Preprints:

• 28. (with Ruth Davidson) A lexicographic shellability characterization of geometric lattices (10 pages -- version of October 9, 2012); also available at arXiv:1108.2056. Submitted.
  
  Gives a new characterization of geometric lattices as those finite atomic lattices such that every atom order induces an EL-labeling.
  
  
• 27. Regular cell complexes in total positivity (60 pages -- version of May 10, 2013). Submitted.
  
  Provides a new criterion for determining whether a finite CW complex is regular (with respect to a chosen collection of characteristic maps). It involves both combinatorics of the closure poset and also the topology of the codimension one cell incidences. This is used to prove a conjecture of Fomin and Shapiro that a collection of stratified spaces from total positivity having the Bruhat intervals (of a finite Weyl group) as their closure posets are regular CW complexes homeomorphic to 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.
  
  
• 26. (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_n-representation on top homology of the partition lattice. Proves PD_n(q) is Cohen-Macaulay, implying the Lefschetz character is the negative of the actual character on top homology, and opening the door for a study of rank-selected homology (in analogy with the partition lattice). Along the way, introduces the lattice of partial partitions of a finite set, proves this is supersolvable, collapsible, a lattice, with PD_n(q) as q-analogue of the lattice of partial partitions. The proof of the Cohen-Macaulay property uses lexicographic discrete Morse functions together with properties of finite geometries to build PD_n(q) out of overlapping copies of the lattice of partial partitions. We have been slow about submitting this paper, in hopes of simplifying the proofs (some of which are extremely technical).
  
  

Publications:

• 25. (with Alex Engstrom and Bernd Sturmfels) Toric cubes (13 pages -- version of February 20, 2012); also available at arXiv:1202.4333. Rendiconti del Circolo Matematico di Palermo (2) 62 (2013), no. 1, 67--78.
  
  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 edge-product 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.
  International Math Research Notices, 2013, (2013), 463--473.
  
  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 Greene-Kleitman 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), 46--53.
  
  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 Gelfand-Tsetlin 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 Gelfand-Tsetlin pattern so as to obtain a linear combination of other Gelfand-Tsetlin patterns. We give an explanation for these rational functions using generalized hook-lengths, and then also give an algorithm for recursively determining these coefficients by moving across the poset.
  
  
• 21. Shelling Coxeter-like complexes and sorting on trees; also available at arXiv:0809.2414.
  Advances in Mathematics, 221 (2009), no. 3, 812--829.
  
  Any finite group together with any minimal generating set gives rise to a Coxeter-like 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 Coxeter-like 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 Coxeter-like 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, 782--808.
  
  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), 471--484. (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), 113--119.
  
  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. ISBN-13: 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, 205--214.
  
  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 Poincare' series for monomial rings
  J. Algebra, 308 (2007), no. 1, 73-90.
  
  Expresses the denominator for Poincare' 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, 339-342.
  
  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 (r-2)-connected.
  
  Update: Russ Woodroofe has proven my conjecture that the (r-1)-skeleton is shellable.
  
  
• 13. On optimizing discrete Morse functions ; also at arXiv:0311270.
  Advances in Appl. Math., 35 (2005), 294-322.
  
  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 Cohen-Macaulayness 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 Polyhedra--geometry, number theory, algebra, optimization, 101--138, 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), 509-534.
  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 Cohen-Macaulay, but which were contractible).
  
  
• 10. Connectivity of h-complexes; also at arXiv:math.CO/0311271.
  J. Combinatorial Theory, Ser. A., 105 (2004), no. 1, 111-126.
  
  Proves a conjecture of Edelman and Reiner that basically says that the homology groups for the h-complex 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, 109-125.
  
  Proves two main results: (1) that the Laplacian for the complex of injective words has integral spectrum iff the random-to-random shuffle operator does, and (2) provides a Hodge-type decomposition for the S_n-module 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. Uyemura-Reyes previously conjectured that the random-to-random 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 rank-selected homology of the partition lattice; also at math.CO/0311264.
  J. Algebra, 266 (2003), no. 2, 521-538.
  
  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, 255-272.
  
  Deduces various properties for this quotient complex: a partitioning, collapsibility and characteristic-dependent Cohen-Macaulayness results. Exhibits 2-torsion 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 Cohen-Macaulay.
  
  
• 6. Lexicographic shellability for balanced complexes; also at math.CO/0311262.
  J. Algebraic Combinatorics, 17 (2003), no. 3, 225-254.
  
  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 rank-selected 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 Cohen-Macaulay, independent of field characteristic.
  
  
• 5. Chain decomposition and the flag f-vector.
  J. Combinatorial Theory, Ser. A, 103 (2003), no. 1, 27-52.
  
  Provides flag f-vector formulas in terms of quasi-symmetric functions for several classes of posets. These turn out to be symmetric functions in the posets we consider, and in some cases we prove Schur-positivity. 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 h-vector and the Upper Bound Theorem; also at math.CO/0111302.
  Discrete and Computational Geometry, 28 (2002), no. 3, 283-289.
  
  Extends the Upper Bound Theorem to odd-dimensional 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 odd-dimensional Eulerian complexes with isolated singularities (hence for triangulations of odd-dimensional 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, 1-26.
  
  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^*S-labelling for the noncrossing partition lattices for classical reflection groups.
  
  
• 1. On exact n-step domination.
  Discrete Mathematics, 205 (1999), 235-239. 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 n-step 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 f-vector".

Unpublished Manuscripts:

• CW posets after the Poincare Conjecture (5 pages -- version of August, 8, 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 Cohen-Macaulay. This is an easy consequence of the Poincare Conjecture which to me looked like it could be useful. This will likely be merged with some work of Sonja Mapes.
  
  
• Shuffle posets of multisets. 13 page extended abstract for FPSAC talk in Toronto, June 1998.
  
  
• A bijective proof of the dimension of the Fuss-Catalan Algebras. Unpublished 4 page note written in 1997. A nicer bijective proof by Przytycki and Sikora appeared in JCTA 92 (2000) No. 1, 68-76. See the paper entitled "Fuss-Catalan algebras and chains of intermediate subfactors" by Zeph Landau in Pacific J. Math 197 (2001), 325-367, for a study of Fuss-Catalan 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 re-latexing. 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)