Welcome to my homepage. I am currently an Assistant Professor at the Department of Mathematics, University of Nebraska - Lincoln (UNL).

Before moving to UNL, I was a Postdoctoral Associate at the 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, including Algebraic and Enumerative Combinatorics, Bijective Combinatorics, Partition Theory and Cluster Algebras.

**My Résumé / Curriculum Vitae (Updated May 2019)**

**My Research Statement (Updated May 2019)**

Most of my papers are available on arXiv.org . However, the preprints on arXiv.org may be **slightly different** from the 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

** 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*. Preprint arXiv:1502.01679v4 .

**18) Beyond Aztec Castles: Toric Cascades in the dP3 Quiver
(with Gregg Musiker )**, * Communications in Mathematical Physics, Volume 356, Issue 3 (2017), pp. 823-881.* Preprint arXiv:1512.00507v2

**19) Perfect Matchings of Trimmed Aztec Rectangles** (33 pages), * Electronic Journal of Combinatorics, Volume 24, Issue 4 (2017), #P4.19.* Preprint arXiv: 1504.00291

** 20) Cyclically Symmetric Tilings of a Hexagon with Four Holes (with Ranjan Rohatgi)**, * Advances in Applied Mathematics, Volume 96 (2018), pp. 249-285 *. Preprint arXiv:1705.01122 .

** 21) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary** (27 pages), * SIAM Journal on Discrete Mathematics, Volume 32, No. 1 (2018), pp. 783-814. *. Preprint arXiv:1610.06284 .

**22) Proof of a Conjecture of Kenyon and Wilson on Semicontiguous Minors **, * Journal of Combinatorial Theory, Series A, Volume 161 (2019), pp. 134-163 *. Preprint arXiv:1507.02611v6 .

**23) Enumeration of lozenge tilings of a hexagon with a shamrock missing on the symmetry axis (26 pages) (with Ranjan Rohatgi )**, * Discrete Mathematics, Volume 342, Issue 2 (2019), pp. 451-472 *. Preprint arXiv:1711.02818 .

** 24) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary, Part II (26 pages)** , * Electronics Journal of Combinatorics, Volume 25, Issue 4 (2018), \# P.4.58 (34 pp) *.

**25) Lozenge Tilings of Doubly-intruded Hexagons (with Mihai Ciucu) (34 pages). Journal of Combinatorial Theory, Series A, Volume 167 (2019), pp. 294-339.**

**26) Dungeons and Dragons: Combinatorics for the dP3 Quiver
(with Gregg Musiker ), 51 pages. Accepted for Publication in Annals of Combinatorics (2019).**

**27) Enumeration of hybrid domino-lozenge tilings III: Symmetric tilings** (29 pages), * Australasian Journal of Combinatorics, Volume 74, Issue 2 (2019), 253--287*. Preprint arXiv:1609.03116

**28) A New Proof for a Triple Product Formula for Plane Partition (13pp).** Preprint arXiv:1710.02241 .

** 29) Tiling Enumeration of Doubly-intruded Halved Hexagons (35 pages). ** Preprint arXiv:1801.00249.

** 30) Lozenge Tilings of Hexagons with Central Holes and Dents (52 pages). ** Preprint arXiv:1803.02792.

** 31) Tiling Enumeration of Hexagons with Off-central Holes (59 pages). ** Preprint arXiv:1905.07119.

** 32) A Shuffling Theorem for Lozenge Tilings of Doubly-Dented Hexagons (with Ranjan Rohatgi) (12 pages). ** Preprint arXiv:1905.08311.

** 33) A Shuffling Theorem for Reflectively Symmetric Tilings (19 pages). ** Preprint arXiv:1905.09268.

** 34) New Aspects of Lozenge Tilings of Hexagons with a Shamrock Hole (with Ranjan Rohatgi). ** Preprint is available soon on arXiv.org.

** 35) A Shuffling Theorem for Centrally Symmetric Tilings. ** Preprint arXiv:1906.03759.

** 36) Tiling Shuffling Theorems and Characters of Classical Groups. ** Preprint is available by September, 2019. I am currently focusing on other projects and plan to type up the final version of the paper by September, 2019.

**37) Beyond Majority Digraphs (with Larry Moss).** In progress

** 38) Generalize Mills-Robbins-Rumsey's (q-)formula for cyclically symmetric plane partitions. ** In progress.

** 39) Generalized Symmetric Shamrock. ** In progress.

** 40) A Factorization Theorem for Hexagons with Three Bowties Removed (with Mihai Ciucu and Ranjan Rohatgi). ** In progress.

** 41) Enumeration of Tilings of Holey Hexagonal Dungeon (Summer 2019 project with grad students). ** In progress.

** 42) A 3-refinement of Tiling Enumeration of Hexagonal Dungeons (Summer 2019 project with grad students). ** In progress.

** 43) Count double-dimer configurations (Summer 2019 project with grad students). ** In progress.

** 44) More q-enumerations (Summer 2019 project with grad students). ** In progress.

** 45) A Proof of a Conjecture of Bauer, Fan and Veldman** (Undergraduate paper) (25 pages). Preprint arXiv:1309.5379

**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, 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 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".

See the review for the paper on Mathscinet "Review" .

Larry Moss has given several talks on this topic: "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.

** 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 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) 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.

**18) Beyond Aztec Castles: Toric Cascades in the dP _{3} Quiver
(with Gregg Musiker ),** Communications in Mathematical Physics, Volume 356, Issue 3 (2017), pp. 823-881. Preprint arXiv:1512.00507v2

**ABSTRACT**: We consider the dP_{3} 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 dP_{3} quiver. Moreover, the latter cluster variable can be written as a closed-form product formula.

**19) Perfect Matchings of Trimmed Aztec Rectangles**, (33 pages). *Electronic Journal of Combinatorics, Volume 24, Issue 4 (2017), #P4.19*, 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.

** 20) Cyclically Symmetric Tilings of a Hexagon with Four Holes (with Ranjan Rohatgi)**, * Advances in Applied Mathematics, Volume 96 (2018), pp. 249-285. *. 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 120^{0}. 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.

** 21) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary** (27 pages), * SIAM Journal on Discrete Mathematics, Volume 32, No. 1 (2018), pp. 783-814. *. 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.

**22) Proof of a Conjecture of Kenyon and Wilson on Semicontiguous Minors ** (33 pages), * Journal of Combinatorial Theory, Series A, Volume 161 (2019), pp. 134-163. *. Preprint arXiv:1507.02611v6 .

**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.

**23) Enumeration of lozenge tilings of a hexagon with a shamrock missing on the symmetry axis (26 pages) (with Ranjan Rohatgi )**, * Accepted in Discrete Mathematics *. Preprint arXiv:1711.02818 .

**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 Eisenkölbl's work about lozenge tilings of a hexagon with two unit triangles missing on the symmetry axis ( Electron. J. Combin. 1999).

** 24) Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary, Part II (26 pages)** , * Accepted in Electronics Journal of Combinatorics *.

**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.

**25) Lozenge Tilings of Doubly-intruded Hexagons (with Mihai Ciucu) (34pages).** * Accepted for Publication in Journal of Combinatorial Theory, Series A (2019)*.

**ABSTRACT**:
Motivated in part by Propp's intruded Aztec diamond regions, we consider hexagonal regions out of which two horizontal chains of triangular holes (called ferns) are removed, so that the chains are at the same height, and are attached to the boundary. By contrast with the intruded Aztec diamonds (whose number of domino tilings contain some large prime factors in their factorization), the number of lozenge tilings of our doubly-intruded hexagons turns out to be given by simple product formulas in which all factors are linear in the parameters. We present in fact *q*-versions of these formulas, which enumerate the corresponding plane-partitions-like structures by their volume. We also pose some natural statistical mechanics questions suggested by our set-up, which should be possible to tackle using our formulas.

**26) Dungeons and Dragons: Combinatorics for the dP3 Quiver
(with Gregg Musiker ), 52 pages. Accepted for Publication in Annals of Combinatorics (2019). ** Preprint arXiv:1805.09280 .

**ABSTRACT**:
In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work by the second author and REU students, and more recently of both authors, analyzed the cluster algebra associated to the cone over dP_{3}, the del Pezzo surface of degree 6 (CP^{2} blown up at three points). By investigating sequences of toric mutations, those occurring only at vertices with two incoming and two outgoing arrows, in this cluster algebra, we obtained a family of cluster variables that could be parameterized by Z^{3} and whose Laurent expansions had elegant combinatorial interpretations in terms of dimer partition functions (in most cases). While the earlier work focused exclusively on one possible initial seed for this cluster algebra, there are in total four relevant initial seeds (up to graph isomorphism). In the current work, we explore the combinatorics of the Laurent expansions from these other initial seeds and how this allows us to relate enumerations of perfect matchings on Dungeons to Dragons.

**27) Enumeration of hybrid domino-lozenge tilings III: Symmetric tilings.*** Accepted for Publication in Australasian Journal of Combinatorics (2019)*. 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.

**28) A New Proof for a Triple Product Formula for Plane Partitions.** Preprint arXiv:1710.02241 .

**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 Kamioka's generalization of Stanley's norm-trace formula.

** 29) Tiling Enumeration of Doubly-intruded Halved Hexagons (35pages).** Preprint arXiv:1801.00249.

**ABSTRACT**:
Inspired by Propp's intruded Aztec diamond regions, we consider halved hexagons in which two aligned arrays of triangular holes
(called ferns) have been removed from the boundary of the halved hexagons. Unlike the intruded Aztec diamonds (whose number of domino tilings contains some large
prime factors in their factorization), the number of lozenge tilings of our doubly-intruded halved hexagons is given
by simple product formulas in which all factors are linear in the parameters.
We present in fact an extensive list of tiling enumerations of sixteen different doubly-intruded halved hexagons.
We also prove that the lozenge tilings of a symmetric hexagon with three ferns removed are always enumerated
by a simple product formula.

** 30) Lozenge Tilings of Hexagons with Central Holes and Dents (52 pages). ** Preprint arXiv:1803.02792.

**ABSTRACT**:
Ciucu introduced a structure, called a `fern', consists of an arbitrary
number of equilateral triangles of alternating orientations lined up along
lattice line. He showed that the number of a hexagon in which a fern has been
removed in the center is given by a simple product formula (*Adv. Math., 2017*).
In this paper, we consider a multi-parameter generalization
of the above work by giving an explicit enumeration for lozenge tilings of
hexagons with three ferns cut off, besides the middle fern located in the
center, we remove two additional ferns from two sides of the hexagons.
Especially, our result generalizes of the `dual' of MacMahon's classical
theorem on plane partitions by Ciucu.

** 31) Tiling Enumeration of Hexagons with Off-central Holes (59 pages). ** Preprint arXiv:1905.07119.

**ABSTRACT**:
In the prequel of the paper ( arXiv:1803.02792), we considered exact enumerations of the cored versions of a doubly-intruded hexagon. The result generalized Ciucu's work about F-cored hexagons (*Adv. Math. 2017*). In this paper, we provide an extensive list of 30 tiling enumerations of hexagons with three collinear chains of triangular holes with alternating orientations. Besides two chains of holes attaching to the boundary of the hexagon, we remove one more chain of triangles that is slightly off the center of the hexagon. Two of our enumerations imply two conjectures posed by Ciucu, Eisenkölbl, Krattenthaler, and Zare (*J. Combin. Theory Ser. A 2001*) as two very special cases.

** 32) A Shuffling Theorem for Lozenge Tilings of Doubly-Dented Hexagons (with Ranjan Rohatgi) (12 pages). ** Preprint arXiv:1905.08311.

**ABSTRACT**:
MacMahon's theorem on plane partitions yields a simple product formula for the tiling number of a hexagon, and Cohn, Larsen and Propp's theorem provides an explicit enumeration for tilings of a dented semihexagon via semi-strict Gelfand-Tsetlin patterns. In this paper, we prove a natural hybrid of the two theorems for hexagons with an arbitrary set of unit triangles removed along a horizontal axis. In particular, we show that `shuffling' removed unit triangles only changes the tiling number of the region by a simple multiplicative factor. Our main result generalizes a number of known enumerations and asymptotic enumerations of tilings. We also reveal connections between the main result and the study of symmetric functions and q-series.

** 33) A Shuffling Theorem for Reflectively Symmetric Tilings (19 pages). ** Preprint arXiv:1905.09268.

**ABSTRACT**:
In ( arXiv:1905.09268), the author and Rohatgi proved a shuffling theorem for doubly--dented hexagons. In this paper, we consider the same phenomenon for the reflectively symmetric tilings of the doubly--dented hexagons. We also prove several shuffling theorems for halved hexagons. These theorems generalize a number of known results in the enumeration of halved hexagons.

** 34) New Aspects of Lozenge Tilings of Hexagons with a Shamrock Hole (with Ranjan Rohatgi). ** Preprint is available soon on arXiv.org.

**ABSTRACT**:
We consider a factorization theorem for tiling generating function of hexagons with a cluster of four triangles (called a `*shamrock*') removed. Our result gives a generalization for the work of Ciucu and Krattenthaler on hexagons with the central shamrock removed (*PNAS U.S.A., 2013*). Our factorization can be used to generalize, in several different ways, the striking asymptotic enumeration of Ciucu and Karattenthaler, called a `dual' of MacMahon's theorem on plane partitions. In the second part of the paper, we consider several exact tiling enumerations in the case when a shamrock is removed slightly off the center of the hexagon. These enumerations generalize two (ex-)conjectures posed by Ciucu, Eisenkölbl, Krattenthaler, and Zare (*J. Combin. Theory Ser. A, 2001*). We also discuss how our main theorem generalizes a number of known tiling enumerations and q-enumerations in the literatures.

** 35) A Shuffling Theorem for Centrally Symmetric Tilings. ** Preprint arXiv:1906.03759.

**ABSTRACT**:
In * arXiv:1905.08311*, Rohatgi and the author proved a shuffling theorem for lozenge tilings of doubly--dented hexagons. The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings: MacMahon's theorem about centrally symmetric hexagons and Cohn-Larsen-Prop's theorem about semihexagons with dents. In this paper, we consider a similar shuffling theorem for the centrally symmetric tilings of the doubly-dented hexagons. Our theorem generalizes a recent conjecture by Ciucu about centrally symmetric tiling of hexagons with `ferns' removed. Our theorem also implies a conjecture posed by the author in * arXiv:1803.02792* about the enumeration of centrally symmetric tilings of hexagons with three arrays of triangular holes. This enumeration, in turn, generalizes (a tiling-equivalent version of) Stanley's enumeration of self-complementary plane partitions and Ciucu's work on symmetries of the shamrock structure.

** 36) Tiling Shuffling Theorems and Characters of Classical Groups. ** Preprint is available by September, 2019. I am currently focusing on other projects with my grad students and plan to type up the final version of the paper by September, 2019.

**ABSTRACT**:
We consider various shuffling theorems for weighted lozenge tilings and the connections to irreducible characters of classical groups.

**37) Beyond Majority Digraphs (with Larry Moss).** In progress

** 38) Generalize Mills-Robbins-Rumsey's (q-)formula for cyclically symmetric plane partitions. ** In progress.

** 39) Generalized Symmetric Shamrock. ** In progress.

** 40) A Factorization Theorem for Hexagons with Three Bowties Removed (with Mihai Ciucu and Ranjan Rohatgi). ** In progress.

** 41) Enumeration of Tilings of Holey Hexagonal Dungeon (Summer 2019 project with grad students). ** In progress.

** 42) A 3-refinement of Tiling Enumeration of Hexagonal Dungeons (Summer 2019 project with grad students). ** In progress.

** 43) Count double-dimer configurations (Summer 2019 project with grad students). ** In progress.

** 44) More q-enumerations (Summer 2019 project with grad students). ** In progress.

**45) 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, 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) Discrete Mathematics Seminar, University of Nebraska-Lincoln, NE (September 20, 2016). Talk: *"Proof of a generalization of the Aztec diamond theorem."*

37) Midwest Combinatorics Conference,
University of Minnesota, Minneapolis, MN (May 23-25, 2017). Talk: *q-Enumeration of lozenge tilings.*

38) Joint Mathematics Meeting 2018,
San Diego, CA (January 10--13, 2018).

39) Geometry Seminar,
Texas A&M University, College Station TX
(Feb 9, 2018). Talk: *"Tilings and More."*

39) CombinaTexas 2018,
Texas A&M University, College Station TX
(Feb 10-11, 2018). Talk: *"Tiling Enumeration of Hexagons with Central Holes."*

40) SIAM Conference on Discrete Mathematics 2018,
Denver, CO (June 4--8, 2018). Talk * "More Duals of MacMahon's Theorem on Plane Partitions" * Slide

41) Combinatory Analysis 2018 -- A Conference in Honor of George Andrews' 80th Birthday,
Penn State University, State College, PA
(June 21-24, 2018). Talk: *"Cyclically Symmetric Lozenge Tilings of a Hexagon with
Four Holes." Slide *

42) FPSAC 2018,
Dartmouth College, Hanover NH
(July 16-20, 2018). Poster

43) Colloquium Lecture,
University of Florida, Gainesville, FL (January 11, 2019). Talk: *"Enumeration of Tilings and Related Problems."*

44) CombinaTexas 2019,
Texas A&M University, College Station TX
(March 23-24, 2018). Talk: *"Factorization Theorems for Tiling Enumerations of Regions with Holes."*

45) (Upcoming Workshop) Dimers, Ising Model, and their Interactions,
Banff, Alberta, CAN
(Nov 17-22, 2019).

46) (Upcoming Conference) Open Problems in Algebraic Combinatorics, University of Minnesota, Minneapolis, MN
(May 18-22, 2020).

**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***Summer 2013:***M118 - Finite Mathematics*-
**Fall 2013:***M119 - Brief Survey of Calculus I*

**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***Fall 2017:***M221 - Differential Equations***Spring 2018:***M958 - Topic in Discrete Mathematics: "Bijective Combinatorics"***Fall 2018:***M850 - Discrete Mathematics I***Spring 2019:***M852 - Discrete Mathematics II***Fall 2019:***M450 - Combinatorics*

Syllabus is available here

Syllabus is available here

Syllabus is available here

Syllabus is available here

Slides about Catalan Numbers that I used in my class: Catalan Objects

**Tri Lai**

Department of Mathematics

University of Nebraska - Lincoln

Office: 219 Avery Hall

Email: "tlai3 (at) unl (dot) edu"

Tel: 402-472-7222

Igor Pak's Collection of Combinatorics Videos .

Igor Pak's Catalan Numbers Page .

Xavier Viennot's Lectures Page .

Federico Ardila's Lectures Page .

Douglas West's Combinatorics Conferences Page .

Math Olympiad Resources: Art of Problem Solving

Good math books for kids: Beast Academy