Welcome to my homepage. I am currently a Postdoctoral Associate at Institute for Mathematics and its Applications (IMA), University of Minnesota.
I got my Ph.D. from Indiana University under the supervision of Professor Mihai Ciucu. I am interested in various aspects of combinatorics, especially: Algebraic and Enumerative Combinatorics, Extremal Combinatorics and Graph Theory.
I am also a member of Combinatorics Group at University of Minnesota.
Our Combinatorics Seminar is currently organized by Michael Chmutov and my mentor, Gregg Musiker.
Most of my papers are available on arXiv.org or upon request. However, the preprints on arXiv.org may contain small typos, and may be slightly different from official journal versions.
1) Enumeration of Hybrid Domino-Lozenge Tilings, Journal of Combinatorial Theory, Series A, Volume 122, 2014, pp. 53-81. Available online at ScienceDirect or arXiv:1309.5376
ABSTRACT: We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglas' Theorem on the number tilings of a family of regions of the square lattice with every second diagonal drawn in.
2) New Aspects of Regions whose Tilings are Enumerated by Perfect Powers, Electronic Journal of Combinatorics Volume 20, Issue 4 (2013), P31 (47 pages). Available online at Combinatorics.org or arXiv:1309.6022v2
ABSTRACT: In 2003, Ciucu presented a unified way to enumerate tilings of lattice regions by using a certain Reduction Theorem (Ciucu, Perfect Matchings and Perfect Powers, Journal of Algebraic Combinatorics, 2003). In this paper we continue this line of work by investigating new families of lattice regions whose tilings are enumerated by perfect powers or products of several perfect powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on fortresses (B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, MA, 1991). On the square lattice with zigzag paths, we consider two particular families of regions whose numbers of tilings are always a power of 3 or twice a power of 3. The latter result provides a new proof for a conjecture of Matt Blum first proved by Ciucu. We also obtain a large number of new lattices by periodically applying two simple subgraph replacement rules to the square lattice. On some of those lattices, we get new families of regions whose numbers of tilings are given by products of several perfect powers. In addition, we prove a simple product formula for the number of tilings of a certain family of regions on a variant of the triangular lattice.
3) Proof of Blum's Conjecture on Hexagonal Dungeons (with Mihai Ciucu), Journal of Combinatorial Theory, Series A, Volume 125, 2014, pp. 273-305. Available online at ScienceDirect or arXiv:1402.7257
ABSTRACT: Matt Blum conjectured that the number of tilings of the Hexagonal Dungeon of sides a, 2a, b, a, 2a, b (where b ≥ 2a) is 13^{2a2}14^{⌊a2⁄2⌋}(J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present a proof for this conjecture using Kuo's Graphical Condensation Theorem (E. Kuo, Applications of Graphical Condensation for Enumerating Matchings and Tilings, Theoretical Computer Science, 2004).
One can download data for the base cases in the paper here
4) A Generalization of Aztec Diamond Theorem, Part I, Electronic Journal of Combinatorics Volume 21, Issue 1 (2014), P1.51 (19 pages). Available online at Combinatorics.org or arXiv:1310.0851
ABSTRACT: We consider a new family of 4-vertex regions with zigzag boundary on the square lattice with diagonals drawn in. By proving that the number of tilings of the new regions is given by a power 2, we generalize both Aztec diamond theorem and Douglas' theorem. The proof extends an idea of Eu and Fu for Aztec diamonds, by using a bijection between domino tilings and non-intersecting Schröder paths avoiding certain barriers, then applying Lindström-Gessel-Viennot methodology.
5) A Simple Proof for the Number of Tilings of Quartered Aztec Diamonds, Electronic Journal of Combinatorics Volume 21, Issue 1 (2014), P1.6 (13 pages). Available online at Combinatorics.org or arXiv:1309.6720
ABSTRACT: Divide an Aztec diamond region by two zigzag paths passing its center give us four quartered Aztec diamonds. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of a quartered Aztec diamond is given by a simple product formula. In this paper we give a visual proof for this result.
6) Enumeration of tilings of quartered Aztec rectangles, Electronic Journal of Combinatorics, Volume 21, Issue 4 (2014), P4.46. (28 pages). Preprint arXiv:1403.4493v3
ABSTRACT: We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the number of tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindström-Gessel-Viennot methodology to find the number of tilings of a quartered lozenge hexagon.
7) A New Proof for the Number of Lozenge Tilings of Quartered Hexagons *, Discrete Mathematics 338 (2015) pp. 1866-1872. Preprint arXiv:1410.8116v2
ABSTRACT: It has been proven that the lozenge tilings of a quartered hexagon on the triangular lattice are enumerated by a simple product formula. In this paper we give a new proof for the tiling formula by using Kuo's graphical condensation. Our result generalizes a Proctor's theorem on enumeration of plane partitions contained in a ``maximal staircase".
This work was motivated by a question of Ranjan Rohatgi in Combinatorics Seminar at Department of Mathematics, Indiana University on October 07, 2014.
(*) Based on advices of several experts in the field, the title has been changed from "A new proof for a generalization of a Proctor's formula on plane partitions" to the current title.
8) A Generalization of Aztec Diamond Theorem, Part II, Volume 339, Issue 3 (2016), 1172--1179. Discrete Mathematics 2015. Preprint arXiv:1310.1156v6
ABSTRACT: We present a new proof for a multi-parameter generalization of the Aztec diamond theorem using Kuo's graphical condensation.
9) Generating Function of the Tilings of an Aztec Rectangle with Holes (15 pages). To appear in Graphs and Combinatorics, August 2015. Preprint arXiv:1402.0825v6
ABSTRACT: We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.
10) Double Aztec Rectangles, Advances in Applied Mathematics, Volume 75 (2016), 1--17. Preprint arXiv:1411.0146v2
ABSTRACT: We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin. 1992). Moreover, we consider the connection between the generating function and MacMahon's q-enumeration of plane partitions fitting in a given box.
11) A Generalization of Aztec Dragons (19 pages). Accepted (with minor revisions) for publication in Graphs and Combinatorics, Preprint arXiv:1504.00303 .
ABSTRACT: Aztec dragons are lattice regions first introduced by James Propp, which have the number of tilings given by a power of 2. This family of regions has been investigated further by a number of authors. In this paper, we consider a generalization of the Aztec dragons to two new families of 6-sided regions. By using Kuo's graphical condensation method, we prove that the tilings of the new regions are always enumerated by powers of 2 and 3.
12) Majority Digraphs (with Larry Moss and Jörg Endrullis). Accepted for publication in Proceeding of the AMS 2015 (16 pages). Preprint arXiv:1509.07567.
ABSTRACT: Let α∈(0, 1). A majority-digraph is a finite simple graph G such that there exist finite sets A_{g} for g ∈ G with the following property: g → h iff "at least α of the A_{g} are A_{h}". That is, g → h iff |A_{g} ∩ A_{h}| > α|A_{g}|. We characterize majority-digraphs as the digraphs with the property that every directed cycle has a back-edge. This characterization is independent of α. When α= 1/2 , we apply the result to obtain a result on the logic of assertions "most X are Y".
Prof. Larry Moss has given several talks on this topic. Here is the link for the slides of the talk "Reasoning about the sizes of sets" given by him at EASLLC 2014.
13) A Note on a 2-enumeration of Antisymmetric Monotone Triangles (11 pages). Accepted for publication in Discrete Mathematics 2015, Preprint arXiv:1410.8112v3
ABSTRACT: In their unpublished work, Jockusch and Propp showed that a 2-enumeration of antisymmetric monotone triangles is given by a simple product formula. On the other hand, the author proved the same formula for the number of domino tilings of a quartered Aztec rectangle. In this paper, we give a direct proof for the equality between the 2-enumeration and the number of domino tilings by extending an idea of Jockusch and Propp.
This work was motivated by a question of Dylan Thurston in Combinatorics Seminar at Department of Mathematics, Indiana University on October 07, 2014.
14) Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-octagonal Regions (25 pages). Submitted for publication, Preprint arXiv:1310.3332v4
ABSTRACT: We use the subgraph replacement method to prove a simple product formula for the tilings of a 8-vertex counterpart of Propp's quasi-hexagon (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.
15) On the Number of Perfect Matchings of Trimmed Aztec Rectangles, (33 pages). Submitted for publication, Preprint arXiv: 1504.00291
ABSTRACT: We consider several new family of graphs obtain from Aztec rectangle and augmented Aztec rectangle graphs by trimming two opposite corners. We prove that the perfect matchings of the new graphs are enumerated by perfect powers of 2,3,5 and 11. In addition, we reveal a hidden relation between our graphs and the hexagonal dungeons introduced by Blum.
16) Proof of a Refinement of Blum's Conjecture on Hexagonal Dungeons (22 pages). Submitted for publication, Preprint arXiv:1403.4481v4
ABSTRACT: Matt Blum conjectured that the number of tilings of a hexagonal dungeon of side-lengths a,2a,b,a,2a,b (for b ≥ 2a) equals 13^{2a2}14^{⌊a2⁄2⌋}. Ciucu and the author proved the conjecture by using Kuo's graphical condensation method. In this paper, we investigate a 3-parameter refinement of the conjecture by assign to each tile a weight. In addition, we apply the result to enumerate tilings of several variations of hexagonal dungeons.
17) Proof of a Conjecture of Kenyon and Wilson on Semicontiguous Minors (26 pages). Preprint arXiv:1507.02611v5 .
ABSTRACT: In their paper on circular planar electrical networks ( arXiv:1411.7425 ), Kenyon and Wilson showed how to test if an electrical network with n nodes is well-connected by checking the positivity of n(n-1)/2 minors of the response matrix. In particular, they proved that any contiguous minor of a matrix can be expressed as a Laurent polynomial in the central minors. Interestingly, the Laurent polynomial is the generating function of domino tilings of an Aztec diamond weighted by the central minors. They conjectured that any semicontiguous minor can also be written in terms of domino tilings of a region on the square lattice. In this paper, we present a proof of the conjecture.
I would like to thank Pavlo (Pasha) Pylyavskyy for introducing the conjecture to me.
18, 19, 20, 21,22) q-enumeration of Lozenge Tilings and Boxed Plane Partitions , In progress
ABSTRACT: MacMahon proved a simple product formula for the generating function of the plane partitions fitting in a given rectangular box. The theorem implies the number of lozenge tilings of a semi-regular hexagon on the triangular lattice. By investigating the lozenge tilings of a hexagon with a hole on the boundary, we generalize the ordinary plane partitions to piles of unit cubes fitting in a union of several adjacent rectangular boxes. We extend MacMahon's classical theorem by proving that the generating function of the generalized plane partitions is given by a simple product formula.
ABSTRACT: We q-enumerate lozenge tilings of a hexagon from which three bowtie-shaped regions have been removed from three non-consecutive sides of the hexagon. The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths 2n,2n+3,2n,2n+3,2n,2n+3 (in cyclic order) with the central unit triangles on the (2n+3)-sides removed.
The third paper: "Enumeration of Lozenge Tilings of a Hexagon with an Array of Holes." (with Mihai Ciucu) In progress.
ABSTRACT: We consider a large family of hexagons with one or two arrays of holes. We show that the lozenge tilings of our region are q-enumerated by a closed form product formula. In addition, we find some interesting connection between the result and statistical mechanics and symmetric function.
The fourth paper: "Symmetric Generalized Plane Partitions." (with Ranjan Rohatgi ), In progress.
ABSTRACT: Inspired by Stanley's classical paper about 10 symmetric classes of plane partitions, we study the symmetric generalized plane partitions.
ABSTRACT: Generalizing MacMahon's classical "norm generating function" formula is an important subject in study of plane partitions. Staley introduced the "trace" of plane partitions and proved a simple product formula for the "norm-trace generating function". In this paper we use techniques in enumeration of tilings to give a new proof for a nice formula of Kamioka on a variation of the norm-trace generating function of boxed plane partitions.
23) Beyond Aztec Castles: Toric Cascades in the dP3 Quiver
(with Gregg Musiker ) (59 pages), Preprint arXiv:1512.00507
ABSTRACT: Given a super-symmetric quiver gauge theory, string theorists can associate a corresponding toric variety (which is a cone over a Calabi-Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface dP_{3}. In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables which are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the dP_{3} brane tiling for these formulas in most cases.
Gregg mentioned the result in his talk "Combinatorics of the Del-Pezzo 3 Quiver: Aztec Dragons, Castles, and Beyond" at Conference on Cluster Algebras in Combinatorics and Topology, KIAS (Korea), December 14, 2014. Gregg also gave a talk about the result at AMS Central Spring Sectional Meeting, Michigan State University, East Lansing, MI (March 14-15, 2015) Slides , and at CRM Workshop: Positive Grassmannians (July 28, 2015) Slides .
24) Beyond Majority Digraphs (with Larry Moss). In progress
ABSTRACT: We consider majority digraphs with additional structures.
25) Enumeration of hybrid domino-lozenge tilings III: Symmetric tilings, In Progress.ABSTRACT: We enumerate centrally symmetric tilings of several families of regions in the square lattice with diagonals drawn in. In particular, we prove a closed form product formula for the number of centrally symmetric tilings of a generalized Douglas region with holes along the southwest-to-northeast symmetry axis. Moreover, we show that the symmetric tilings of a quasi-hexagon are also enumerated by a simple product formula.
26) Aztec Dragon, Aztec Dungeon and perfect power (with Gregg Musiker), In Progress.ABSTRACT: We continue the work in [23] by investigating the connection between several families: Aztec Dungeons, Hexagonal Dungeons, Aztec Castles, Aztec Dragons, and Needle regions.
27) A Proof of a Conjecture of Bauer, Fan and Veldman (Undergraduate paper) (25 pages). Preprint arXiv:1309.5379
ABSTRACT: For a 1-tough graph G we define σ_{3}(G) = min{deg(u) + deg(v)+deg(w): {u; v; w} is an independent set of vertices} and NC2(G)=min {|N(u)∪ N(v)|: d(u,v)=2}. D. Bauer, G. Fan and H.J.Veldman proved that c(G)≥ min{n,2NC2(G)} for any 1-tough graph G with σ_{3}(G)≥ n ≥ 3, where c(G) is the circumference of G (D. Bauer, G. Fan and H.J.Veldman, Hamiltonian properties of graphs with large neighbourhood unions, Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: c(G)≥ min{n,2NC2(G)+4}. In this paper, we present a case-by-case proof for this conjecture.
1) The 50th Anniversary Conference at The Department of Mathematics, Mechanics and Informatics, Hanoi University of Science, Vietnam 2006. Invited talk: "A conjecture of Bauer."
2)The 14th Midrasha Mathematicae: Probability and Geometry: The Mathematics of Oded Schramm, Jerusalem, Israel Dec 2009.
3)FPT Technology Center for Young Talents Seminar, Hanoi, Vietnam, May 2010. Invited talk: "Probabilities on Trees and Graphs",
4) Vietnam Education Foundation annual conferences 2009, 2010, 2011, 2012, 2013; poster presenter in Jan 3, 2013. Poster file
5) AMS Southeast Spring Sectional Meeting, March 1-3, 2013, at the University of Mississippi, Oxford, MS. Invited talk: "Enumeration of Hybrid Domino-lozenge Tilings."
6) Graduate Student Combinatorics Conference 2013, April 19-21, 2013, at the University of Minnesota, Minneapolis, MN. Invited talk: "Subgraph Replacements in Enumeration of Tilings." Slide
7) AMS Fall Southeastern Sectional Meeting, University of Louisville, Louisville, KY, October 5-6, 2013.
8) Combinatorics Seminar, Department of Mathematics, Indiana University, Oct 7, 2013. Talk title: "Enumeration of Hybrid Domino-Lozenge Tilings."
9) Combinatorics Seminar, Department of Mathematics, Indiana University, Oct 14, 2013. Talk title: "Proof of Blum's Conjecture on Hexagonal Dungeons." Slide
10) IMA Postdoc Show and Tell Seminar, September 16, 2014. Talk: "Exact enumeration of
tilings". Poster: "Enumeration of tilings of quartered Aztec rectangles.
11) AMS Central Fall Section Meeting, University of Wisconsin-Eau Claire, Eau Claire, WI, September 20-21, 2014. Talk: "Proof of Blum's Conjecture on Hexagonal Dungeons."
12) IMA Thematic Year on Discrete Structures: Analysis and Applications (September 2014-June, 2015).
13) Combinatorics Seminar, Department of Mathematics, Indiana University (October 7, 2014). Talk: ``Enumeration of tilings of quartered Aztec rectangles". Slide
14) Combinatorics Seminar, School of Mathematics, University of Minnesota, December 05, 2014. Talk: ``Proof of a generalization of Aztec diamond theorem". Slide
15) Conference to Celebrate the Mathematics of Michelle Wachs, University of Miami (January 5 - 9, 2015)
16) IMA postdoc seminar February 3, 2015. Talk: Enumeration of lozenge tilings of a hexagon with holes on boundary. Slide
17) Central Spring Sectional Meeting, Michigan State University, East Lansing, MI (March 14-15, 2015). Talk: Lozenge tilings of a hexagon with holes on boundary and plane partitions that fit in a special box.
18) Combinatorics Seminar, Department of Mathematics, Indiana University, (March 2015) Talk title: Enumeration of lozenge tilings of a hexagon with a shamrock hole on boundary
19) ICERM Workshop Limit Shapes (April 13-17, 2015).
20) Spring Western Sectional Meeting, University of Nevada, Las Vegas NV (April 18-19, 2015). Talk: Enumeration of lozenge tilings of a hexagon with holes on boundary. Slides
21) 28th Cumberland Conference on Combinatorics, Graph Theory & Computing, University of South Carolina
Columbia, SC (May 15-17,2015). Talk: "Lozenge tilings of hexagon with holes and plane partitions fitting in a special box".
22) The 2015 Midwest Combinatorics Conference, University of Minnesota, Minneapolis MN (May 19-21, 2015)
23) 8th International Conference on Lattice Path Combinatorics and Applications, California State Polytechnic University, Pomona, CA (August 17-20, 2015) Talk: "Lozenge tilings of a hexagon with three holes".
24) IMA Postdoc Show and Tell (September 15, 2015) Talk: "Enumeration of tilings and related problems" .
25) Combinatorics Seminar, Indiana University, Bloomington IN (October 6, 2015) Talk: "Proof of a conjecture of Kenyon and Wilson on semicontiguous minors".
26) 12th ALGECOM, University of Michigan, Ann Arbor MI (October 24, 2015) Invited talk: "Proof of a conjecture of Kenyon and Wilson on semicontiguous minors".
27) Algebra and Combinatorics Seminar, North Carolina State University, Raleigh NC (November 2, 2015) Talk: "Proof of a conjecture of Kenyon and Wilson on semicontiguous minors".
28) Combinatorics Seminar, University of Minnesota, Minneapolis MN (November 6, 2015) Talk: "Proof of a conjecture of Kenyon and Wilson on semicontiguous minors".
29) Integrability and Representation Theory Seminar, University of Illinois at Urbana-Champaign, IL (November 11, 2015). Talk: "Proof of a conjecture of Kenyon and Wilson on semicontiguous minors".
30) Combinatorics Seminar, University of California, Los Angeles CA (November 19, 2015). Talk: "Proof of a conjecture of Kenyon and Wilson on semicontiguous minors".
31) (Upcoming talk) Colloquium, University of Nebraska, Lincoln, NE (January 20, 2016) . Talk title: TBA.
32) (Upcoming conference) AMS Spring Central Sectional Meeting, North Dakota State University, Fargo, ND (April 16--17, 2016) . Talk title: TBA.
33) (Upcoming conference) 18th SIAM Conference on Discrete Mathematics, Georgia State University, Atlanta, GA (June 6-10, 2016).
2005-2007: Lecturer at Hanoi National University of Education, Vietnam
2006-2008: Lecturer at FPT University, Vietnam
2008-(August) 2014: Associate Instructor at Indiana University Bloomington:
Here are the slides that I used in my class (based on Applied Calculus, 4th Edition, Hughes-Hallett et al.):
Any semicontiguous minor is the generating function of domino tilings of a region on the square lattice.