Plane Partitions Research Articles

Lozenge tilings with free boundary

We study tilings with lozenges of a domain with free boundary conditions on one side. These correspondto boxed symmetric plane partitions. We show that the positions of the horizontal lozenges near the left flatboundary, in the limit, have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (theGUE-corners/minors process). We also prove the existence of a limit shape of the height function (the symmetricplane partition). We also consider domains where the sides converge to $\infty$ at different rates and recover again theGUE-corners process. Nous étudions les pavages par losanges d’un domaine dont le bord vertical est “libre”. Nous montrons queles positions des losanges horizontaux proches du bord gauche ont la même distribution que les valeurs propres del’ensemble gaussien unitaire. Nous montrons aussi l’existence d’une limite de la forme de la fonction de hauteur (unepartition plane symétrique). Nous considérons aussi des domaines ou des bords différents convergent vers $\infty$ destaux différents et nous retrouvons nouveau les processus EGU au bord.

Subwords and Plane Partitions

Using the powerful machinery available for reduced words of type $B$, we demonstrate a bijection between centrally symmetric $k$-triangulations of a $2(n + k)$-gon and plane partitions of height at most $k$ in a square of size $n$. This bijection can be viewed as the type $B$ analogue of a bijection for $k$-triangulations due to L. Serrano and C. Stump. En utilisant la machinerie puissante pour mots réduits de type $B$, nous démontrons une bijection entre les $k$-triangulations centralement symétriques d’un $2(n + k)$-gon et les partitions de plans de hauteur inférieure ou égale à $k$ dans un carré de taille $n$. Cette bijection peut être considérée comme l’analogue de type $B$ d’une bijection de $k$-triangulations due à L. Serrano et C. Stump.

Estimating the Asymptotics of Solid Partitions

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $\lim_{n\rightarrow\infty} n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001$. This shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $\log p_3(n)$. In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to $n^{1/4}$, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.

Enumeration of plane partitions with a restricted number of parts

We use the quantum statistical approach to estimate the number of restricted plane partitions of an integer n with the number of parts not exceeding some finite N. We use the analogy between this number theory problem and the enumeration of microstates of the ideal two-dimensional Bose gas. The numbers of restricted plane partitions calculated with the conjectured expression agree well with the exact values for n from 10 to 20.

K-theoretic boson–fermion correspondence and melting crystals

We study non-Hermitian integrable fermion and boson systems from the perspectives of Grothendieck polynomials. The models considered in this article are the five-vertex model as a fermion system and the non-Hermitian phase model as a boson system. Both models are characterized by different solutions satisfying the same Yang–Baxter relation. From our previous works on the identification between the wavefunctions of the five-vertex model and Grothendieck polynomials, we introduce skew Grothendieck polynomials and derive the addition theorem among them. Using these relations, we derive the wavefunctions of the non-Hermitian phase model as a determinant form, which can also be expressed as Grothendieck polynomials. Namely, we establish a K-theoretic boson–fermion correspondence at the level of wavefunctions. As a by-product, the partition function of the statistical mechanical model of a three-dimensional (3D) melting crystal is exactly calculated by use of the scalar products of the wavefunctions of the phase model. The resultant expression can be regarded as a K-theoretic generalization of the MacMahon function describing the generating function of the plane partitions, which interpolates the generating functions of two-dimensional (2D) and (3D) Young diagrams.

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

Combinatorics of a strongly coupled boson system

We introduce a quantum phase model as a limit for very strong interactions of a strongly correlated q-boson hopping model. We describe the general solution of the phase model and express scalar products of state vectors in determinant form. The representation of state vectors in terms of Schur functions allows obtaining a combinatorial interpretation of the scalar products in terms of nests of self-avoiding lattice paths. We show that under a special parameterization, the scalar products are equal to the generating functions of plane partitions confined in a finite box. We consider the two-dimensional vertex model related to the phase model and express the vertex model partition function with special boundary conditions in terms of the scalar product of the phase model state vectors.

A Combinatorial Interpretation of the Scalar Products of State Vectors of Integrable Models

The representation of the Bethe wave functions of certain integrable models via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The algebraic relations arising in the calculation of the scalar products and the correlation functions are based on the Binet-Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of the Bethe state-vectors in terms of nests of the self-avoiding lattice paths constituting the so-called watermelon configurations. The interpretation proposed is, in its turn, related to the enumeration of the boxed plane partitions.

Plane partition polynomial asymptotics

The plane partition polynomial \(Q_n(x)\) is the polynomial of degree \(n\) whose coefficients count the number of plane partitions of \(n\) indexed by their trace. Extending classical work of E. M. Wright, we develop the asymptotics of these polynomials inside the unit disk using the circle method.

Plane Partitions with Two-Periodic Weights

We study scaling limits of skew plane partitions with periodic weights under several boundary conditions. We compute the correlation kernel of the limiting point process in the bulk and near turning points on the frozen boundary. The turning points that appear in the homogeneous case split in our model into pairs of turning points macroscopically separated by a "semi-frozen" region. As a result the point process at a turning point is not the GUE minor process, but rather a pair of GUE minor processes, non-trivially correlated. We also study an intermediate regime when the weights are periodic but all converge to 1. In this regime the limit shape and correlations in the bulk are the same as in the case of homogeneous weights and periodicity is not visible in the bulk. However, the process at turning points is still not the GUE minor process.

Interaction of streamers and stationary corrugated ionization waves in semiconductors.

A numerical simulation of evolution of an identical interacting streamers array in semiconductors has been performed using the diffusion-drift approximation and taking into account the impact and tunnel ionization. It has been assumed that the external electric field E0 is static and uniform, the background electrons and holes are absent, the initial avalanches start simultaneously from the nodes of the plane hexagonal lattice, which is perpendicular to the external field, but the avalanches and streamers are axially symmetric within a cylinder of radius R. It has been shown that under certain conditions, the interaction between the streamers leads finally either to the formation of two types of stationary ionization waves with corrugated front or to a stationary plane ionization wave. A diagram of different steady states of this type of waves in the plane of parameter E0,R has been presented, and a qualitative explanation of the plane partition into four different regions has been given. Characteristics of corrugated waves have been studied in detail and discussed in the region of R and E0 large values, in which the maximum field strength at the front is large enough for the tunnel ionization implementation. It has been shown that corrugated waves ionize semiconductors more efficiently than flat ones, especially in relatively weak external fields.

Counting proper mergings of chains and antichains

A proper merging of two disjoint quasi-ordered sets (P,←P) and (Q,←Q), denoted by P and Q, respectively, is a quasi-order on the union of P and Q such that the restriction to P or Q yields the original quasi-orders on those sets and such that no elements of P and Q are identified. In this article, we consider the cases where P and Q are chains, where P and Q are antichains, and where P is an antichain and Q is a chain. We give formulas that determine the number of proper mergings in all three cases. We also introduce two new bijections from proper mergings of two chains to plane partitions and from proper mergings of an antichain and a chain to monotone colorings of complete bipartite digraphs.

NEW INFINITE FAMILIES OF CONGRUENCES MODULO 4 AND 8 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

AbstractIn 2012, Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’,Util. Math. 88(2012), 223–235] introduced a special class of totally symmetric plane partitions, called$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$-shell totally symmetric plane partitions. Let$f(n)$denote the number of$1$-shell totally symmetric plane partitions of weight$n$. More recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’,Bull. Aust. Math. Soc.to appear. Published online 27 September 2013] discovered a number of arithmetic properties satisfied by$f(n)$. In this paper, employing some results due to Cui and Gu [‘Arithmetic properties of$l$-regular partitions’,Adv. Appl. Math. 51(2013), 507–523], and Hirschhorn and Sellers, we prove several new infinite families of congruences modulo 4 and 8 for$1$-shell totally symmetric plane partitions. For example, we find that, for$n\geq 0$and$\alpha \geq 1$,$$\begin{equation*} f(8 \times 5^{2\alpha } n+39\times 5^{2\alpha -1})\equiv 0 \pmod 8. \end{equation*}$$

Characters of representations of the quantum toroidal algebra : Plane partitions with “stands”

An expression for the generating function of plane partitions a i,j subject to the constraints a m,n = 0 and a i,j ⩾ k j , 1 ⩽ j ⩽ n, which is the character of an irreducible representation of the quantum toroidal algebra Open image in new window , is obtained.

The $q = -1$ phenomenon via homology concentration

We introduce a homological approach to exhibiting instances of Stembridge's q=-1 phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of fixed size and that of plane partitions fitting in a box of fixed size. A more general framework of invariant and coinvariant complexes with coefficients taken mod 2 is developed, and as a part of this story an analogous homological result for necklaces is conjectured.

Lozenge tiling constrained codes

While the field of one-dimensional constrained codes is mature, with theoretical as well as practical aspects of code- and decoder-design being well-established, such a theoretical treatment of its two-dimensional (2D) counterpart is still unavailable. Research has been conducted on a few exemplar 2D constraints, e.g., the hard triangle model, run-length limited constraints on the square lattice, and 2D checkerboard constraints. Excluding these results, 2D constrained systems remain largely uncharacterized mathematically, with only loose bounds of capacities present. In this paper we present a lozenge constraint on a regular triangular lattice and derive Shannon noiseless capacity bounds. To estimate capacity of lozenge tiling we make use of the bijection between the counting of lozenge tiling and the counting of boxed plane partitions.

Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks

The XX0 Heisenberg model on a cyclic chain is considered. The representation of the Bethe wave functions via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The determinantal expressions of the form-factors and of the thermal correlation functions are obtained. The q-binomial determinants enable the connection of the form-factors with the generating functions both of boxed plane partitions and of self-avoiding lattice paths. The asymptotical behavior of the thermal correlation functions is studied in the limit of low temperature provided that the characteristic parameters of the system are large enough.

Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

A Gelfand–Tsetlin scheme of depth $$N$$ is a triangular array with $$m$$ integers at level $$m$$ , $$m=1,\ldots ,N$$ , subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed $$N$$ th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its $$q$$ -deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

ARITHMETIC PROPERTIES OF 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

AbstractBlecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’,Util. Math. 88(2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight$n$. He also proved that the generating function for$f(n), $the number of 1-shell totally symmetric plane partitions of weight$n$, is given by$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$In this brief note, we prove a number of arithmetic properties satisfied by$f(n)$using elementary generating function manipulations and well-known results of Ramanujan and Watson.

The Hankel transform of aerated sequences

This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Two different aerating transforms are introduced and closed-form expressions are derived for the Hankel transform of such aerated sequences. Combinations of both aerating and the Hankel transforms are also considered. Our results are general and can be applied to a wide class of integer sequences. As an application, we use our tools on the sequence of shifted Catalan numbers (Cn+t)n∈ℕ 0. For that purpose, we need to evaluate the Hankel and Hankel-like determinants based on the Catalan numbers. Our approach is based on the results of Gessel and Viennot [Determinants, paths, and plane partitions preprint. Available from: http://www.cs.brandeis.edu/ ira] and more recent results of Krattenthaler [Determinants of (generalised) Catalan numbers. J. Statist. Plann. Inference 2010; 240: 2260–2270]. We generalize a sequence obtained by the series reversion of Q(x)=x/(1+α x+β x 2) (studied in our previous paper [Bojičić R, Petković MD, Barry P. Hankel transform of a sequence obtained by series reversion. Integral Transforms Spec. Funct. DOI: 10.1080/10652469.2011.640326]) and provide the Hankel transform evaluation of that sequence and its shifted sequences.