Bio information
Welcome to my homepage. I am currently an Assistant Professor at Department of Mathematics, University of Nebraska - Lincoln (UNL).
Before moving to UNL, I was a Postdoctoral Associate at Institute for Mathematics and its Applications (IMA), University of Minnesota. I was also a member of Combinatorics Group at the 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, Bijective Combinatorics, Cluster Algebras, and Electrical Networks.
List of Publications
Most of my papers are available on arXiv.org . However, the preprints on arXiv.org may be slightly different from the official journal versions.
Published and Accepted Papers
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
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
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
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
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
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
7) A New Proof for the Number of Lozenge Tilings of Quartered Hexagons *, Discrete Mathematics, Volume 338, Issue 11 (2015), pp. 1866-1872. Preprint arXiv:1410.8116v2
8) A Generalization of Aztec Diamond Theorem, Part II, Discrete Mathematics, Volume 339, Issue 3 (2016), pp. 1172-1179. Preprint arXiv:1310.1156v5
9) Generating Function of the Tilings of an Aztec Rectangle with Holes , Graphs and Combinatorics, Volume 32, Issue 3 (2016), pp. 1039-1054. Preprint arXiv:1402.0825v6
10) Double Aztec Rectangles, Advances in Applied Mathematics, Volume 75 (2016), pp. 1-17 Preprint arXiv:1411.0146v2
11) A Generalization of Aztec Dragons , Graphs and Combinatorics, Volume 32, Issue 5 (2016), pp. 1979-1999 Preprint arXiv:1504.00303 .
12) Majority Digraphs (with Larry Moss and Jörg Endrullis), Proceeding of the AMS, Volume 144, Number 9 (2016), pp. 3701-3715. Preprint arXiv:1509.07567.
13) Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-octagonal Regions, Electronic Journal of Combinatorics, Volume 23, Issue 2 (2016), P2.2 (25 pages). Preprint arXiv:1310.3332v4
14) Enumeration of Antisymmetric Monotone Triangles and Domino Tilings of Quartered Aztec Rectangles, Discrete Mathematics, Volume 339, Issue 5 (2016), pp. 1512-1518. Preprint arXiv:1410.8112v3
15) A q-enumeration of Lozenge Tilings of a Hexagon with Three Dents, Advances in Applied Mathematics, Volume 82 (2017), pp. 23-57. Preprint arXiv:1502.05780v5
16) Proof of a Refinement of Blum's Conjecture on Hexagonal Dungeons . Discrete Mathematics, Volume 340, Issue 7 (2017), pp. 1617-1632. Preprint arXiv:1403.4481v4
17) A q-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary. European Journal of Combinatorics, Volume 64 (2017), pp. 66-87. The initial arxiv version was entitled "A q-enumeration of generalized plane partitions", arXiv:1502.01679v4 .
Preprints and Work in Progress
18) On the Number of Perfect Matchings of Trimmed Aztec Rectangles, (33 pages). Submitted for publication, Preprint arXiv: 1504.00291
19) Proof of a Conjecture of Kenyon and Wilson on Semicontiguous Minors (26 pages). Preprint arXiv:1507.02611v5 .
20) Beyond Aztec Castles: Toric Cascades in the dP3 Quiver
(with Gregg Musiker ), Preprint arXiv:1512.00507v2
21) Enumeration of hybrid domino-lozenge tilings III: Symmetric tilings. Preprint arXiv:1609.03116
22) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed , Preprint arXiv:1610.06284 .
23) Cyclically Symmetric Tilings of a Hexagon with Four Holes (with Ranjan Rohatgi), Preprint arXiv:1705.01122 .
24) Enumeration of Lozenge Tilings of a Hexagon with Two Ferns Removed (with Mihai Ciucu). In progress.
25) Use Kuo Condensation to Prove a Trace-Generating-Function Formula of Boxed Plane Partitions. In progress.
26) Lozenge tilings of a symmetric hexagon with a shamrock missing on the symmetry axis (with Ranjan Rohatgi ). In progress.
27) Beyond Majority Digraphs (with Larry Moss). In progress
28) Beyond Aztec Castles: Toric Cascades in the dP3 Quiver, Part II
(with Gregg Musiker ). In progress.
29) Centrally and Reflectively Symmetric Tilings of a Hexagon with Three Ferns Removed, In progress.
30) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed II, In progress.
31) Generalize Mills - Robbins- Rumsey's (q-)formula for cyclically symmetric plane partitions. In progress.
32) Lozenge tilings of a hexagon with three dents/holes. In progress.
Other Papers
33) A Proof of a Conjecture of Bauer, Fan and Veldman (Undergraduate paper) (25 pages). Preprint arXiv:1309.5379
Research: Annotated list Papers and Preprints (with Abstracts and Illustrative Pictures)
Published and Accepted Papers
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 132a214⌊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, Volume 338, Issue 11 (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, Discrete Mathematics, Volume 339, Issue 3 (2016), pp. 1172-1179. Preprint arXiv:1310.1156v5
ABSTRACT: The author gave a proof of a generalization of the Aztec diamond theorem for a family of 4-vertex regions on the square lattice with southwest-to-northeast diagonals drawn in (Electron. J. Combin., 2014) by using a bijection between tilings and non-intersecting lattice paths. In this paper, we use Kuo graphical condensation to give a new proof.
9) Generating Function of the Tilings of an Aztec Rectangle with Holes , Graphs and Combinatorics, Volume 32, Issue 3 (2016), pp. 1039-1054. 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), pp. 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 , Graphs and Combinatorics, Volume 32, Issue 5 (2016), pp. 1979-1999 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), Proceeding of the AMS, Volume 144, Number 9 (2016), pp. 3701-3715. Preprint arXiv:1509.07567.
ABSTRACT: Let α∈(0, 1). A majority-digraph is a finite simple graph G such that there exist finite sets Ag for g ∈ G with the following property: g → h iff "at least α of the Ag are Ah". That is, g → h iff |Ag ∩ Ah| > α|Ag|. 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".
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) Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-octagonal Regions, Electronic Journal of Combinatorics, Volume 23, Issue 2 (2016), P2.2 (25 pages). 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.
14) Enumeration of Antisymmetric Monotone Triangles and Domino Tilings of Quartered Aztec Rectangles, Discrete Mathematics, Volume 339, Issue 5 (2016), pp. 1512-1518. 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.
15) A q-enumeration of Lozenge Tilings of a Hexagon with Three Dents, Advances in Applied Mathematics, Volume 82 (2017), pp. 23-57. Preprint arXiv:1502.05780v5
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.
16) Proof of a Refinement of Blum's Conjecture on Hexagonal Dungeons . Discrete Mathematics, Volume 340, Issue 7 (2017), pp. 1617-1632. 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 132a214⌊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) A q-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary. European Journal of Combinatorics, Volume 64 (2017), pp. 66-87. The initial arxiv version was entitled "A q-enumeration of generalized plane partitions", arXiv:1502.01679v4 .
ABSTRACT: MacMahon proved a simple product formula for the generating function of plane partitions fitting in a given box. The theorem implies a q-enumeration of lozenge tilings of a semi-regular hexagon on the triangular lattice. In this paper we generalize MacMahon's classical theorem by q-enumerating lozenge tilings of a new family of hexagons with four adjacent triangles removed from their boundary.
Preprints and Work in Progress
18) 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.
19) 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.
20) Beyond Aztec Castles: Toric Cascades in the dP3 Quiver
(with Gregg Musiker ), Preprint arXiv:1512.00507v2
ABSTRACT: We consider the dP3 quiver, and construct a family of subgraphs of the brane tiling restricted by certain 6-sided oriented contours (the direction the contour depends on its `signed side-lengths'). Our family of graphs generalizes many known families, including the Aztec Dragons, Aztec Castles, and Dragon regions. We showed that the weighted sums of perfect matchings of our graphs are equal to cluster variables arising from sequences of toric mutations in the dP3 quiver. Moreover, the latter cluster variable can be written as a closed-form product formula.
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 .
21) Enumeration of hybrid domino-lozenge tilings III: Symmetric tilings. Preprint arXiv:1609.03116
ABSTRACT: We use the subgraph replacement method to investigate new properties of regions on the square lattice with diagonals drawn in. In particular, we show that cyclically symmetric tilings of a generalization of the Aztec diamond are always enumerated by a simple product formula. We also prove an explicit product formula for the number of cyclically symmetric tilings of a quasi-hexagon.
22) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed , Preprint arXiv:1610.06284 .
ABSTRACT: Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Recently, Rohatgi extended this tiling enumeration by proving an exact tiling formula for a halved hexagon with a triangle removed from the boundary. In this paper we prove a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon with an array of adjacent triangles removed from the boundary.
23) Cyclically Symmetric Tilings of a Hexagon with Four Holes (with Ranjan Rohatgi), Preprint arXiv:1705.01122 .
ABSTRACT: The work of Mills, Robbins, and Rumsey on cyclically symmetric plane partitions yields a simple product formula for the number of lozenge tilings of a regular hexagon, which are invariant under rotation by 1200. In this paper we generalize this result by enumerating the cyclically symmetric lozenge tilings of a hexagon in which four triangles have been removed in the center.
24) Enumeration of Lozenge Tilings of a Hexagon with Two Ferns Removed (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.
25) Use Kuo Condensation to Prove a Trace-Generating-Function Formula of Boxed Plane Partitions. In progress.
ABSTRACT: Generalizing MacMahon's classical "norm generating function" formula is an important subject in study of plane partitions. Stanley 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.
26) Lozenge tilings of a symmetric hexagon with a shamrock missing on the symmetry axis (with Ranjan Rohatgi ). In progress.
ABSTRACT: In their paper about a dual of MacMahon's classical theorem about plane partitions, Ciucu and Krattenthaler proved a closed form product formula for the tiling number of a hexagon with a ``shamrock", an union of four adjacent triangles, removed in the center (Proc. Natl. Acad. Sci. USA 2013). The first author later presented a nice q-enumeration for lozenge tilings of hexagon with a shamrock removed from the boundary ( arXiv:1502.01679 ). However, these are only two positions of the shamrock hole that yield nice tiling enumerations. In this paper we show that in the case of symmetric hexagons, we always have a closed form tiling formula when removing a shamrock at any position along the symmetry axis. Our result also generalizes Eisenk\"{o}lbl's work about lozenge tilings of a hexagon with two unit triangles missing on the symmetry axis ( Electron. J. Combin. 1999).
27) Beyond Majority Digraphs (with Larry Moss). In progress
28) Beyond Aztec Castles: Toric Cascades in the dP3 Quiver, Part II
(with Gregg Musiker ). In progress.
ABSTRACT: Continuing our previous work arXiv:1512.00507v2 , we consider a number of six-sided graphs and regions, including Aztec Castles, Trimmed Aztec Rectangles, and Hexagonal Dungeons, under one roof. In particular, we show that the weighted sums of their matchings and tilings are combinatorial interpretations of certain toric mutations.
29) Centrally and Reflectively Symmetric Tilings of a Hexagon with Three Ferns Removed, In progress.
30) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed II, In progress.
ABSTRACT: Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi later extended this tiling enumeration by proving an exact tiling formula for a halved hexagon with a triangle removed from the boundary. In the previous paper ( arXiv:1610.06284 ) we proved a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon with an array of adjacent triangles removed from the non-staircase boundary. This paper is devoted to the study of lozenge tilings of a halved hexagon in which an array of adjacent triangular holes has been remove from the staircase boundary. We also investigate the case two aligned arrays of holes have been removed simultaneously from the halved hexagon. The latter yields the number of tilings of a symmetric hexagon with three arrays of holes removed.
31) Generalize Mills - Robbins- Rumsey's (q-)formula for cyclically symmetric plane partitions. In progress.
32) Lozenge tilings of a hexagon with three dents/holes. In progress.
Other Papers
33) 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 σ3(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 σ3(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.
Conferences and Talks
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, University of Mississippi, Oxford, MS (March 1-3, 2013). Invited talk: "Enumeration of Hybrid Domino-lozenge Tilings."
6) Graduate Student Combinatorics Conference 2013, University of Minnesota, Minneapolis, MN (April 19-21, 2013). 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 (October 7, 2013). Talk title: "Enumeration of Hybrid Domino-Lozenge Tilings."
9) Combinatorics Seminar, Department of Mathematics, Indiana University (October 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
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."
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). 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 about semicontiguous minors."
28) Combinatorics Seminar, University of Minnesota, Minneapolis MN (November 6, 2015) . Talk: "Proof of a conjecture of Kenyon and Wilson about 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 about semicontiguous minors."
30) Combinatorics Seminar, University of California, Los Angeles CA (November 19, 2015). Talk: "Proof of a conjecture of Kenyon and Wilson about semicontiguous minors."
31) Colloquium Lecture, University of Nebraska-Lincoln, NE (January 20, 2016). Talk: "Tiling expression of minors."
32) Discrete Mathematics Seminar, University of British Columbia, Vancouver Canada (February 2, 2016). Talk: "Enumeration of lozenge tilings of a hexagon with three dents." Slides
33) Colloquium Lecture, University of British Columbia, Vancouver Canada (February 3, 2016). Talk: "Tiling expression of minors." Slides
34) Discrete Mathematics Seminar, University of Nebraska-Lincoln, NE (September 20, 2016). Talk: "Proof of a generalization of the Aztec diamond theorem."
35) AMS Central Sectional Meeting, University of St. Thomas, Minneapolis, MN (October 28-30, 2016). Talk: "Enumeration of domino tilings of a double Aztec rectangle." Slides
36) Midwest Combinatorics Conference,
University of Minnesota, Minneapolis, MN (May 23-25, 2017). Talk: q-Enumeration of lozenge tilings.
Teaching
2005-2007: Lecturer at Hanoi National University of Education, Vietnam
2006-2008: Lecturer at FPT University, Vietnam
2008-2014: Associate Instructor at Indiana University Bloomington:
- Spring 2012: M025 - Precalculus Mathematics (2 classes),
- Summer 2012: M118 - Finite Mathematics,
- Spring 2013: M118 - Finite Mathematics,
- Fall 2013: M119 - Brief Survey of Calculus I
Here are the slides that I used in my class (based on Applied Calculus, 4th Edition, Hughes-Hallett et al.):
2016-present: Assistant Professor at University of Nebraska -Lincoln:
- Fall 2016: M314 - Linear Algebra (2 classes)
- Spring 2017: M314 - Linear Algebra
- Fall 2017: M450 Combinatorics
- Spring 2018: M958 - Topic in Discrete Mathematics
Contact information
Tri Lai
Department of Mathematics
University of Nebraska - Lincoln
Office: 339 Avery Hall
Email: "tlai3 (at) unl (dot) edu"
Tel: 402-472-7001
Useful Links
Igor Pak's Collection of Combinatorics Videos
Igor Pak's Catalan Number Page
Xavier Viennot's Lecture Page
Good math books for kids: Beast Academy
Recent project: Domino representation of semicontiguous minors
Any semicontiguous minor is the generating function of domino tilings of a region on the square lattice.