Mochizuki's indigenous bundles and Ehrhart polynomials, with Brian Osserman (PDF) Spring 2004 Journal of Algebraic Combinatorics23 (2006), no. 2, 125-136.
A short paper applying certain finite-flatness results of Mochizuki to
obtain identities for numbers of lattice points in different polytopes,
and conversely applying the theory of Ehrhart polynomials to show that
Mochizuki's indigenous bundles are counted by polynomials in the
characteristic of the base field.
polynomials of cyclic polytopes (arXiv) Fall 2004 Journal of Combinatorial Theory Ser. A111 (2005), no. 1, 111-127.
A paper proving the conjecture made in math.CO/0402148 stating that the coefficients of the Ehrhart polynomial of an integral cyclic polytope are given by the volume of its lower envelopes.
(k,m)-Catalan numbers and hook length polynomials for plane trees, with Rosena Du (arXiv) Winter 2005 European Journal of Combinatorics28 (2007), no. 4, 1312-1321.
A short paper defining the hook length polynomials for m-ary trees and
plane forests, and introducing a new generalization of Catalan numbers, by
using which we prove our hook length polynomials have a simple binomial
Ehrhart polynomials of lattice-face polytopes (arXiv) Fall 2005 Transactions of the American Mathematical Society360 (2008), no. 6, 3041-3069.
A paper defining a new family of polytopes, lattice-face polytopes, which
is a generalization of cyclic polytopes. We show that the Ehrhart polynomial
of a lattice-polytope has the same simple form as cyclic polytopes.
The irreducibility of certain pure-cycle Hurwitz spaces, with Brian Osserman (arXiv) Fall
2006 American Journal of Mathematics130 (2008), no. 6, 1687-1708.
We use a combination of geometric and group-theoretic techniques to prove
that Hurwitz spaces of genus-0 covers of the projective line having a single
ramified point over each branch point are irreducible.
A generating function for all semi-magic squares and the volume of the Birkhoff polytope, with Jesus A. De Loera and Ruriko Yoshida (arXiv) Fall 2006 - Winter 2007 Journal of Algebraic Combinatorics30 (2009), no. 1, 113-139.
We provide an explicit combinatorial formula for the volume of the polytope of n by n doubly-stochastic matrices, also known as the Birkhoff polytope. We do this through finding the multivariate generating function for the lattice points of the polytope.
Combinatorial bases for multilinear parts of free algebras with two compatible brackets (arXiv) Spring - Summer 2008 Journal of Algebra323 (2010), no. 1, 132-166.
We construct bases for Lie_2(n) and P_2(n) from combinatorial objects, the set of rooted trees, then prove the dimension formulas for these two algebras conjectured by B. Feigin. We also define a complementary space Eil_2(n) to Lie_2(n), give a pairing between them, and show that the pairing is perfect.
A note on lattice-face polytopes and their Ehrhart polynomials (arXiv) Summer - Fall 2008 Proceedings of the American Mathematical Society137 (2009), no. 10, 3247-3258.
We redefine lattice-face polytope by removing an unnecessary restriction in the old definition and show that the Ehrhart polynomial of a new lattice-face poltyope has the same simple form as the old ones. Furthermore, we show that the new family of lattice-face polytopes contains all possible combinatorial types of rational polytopes.
Moduli of crude limit linear series (arXiv) Winter 2009 International Mathematics Research Notices2009 (2009), no. 21, 4032-4050.
We answer a combinatorial question posed by Osserman, giving a description of the dimensions of the spaces of crude limit linear series that he introduced.
Higher integrality conditions, volumes and Ehrhart polynomials (arXiv) Fall 2009 Advances in Mathematics226 (2011), no. 4, 3467-3494.
We introduce the definition of k-integral polytopes, and show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees of less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.
Factorizations of cycles and multi-noded rooted trees, with Rosena Du (arXiv) Summer 2010 Graphs and Combinatorics31 (2015), no. 3, 551-575.
Pure-cycle Hurwitz numbers count the number of connected branched covers of the projective line where each branch point has only one ramification point over it. We prove that when the genus is 0 and one of the ramification indices is d, the degree of the covers, the pure-cycle Hurwitz number is dr-3. We give the first desymmetrized bijective proof of this result by constructing a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees.
Perturbation of central transportation polytopes of order kn x n (PDF) Fall 2011 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 961-973, Discrete Math. Theor. Comput. Sci. Proc., AR, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012.
journal online version.
We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation, we obtain combinatorial formulas for the MGF of the central transportation polytopes of order kn x n. We also recover the formula for the maximum possible number of vertices of transportation polytopes of order kn x n.
Perturbation of transportation polytopes (arXiv) Fall 2011 - Winter 2012 Journal of Combinatorial Theory Ser. A120 (2013), no. 7, 1539--1561.
An extended and improved version of "Perturbation of central transportation polytopes of order kn x n". An unnecessary condition in the perturbation method is removed.
The lecture hall parallelepiped, with Richard P. Stanley (arXiv) Winter - Spring 2012 Annals of Combinatorics18 (2014), no. 3, 473-488.
We introduce the s-lecture hall parallelepiped, which we show can be used to find the Ehrhart polynomial of an s-lecture hall polytope. We define bijections between the lattice points inside s-lecture hall parallelepiped and fundamental combinatorial sets. Using these, we are able to show that the s-lecture hall polytope has the same Ehrhart polynomial as the unit cube when s = (n, n-1, ... , 1) or s = (1, 2, ... , n). (The latter case was first proved by Savage and Schuster.)
A combinatorial analysis of Severi degrees (arXiv) Summer 2013 Advances in Mathematics298 (2016), 1-50. journal online version. Extended abstract (PDF),
DMTCS proceedingsBC (2016), 779-790.
Based on results by Brugalle and Mikhalkin, Fomin and Mikhalkin give formulas for computing the classical Severi degree using long-edge graphs. Motivated by a conjecture of Block-Colley-Kennedy, we consider a special multivariate function associated to long-edge graphs, and show that this function is always linear.
Severi degrees on toric surfaces, with Brian Osserman (arXiv) Winter 2014 Journal fur die reine und angewandte Mathematik (Crelle's journal)i739 (2018), 121-158.
Builds on work of Brugalle and Mikhalkin, Ardila and Block, and the author to give universal formulas for the number of nodal curves in a linear system on a certain family of (possibly singular) toric surfaces. These formulas are explicitly related to the Goettsche-Yau-Zaslow formula, and are used to give combinatorial expressions for the coefficients arising in the latter.
On bijections between monotone rooted trees and the comb basis (PDF) Summer - Fall 2014 DMTCS proceedingsFPSAC'15 (2015), 453-464.
Gozalez D'Leon and Wachs, in their study of (co)homology of the poset of weighted partitions, asked whether there are nice bijections between RA,i, the set of rooted trees on A with i decreasing edges, and the comb basis or the Lyndon basis (for the cohomology). We give a natural definition for "nice bijections", and conjecture that there is a unique nice bijection between RA,i and the comb basis. We confirm the conjecture for the extreme cases where i=0 or n-1.
Ehrhart positivity for generalized permutohedra, with Federico Castillo (PDF) Summer - Fall 2014 DMTCS proceedingsFPSAC'15 (2015), 865-876.
We conjecture that the generalized permutohedra have positive Ehrhart coefficients, generalizing a conjecture by DeLoera-Haws-Koeppe. Using the combination of perturbation methods and a valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne, we reduce the conjecture to a new conjecture: the Berlne-Vergne's valuation is positive on regular permutohedra. We then show our two conjectures hold for small dimension cases.
Berline-Vergne valuation and generalized permutohedra, with Federico Castillo (arXiv) Winter 2015 -- Fall 2015 Discrete and Computational Geometry, to appear. journal online version.
Continuing work in "Ehrhart positivity for generalized permutohedra", we use Berline-Vergne's valuation to study our conjecture on the Ehrhart positivity of generalized permutohedra, and the stronger conjecture that the Berline-Vergne valuation is positive on regular permutohedra. We show our conjectures hold for dimension up to 6, and for faces of codimension up to 3. We also give two equivalent statements to the second conjecture in terms of mixed valuations and Todd class, respectively.
On positivity of Ehrhart polynomials (arXiv) Fall 2017
To appear in Recent Trends in Algebraic Combinatorics, a volume of the Association for Women in Mathematics Series, Springer International Publishing.
Ehrhart polynomial counts the number of lattice points inside dilations of integral polytopes. We say a polytope is Ehrhart positive if all the coefficients of its Ehrhart polynomial is positive, and ask when a polytope is Ehrhart positive. In this article, we discuss techniques attacking this problem, and survey known results (positive and negative) on interesting famiies of polytopes.