Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings of doubly-dented hexagons, which generalized the earlier work of Ciucu. Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized some other previous work of Ciucu. In this paper, we give a unified proof of these two shuffling theorems, which also covers the weighted case. Unlike the original proofs, our arguments do not use the graphical condensation method but instead rely on a well-known tiling enumeration formula due to Cohn, Larsen, and Propp. Fulmek independently found a similar proof of Lai and Rohatgi's original shuffling theorem. Our proof also gives a combinatorial explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings.
In arXiv:1905.08311, the author and Rohatgi proved a shuffling theorem for doubly-dented hexagons. In particular, we showed that shuffling removed unit triangles along a horizontal axis in a hexagon only changes the tiling number by a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, 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.
In a recent paper, Lai and Rohatgi proved a "shuffling theorem" for lozenge tilings of a hexagon with "dents" (i.e., missing triangles). Here, we shall point out that this follows immediately from the enumeration of Gelfand--Tsetlin patterns with given bottom row. This observation is also contained in a recent preprint of Byun.
Rohatgi and the author recently proved a shuffling theorem for lozenge tilings of `doubly-dented hexagons' (arXiv:1905.08311). 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 also implies a conjecture posed by the author in arXiv:1803.02792 about the enumeration of centrally symmetric tilings …
MacMahon's theorem on plane partitions yields a simple product formula for 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 the `shuffling' of removed unit triangles only changes the tiling number of the region by a simple multiplicative factor. Our main result generalizes a number of known enume…
In this article we define a generalization of the domino shuffling algorithm for tilings of the Aztec diamond to the interacting $k$-tilings recently introduced by S. Corteel, A. Gitlin, and the first author. We describe the algorithm both in terms of dynamics on a system of colored particles and as operations on the dominos themselves.
We explain the relation between certain random tiling models and interacting particle systems belonging to the anisotropic KPZ (Kardar-Parisi-Zhang) universality class in 2+1-dimensions. The link between these two \emph{a priori} disjoint sets of models is a consequence of the presence of shuffling algorithms that generate random tilings under consideration. To see the precise connection, we represent both a random tiling and the corresponding particle system through a set of non-intersecting lines, whose dynamics is induced by the shuffling algorithm or the particle dynamics. The resulting cl…
We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.
We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{λ/μ}$, where $f^{λ/μ}$ is the number of standard Young tableaux of skew shape $λ/μ$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.
In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents", which differ only in width, factors nicely, and the same is true for the quotient of generating functions of weighted lozenge tilings of two "quarter hexagons with lateral dents". Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice …
In a recent preprint, Lai worked out the quotient of generating functions of weighted lozenge tilings of two "half hexagons with lateral dents" which differ only in width. Lai achieved this by using "graphical condensation" (i.e., application of a certain Pfaffian identity to the weighted enumeration of matchings). The purpose of this note is to exhibit how this can be done by the Lindström--Gessel--Viennot method for nonintersecting lattice paths in a quite simple way. Basically the same observation, but restricted to mere enumeration (i.e., all weights of lozenge tilings are equal to $1$), i…
Proctor proved a formula for the number of lozenge tilings of a hexagon with side-lengths $a,b,c,a,b,c$ after removing a "maximal staircase." Ciucu then presented a weighted version of Proctor's result. Here we present weighted and unweighted formulas for a similar region which has an additional unit triangle removed. We use Kuo's graphical condensation method to prove the results. By applying the factorization theorem of Ciucu, we obtain a formula for the number of lozenge tilings of a hexagon with three holes on consecutive edges.
We generalize a special case of a theorem of Proctor on the enumeration of lozenge tilings of a hexagon with a maximal staircase removed, using Kuo's graphical condensation method. Additionally, we prove a formula for a weighted version of the given region. The result also extends work of Ciucu and Fischer. By applying the factorization theorem of Ciucu, we are also able to generalize a special case of MacMahon's boxed plane partition formula.
Straight polyomino tilings of rectangles and special rim-hook tableaux
University of British Columbia（不列颠哥伦比亚大学）；Indian Statistical Institute（印度统计研究所）；Indraprastha Institute of Information Technology Delhi (IIIT-Delhi)（德里因德拉普拉斯塔信息技术学院）
We obtain an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. The complexity of our expression depends on the distance from the hole to the center of the hexagon. This proves and generalizes conjectures of Ciucu et al., who considered the case of plain enumeration when the triangle is located at or very near the center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete and continuous Selberg-type integrals.
In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating fu…
Kenyon and Wilson showed how to test if a circular planar electrical network with $n$ nodes is well-connected by checking the positivity of $\binom{n}{2}$ central minors of the response matrix. Their test is based on the fact that any contiguous minor of a matrix can be expressed as a Laurent polynomial in the central minors. Moreover, the Laurent polynomial is the generating function of domino tilings of a weighted Aztec diamond. They conjectured that a larger family of minors, semicontiguous minors, can also be written in terms of domino tilings of a region on the square lattice. In this pap…
We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entro…
