On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340-359] that, for any n, k, m and p, the number of n×n alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m −1?s and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of n×n matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindström-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.     

Chain enumeration and non-crossing partitions

A formula is established for the number of chains with designated ranks in the non-crossing partition lattice. As corollaries, certain results of Kreweras are obtained. Non-crossing partitions are then generalized in two ways, and similar problems are solved.     

The five-vertex model and boxed plane partitions

Boxed plane partitions are considered in terms of the five-vertex model on a finite lattice with fired boundary conditions. Assuming that all weights of the model have the same value, the one-point correlation function. describing the probability of having a given state on an arbitrary horizontal edge of the lattice is calculated. This is equivalent to the enumeration of boxed plane partitions that correspond to rhombus tilings of a hexagon with one fired rhombus of a particular type. The solution of the problem is given for the case of a boar of generic size. Bibliography: 24 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 162–179.     

The lattices of ideals of multizigzags and the enumeration of Fibonacci partitions

Let u₁=1, u₂=2, u₃,... be the sequence of Fibonacci numbers. A Fibonacci partition of a natural number n is a partition of n into different Fibonacci numbers. In this paper it is proved that the set of Fibonacci partitions of a natural number, partially ordered with respect to refinement is the lattice of ideals of a multizigzag. On the basis of this theorem we obtain some results concerning the coefficients of the Taylor series of infinite products

Symmetries of plane partitions

We introduce a new symmetry operation, called complementation, on plane partitions whose three-dimensional diagram is contained in a given box. This operation was suggested by work of Mills, Robbins, and Rumsey. There then arise a total of ten inequivalent problems concerned with the enumeration of plane partitions with a given symmetry. Four of these ten problems had been previously considered. We survey what is known about the ten problems and give a solution to one of them. The proof is based on the theory of Schur functions, in particular the Littlewood-Richardson rule. Of the ten problems, seven are now solved while the remaining three have conjectured simple solutions.     

Enumeration of plane partitions

Using some recent results involving Young tableaux and matrices of non-negative integers [10], it is possible to enumerate various classes of plane partitions by actual construction. One of the results is a simple proof of MacMahon's [12] generating function for plane partitions. Previous results of this type [12, 4, 3, 8, 7] involved complicated algebraic methods which did not reveal any intrinsic “reason” why the corresponding generating functions have such a simple form.     

Discrete Tomography and plane partitions

A plane partition   is a p×q$p\times q$ matrix A=(_aij)$A=(a_{ij})$, where 1?i?p$1?i?p$ and 1?j?q$1?j?q$, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids  , subsets of the integer lattice Z³${\mathbb{Z}}^3$ which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations.     

A bijective proof of the generating function for the number of reverse plane partitions via lattice paths

Recently, Hillman and Grassl gave a bijective proof for the generating function for the number of reverse plane partitions of a fixed shape λ. We give another bijective proof for this generating function via completelv different methods. Our bijection depends on a lattice path coding of reverse plane partitions and a new method for constructing bisections out of certain pairs of involutions due to Garsia and Milne.     

A bijective proof of the generating function for the number of reverse plane partitions via lattice paths

Recently, Hillman and Grassl gave a bijective proof for the generating function for the number of reverse plane partitions of a fixed shape λ. We give another bijective proof for this generating function via completelv different methods. Our bijection depends on a lattice path coding of reverse plane partitions and a new method for constructing bisections out of certain pairs of involutions due to Garsia and Milne.     

On free fermions and plane partitions

We use free fermion methods to re-derive a result of Okounkov and Reshetikhin relating charged fermions to random plane partitions, and to extend it to relate neutral fermions to strict plane partitions.     

Gray code enumeration of families of integer partitions

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes P_δ(n, k) and D(n, k) of partitions of n into parts of size at most k in which, for P_δ(n, k), the parts are congruent to one modulo δ and, for D(n, k), the parts are distinct. It is shown that the elements of these classes can be listed so that the only change between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n, k), δ = 1.     

The bijection between plane partitions and nonnegative integer matrices

The one-to-one correspondence between the set of plane partitions withr rows andm columns and the set of matrices of nonnegative integers with the same numbers of rows and columns has been constructed. Published in Lietuvos Matematikes Rinkinys, Vol. 35, No. 2, pp. 204–210, April–June, 1995.     

q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.     

Self-complementary totally symmetric plane partitions

A totally symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box X_n = [1, n] × [1, n] × [1, n] and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the box X_n that do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex (n + 1, n + 1, n + 1). A totally symmetric plane partition is self-complementary if it is congruent (in the geometrical sense) to its complement. This cannot occur unless n = 2m is even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship with m by m alternating sign matrices. In an earlier paper we described the close connection between m by m alternating sign matrices and descending plane partitions with no parts exceeding m. We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.     

Notes on plane partitions VI

We consider the problem of plane partitions with given minimal differences between parts along rows and columns. We obtain a formula for the generating function in the two row case.     

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux – ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.     

Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence

A bijection is presented between (1): partitions with conditions f_j+f_j+1≤k−1 and f₁≤i−1, where f_j is the frequency of the part j in the partition, and (2): sets of k−1 ordered partitions (n⁽¹⁾,n⁽²⁾,…,n^(k−1)) such that and , where m_j is the number of parts in n^(j). This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k−1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud’s version of the Burge correspondence.     

Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions

Aplane quadrangulation G is a simple plane graph such that each face ofG is quadrilateral. A (^*) -orientation D ^*(G) ofG is an orientation ofG such that the outdegree of each vertex on G is 1 and the outdegrees of other vertices are all 2, where G denotes the outer 4-cycle ofG. In this paper, we shall show that every plane quadrangulationG has at least one (^*)-orientation. We also show that any two (*)-orientations ofG can be transformed into one another by a sequence of 4-cycle reversals. Moreover, we apply this fact toorthogonal plane partitions, which are partitions of a square into rectangles by straight segments.A research fellow of the Japan Society for the Promotion of Science.     

Enumeration of plane partitions and the algebraic Bethe anzatz

We establish a relation between an exactly solvable boson model and plane partitions, i.e., three-dimensional Young diagrams enclosed in a box of finite size, which allows representing the partition generating functions as correlation functions of an integrable model and deriving the MacMahon formulas for enumerating partitions using the quantum inverse scattering method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 193–203, February, 2007.