Proves a conjecture of Alavi, that a graph where each vertex has a unique
vertex at distance n from it either has diameter n or is a path on 2n
vertices. Also raises the question of how small an "exact nstep domination
set" can be in terms of n and provides a lower bound of approximately log_2 n.
Ph. D. Thesis:
 Decomposition and Enumeration in Partially Ordered Sets, Ph.D.
thesis, Massachusetts Institute of Technology, May 1999.
(Advisor: Richard Stanley)
Most of the results of my thesis later appeared in
the papers "Two generalizations of posets of shuffles",
"Deformation of chains via a local symmetric group action" and "Chain
decomposition and the flag fvector".
Books edited:
 The Mathematical Legacy of Richard P. Stanley (coedited with Thomas Lam, Pavlo Pylyavskyy and Victor Reiner). Published by the American Mathematical Society, 2017.
 Selected Works of Richard P. Stanley (coedited with Thomas Lam, Pavlo Pylyavskyy and Victor Reiner). Published by the American Mathematical Society, 2017.
Unpublished Manuscripts:
 (with
Phil Hanlon
and John Shareshian)
A
GL_n(q)analogue of the partition lattice (51 pages  version of June 24, 2009). Preliminary version.
Introduces the poset PD_n(q) of partial direct sum
decompositions of a finite vector space as a poset whose GL_n(F_q)Lefschetz character is
the natural analogue of the S_nrepresentation on top homology
of the partition lattice. Proves PD_n(q) is CohenMacaulay, implying the Lefschetz character is the negative of the actual character on
top homology, and opening the door for a study of rankselected homology (in analogy with the partition lattice). The proof of the CohenMacaulay property uses lexicographic discrete Morse functions. We have been slow about submitting this paper, in hopes of simplifying the proofs (some of which are extremely technical).

CW posets after the
Poincare Conjecture. Unpublished 5 page note written in summer 2010.
This gives new sufficient conditions for a finite graded poset with
unique minimal element to be the closure poset of a regular CW complex,
namely that the poset is thin and homotopy CohenMacaulay. This is an
easy consequence of the Poincare Conjecture which to me looked like it
could be useful. This was also proven independently by Anders Björner.
 A
bijective proof of the dimension of the FussCatalan Algebras.
Unpublished 4 page note written in 1997. A nicer bijective proof by
Przytycki and Sikora appeared in JCTA 92 (2000) No. 1, 6876.
See the paper entitled "FussCatalan algebras and chains of
intermediate subfactors" by Zeph Landau in Pacific J. Math 197
(2001), 325367, for a study of FussCatalan algebras. (My motivation
was a discussion with Landau about his work.)

Approximate counting of contingency tables using Markov chains.
Undergraduate (mostly expository) thesis, completed in 1995, despite
what the title page says, due to more recent relatexing. It includes
background on Gröbner bases and combinatorics related to magic
squares. It was for a math/computer science double major and so was
aimed at either audience.
(Advisor: Persi Diaconis)
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