Class Of Partitions Research Articles

The Structure of the Young Symmetrizers for Spin Representations of the Symmetric Group, II

The first paper in this series established that the projective analogue of the Young symmetrizer recently introduced by Nazarov has a naturalPxQ-structure comparable with thepq-form of the classical symmetrizer. This second paper develops the theory on this decomposition further. A more efficient construction of the projective symmetrizer is presented for certain classes of partitions; this approach yields a “compact” expression for the intermediate factorxλin the projective decomposition.

Synchronic information, knowledge and common knowledge in extensive games

Restricting attention to the class of extensive games defined by von Neumann and Morgenstern (1944) with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of “maximum information” we are able to provide an alternative characterization of the meet of the information partitions.

An Exploration of the Permanent-Determinant Method

The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with $4n$ vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edge graph of a graph with vertices of degree at most 3 is a power of 2. 3. The three Carlitz matrices whose determinants count $a \times b \times c$ plane partitions all have the same cokernel. 4. Two symmetry classes of plane partitions can be enumerated with almost no calculation.

Representations and characterizations of vertices of bounded-shape partition polytopes

Consider a finite set whose elements are associated with vectors of common dimension. A partition of such a set is associated with a matrix whose columns are the sums of the vectors corresponding to each part. The partition polytope associated with a class of partitions (that share the number of parts) is then the convex hull of the corresponding matrices. We derive representations and characterizations of these polytopes and their vertices.

Partitions of Partitions

The paper presents methodology for analyzing a set of partitions of the same set of objects, by dividing them into classes of partitions that are similar to one another. Two different definitions are given for the consensus partition which summarizes each class of partitions. The classes are obtained using either constrained or unconstrained clustering algorithms. Two applications of the methodology are described.

Sortabilities of Partition Properties

Consider the partition of a set of integers into parts. Various partition properties have been proposed in the literature to facilitate the restriction of the focus of attention to some small class of partitions. Recently, Hwang, Rothblum and Yao defined and studied the sortability of these partition properties as a tool to prove the existence of a partition with such a property in a given family. In this paper we determine the sortability indices of the seven most interesting properties of partitions providing a complete solution to the sortability issue.

Optimality of consecutive and nested tree partitions

We consider the problem of partitioning the vertex-set of a tree to p parts to minimize a cost function. Since the number of partitions is exponential in the number of vertices, it is helpful to identify small classes of partitions which also contain optimal partitions. Two such classes, called consecutive partitions and nested partitions, have been well studied for the set partition problem, which is a special case of the tree-partition problem when the tree is a path. We give conditions on the optimality of these classes on tree partitions and also extend our results to tree networks. © 1997 John Wiley & Sons, Inc. Networks 30: 75–80, 1997

Chains of controllable partitions of the m-dimensional cube

Controllable partitions, which arise in approximation theory, are finite partitions of compact metric spaces into subsets whose sizes fulfil a uniformity condition depending on the entropy numbers of the underlying space. We characterize a class of partitions of the cube ([0,2] m , d max) which possess a controllable refinement and, in the end, give an ascending chain of controllable partitions of [0,2] m .

Critical Path Concept for Multi-Tool Cutting Processes Optimization

in a cutting process an ordered set of pairs (an elementary cutting operation on an elementary machined surface, datum surfaces set) can be presented as a directed cycle free graph denoted as a KL-graph. Some partitions of the KL-graph lead to an assignment of machining stations to the partition classes of the elementary operations set. Problems similar to the (assembly) line balancing appear when some optimization tasks are formulated. Such tasks differ from the classical balancing problems because times of elementary operations are not constant but they are the functions of the cutting parameters. Since the tools actions may be mapped by a network (PERT methods) balancing and coordination constraints can be formulated as the conditions defining the critical path placement.

Enumeration of Perfect Matchings in Graphs with Reflective Symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch's squarishness theorem. As applications of our result, we enumerate the perfect matchings of several families of graphs and we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions (namely, transposed complementary and cyclically symmetric, transposed complementary) contained in a given box. Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees.

Localizing combinatorial properties of partitions

The purpose of this paper is to develop a framework for the analysis of combinatorial properties of partitions. Our focus is on the relation between global properties of partitions and their localization to subpartitions. First, we study properties that are characterized by their local behavior. Second, we determine sufficient conditions for classes of partitions to have a member that has a given property. These conditions entail the possibility of being able to move from an arbitrary partition in the class to one that satisfies the given property by sequentially satisfying local variants of the property. We apply our approach to several properties of partitions that include consecutiveness, nestedness, order-consecutiveness, full nestedness and balancedness, and we demonstrate its usefulness in determining the existence of optimal partitions that satisfy such properties.

Four Symmetry Classes of Plane Partitions under One Roof

In a previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian–Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2, C). The result is a unified proof of the four enumerations.

Further determinants with the averaging property of Andrews-Burge

The determinants which arise in the enumeration of symmetry classes of plane partitions are difficult to evaluate. Recently, Andrews and Burge have shown that the determinant of one of these classes has a surprising property, which we call the averaging property. We obtain a class of determinants under reasonably general conditions which also possess this property. This class includes the Andrews and Burge example. We also consider conditions under which determinants in this class can be evaluated.

Further Determinants with the Averaging Property of Andrews–Burge

The determinants which arise in the enumeration of symmetry classes of plane partitions are difficult to evaluate. Recently, Andrews and Burge have shown that the determinant of one of these classes has a surprising property, which we call the averaging property. We obtain a class of determinants under reasonably general conditions which also possess this property. This class includes the Andrews and Burge example. We also consider conditions under which determinants in this class can be evaluated.

A Note on the Number of $t$-Core Partitions

A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a partition in the natural way. Fix a positive integer t. A partition of n is called a t−core partition of n if none of its hook numbers are multiples of t. Let ct(n) denote the number of t−core partitions of n. It has been conjectured that if t ≥ 4, then ct(n) > 0 for all n ≥ 0. In [7], the author proved the conjecture for t ≥ 4 even and for those t divisible by at least one of 5, 7, 9, or 11. Moreover if t ≥ 5 is odd, then it was shown that ct(n) > 0 for n sufficiently large. In this note we show that if k ≥ 2, then c3k(n) > 0 for all n using elementary arguments. A partition of a positive integer n is a nonincreasing sequence of positive integers with sum n. Here we define a special class of partitions. Definition 1. Let t ≥ 1 be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t−core partition of n. The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,4,5]. If t ≥ 1 and n ≥ 0, then we define ct(n) to be the number of partitions of n that are t−core partitions. The arithmetic of ct(n) is studied in [3,4]. The power series generating function for ct(n) is given by the infinite product: (1) ∞ ∑

Effective versions of Ramsey's Theorem: Avoiding the cone above 0′

AbstractRamsey's Theorem states that if P is a partition of [ω]k into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for P. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of [ω]2 into finitely many pieces, there exists an infinite homogeneous set A such that ∅′ ≰TA. Two technical extensions of this result are given, establishing arithmetical bounds for such a set A. Applications to reverse mathematics and introreducible sets are discussed.

On Mullineux symbols

The Mullineux symbols of a special class of p-regular partitions are classified. As a consequence it is shown that two apparently very difficult conjectures in the modular representation theory of finite symmetric groups are compatible.

Some hidden relations involving the ten symmetry classes of plane partitions

Let B be a partially ordered product of three finite chains. For any group G of automorphisms of B, let N G ( B, q) denote the rank generating function for G-invariant order ideals of B. If we regard B as a rectangular prism, N G ( B, q) can be viewed as a generating function for plane partitions that fit inside B. Similarly, define N G ′( B, q) to be the rank generating function for order ideals of the quotient poset B G . We prove that N G ( B, − 1) and N G ′( B, − 1) count the number of plane partitions (i.e., order ideals of B) that are invariant under certain automorphisms and complementation operations on B. Consequently, one discovers that the number of plane partitions belonging to each of the ten symmetry classes identified by Stanley is of the form N G ( B, ± 1) or N G ′( B, ± 1) for some subgroup G of S 3, and conversely. We also discuss the occurrence of this phenomenon in general partially ordered sets, and use the theory of P-partitions to derive a criterion for one aspect of it.

Symmetries of plane partitions and the permanent—determinant method

R. P. Stanley (1986, J. Combin. Theory Ser. A 43, 103–113) gives formulas for the number of plane partitions in each of 10 symmetry classes. This paper together with papers of G. Andrews ( J. Combin. Theory Ser. A, to appear) and J. Stembridge (The enumeration of totally symmetric plane partitions, preprint) completes the project of proving all 10 formulas. We enumerate cyclically symmetric, self—complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent—determinant method or a variant, the Hafnian-Pfaffian method, to obtain the answer as the determinant or Pfaffian of a matrix in each of the 10 cases. We row-reduce the resulting matrix in the case under consideration to prove the formula. A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison.

Partition identities and labels for some modular characters

In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups Sn, of the finite symmetric groups S,n in characteristic 5 . One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. 0. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of Sn, in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be natural character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and B * below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture B *, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels. In ? 1 we give a brief description of the groups Sn and their representations, leading up to Conjectures A and B * as they were formulated in [BMO]. That section also presents the background for Conjecture B as stated in [A3] and the equivalence of Conjectures B and B * is explained. Sections 2 and 3 are devoted to the proof of Conjecture B, and ?4 to the proof of Conjecture A. 1. THE CONJECTURES AND THEIR BACKGROUND For facts concerning the general representation theory of finite groups needed in the following, the reader is referred to [F, NT]. In 191 1 Schur [S1] proved that the finite symmetric groups S, have covering groups S, of order 2IS, I = 2 * n ! This means that there is an exact sequence