Fu Liu 刘拂

Contact info:
3220 Mathematical Science Building
(530) 754-0496
Mailing address:
Department of Mathematics
University of California, Davis
One Shields Avenue
Davis, CA 95616 USA
Research interests: Enumerative and algebraic combinatorics.

Vital biographical info: I am currently an Associate Professor in the math department at the University of California, Davis. I received my PhD in June 2006 from MIT under the guidance of Richard Stanley. Before joining Davis, I spent the Fall of 2006 at MSRI as a postdoc fellow.

Here is my curriculum vitae.

Papers and preprints

  1. Mochizuki's indigenous bundles and Ehrhart polynomials, with Brian Osserman (PDF) Spring 2004
    Journal of Algebraic Combinatorics 23 (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.

  2. Ehrhart polynomials of cyclic polytopes (arXiv) Fall 2004
    Journal of Combinatorial Theory Ser. A 111 (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.

  3. (k,m)-Catalan numbers and hook length polynomials for plane trees, with Rosena Du (arXiv) Winter 2005
    European Journal of Combinatorics 28 (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 expression.

  4. Hook length polynomials for plane forests of a certain type (arXiv) Summer 2005
    Annals of Combinatorics 13 (2009), no. 3, 315-322.
    A short note defining the hook length polynomials for plane forests of a given degree sequence type. Some other enumerative results on trees are also given.

  5. Ehrhart polynomials of lattice-face polytopes (arXiv) Fall 2005
    Transactions of the American Mathematical Society 360 (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.

  6. The irreducibility of certain pure-cycle Hurwitz spaces, with Brian Osserman (arXiv) Fall 2006
    American Journal of Mathematics 130 (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.

  7. 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 Combinatorics 30 (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.

  8. Combinatorial bases for multilinear parts of free algebras with two compatible brackets (arXiv) Spring - Summer 2008
    Journal of Algebra 323 (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.

  9. A note on lattice-face polytopes and their Ehrhart polynomials (arXiv) Summer - Fall 2008
    Proceedings of the American Mathematical Society 137 (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.

  10. Moduli of crude limit linear series (arXiv) Winter 2009
    International Mathematics Research Notices 2009 (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.

  11. Higher integrality conditions, volumes and Ehrhart polynomials (arXiv) Fall 2009
    Advances in Mathematics 226 (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.

  12. Factorizations of cycles and multi-noded rooted trees, with Rosena Du (arXiv) Summer 2010
    Graphs and Combinatorics 31 (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.

  13. 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.

  14. Perturbation of transportation polytopes (arXiv) Fall 2011 - Winter 2012
    Journal of Combinatorial Theory Ser. A 120 (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.

  15. The lecture hall parallelepiped, with Richard P. Stanley (arXiv) Winter - Spring 2012
    Annals of Combinatorics 18 (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.)

  16. A combinatorial analysis of Severi degrees (arXiv) Summer 2013
    Advances in Mathematics 298 (2016), 1-50. journal online version.
    Extended abstract (PDF), DMTCS proceedings BC (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.

  17. A distributive lattice connected with arithmetic progressions of length three, with Richard P. Stanley (arXiv) Summer 2013
    Ramanujan Journal 36 (2015), no.1, 203-226.
    Proof of two enumerative conjectures of Noam Elkies arising from a problem contributed by Ron Graham to the Numberplay subblog of the New York Times Wordplay blog.

  18. Severi degrees on toric surfaces, with Brian Osserman (arXiv) Winter 2014
    Journal fur die reine und angewandte Mathematik (Crelle's journal)i 739 (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.

  19. On bijections between monotone rooted trees and the comb basis (PDF) Summer - Fall 2014
    DMTCS proceedings FPSAC'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.

  20. Ehrhart positivity for generalized permutohedra, with Federico Castillo (PDF) Summer - Fall 2014
    DMTCS proceedings FPSAC'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.

  21. 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.

  22. Limit linear series and ranks of multiplication maps, with Brian Osserman, Montserrat Teixidor i Bigas and Naizhen Zhang (arXiv) Summer 2016 -- Fall 2016
    Submitted for publication.

  23. Smooth polytopes with negative Ehrhart coefficients, with Federico Castillo, Benjamin Nill, and Andreas Paffenholz (arXiv) Winter 2017 -- Spring 2017
    To appear in Journal of Combinatorial Theory Ser. A.

  24. Deformation cones of nested Braid fans, with Federico Castillo (arXiv) Summer 2017
    Submitted for publication.

  25. 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.

  26. On the relationship between Ehrhart unimodality and Ehrhart positivity, with Liam Solus (arXiv) Fall 2017 -- Spring 2018
    Submitted for publication.

  27. Stanley's non-Ehrhart-positive order polytopes, with Akiyoshi Tsuchiya (arXiv) Winter 2018 -- Spring 2018
    Submitted for publication.

  28. h*-polynomials with roots on the unit circle, with Ben Braun (arXiv) Fall 2017 -- Summer 2018
    Submitted for publication.

  29. The strong maximal rank conjecture and moduli spaces of curves, with with Brian Osserman, Montserrat Teixidor i Bigas and Naizhen Zhang (arXiv) Summer 2017 -- Summer 2018
    Submitted for publication.