Lattice Of Partitions Research Articles

Homomorphisms on Lattices of Measures

Problem statement: Homomorphisms on lattices of measures defined on the quotient spaces of the integers were considered. These measures were defined in terms of Sharma-Kaushik partitions. The homomorphisms were studied in terms of their relationship with the underlying Sharma-Kaushik partitions. Approach: We defined certain mappings between lattices of Sharma-Kaushik partitions and showed that they are homomorphisms. These homomorphisms were mirrored in homorphisms between related lattices of measures. Results: We obtained the structure of certain homomorphisms of measures. Conclusion: Further information about homomorphisms between lattices of measures of the type considered here can be obtained by investigating the underlying lattices of Sharma-Kaushik partitions.

Noncrossing partitions and the shard intersection order

We define a new lattice structure (W,\preceq ) on the elements of a finite Coxeter group W. This lattice, called the \emphshard intersection order, is weaker than the weak order and has the noncrossing partition lattice \NC (W) as a sublattice. The new construction of \NC (W) yields a new proof that \NC (W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of (W,\preceq ), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of \NC (W). There is a natural dimension-preserving bijection between simplices in the order complex of (W,\preceq ) (i.e. chains in <mbox>(W,\preceq )</mbox>) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of \NC (W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The lattice (W,\preceq ) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W\!. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.

Polynomials, meanders, and paths in the lattice of noncrossing partitions

For every polynomial f f of degree n n with no double roots, there is an associated family C ( f ) \mathcal {C}(f) of harmonic algebraic curves, fibred over the circle, with at most n − 1 n-1 singular fibres. We study the combinatorial topology of C ( f ) \mathcal {C}(f) in the generic case when there are exactly n − 1 n-1 singular fibres. In this case, the topology of C ( f ) \mathcal {C}(f) is determined by the data of an n n -tuple of noncrossing matchings on the set { 0 , 1 , … , 2 n − 1 } \{0,1,\ldots ,2n-1\} with certain extra properties. We prove that there are 2 ( 2 n ) n − 2 2(2n)^{n-2} such n n -tuples, and that all of them arise from the topology of C ( f ) \mathcal {C}(f) for some polynomial f f .

Exponential Dowling structures

The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures leads to a similar vein of study for exponential Dowling structures. In particular, for the extended r -divisible partition lattice we show that the Möbius function is, up to a sign, the number of permutations in the symmetric group on r n + k elements having descent set { r , 2 r , … , n r } . Using Wachs’ original EL -labeling of the r -divisible partition lattice, the extended r -divisible partition lattice is shown to be EL -shellable.

The lattice of integer partitions and its infinite extension

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of a positive integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an encoding of the obtained lattice which makes it possible to enumerate easily and efficiently all the partitions of a given integer. This approach also gives a recursive formula for the number of partitions of an integer, and some informations on special sets of partitions, such as length bounded partitions.

The lattice of partitions of a positive integer

The lattice of partitions of a positive integer

Reconstructing masks from markers in non-distributive lattices

In mathematical morphology for sets (binary images), the geodesic reconstruction associates to a set called mask and a subset of it called marker, the union of all connected components of the mask intersected by the marker. It is obtained by iteration of a geodesic dilation applied to the marker inside the mask. This operation extends naturally to numerical functions (grey-level images), where it allows us to reconstruct flat zones. Considering that the family of images constitutes a complete lattice under some partial order relation, a general theory of geodesic dilations and reconstructions in a complete lattice was given in Ronse and Serra [Fundam Inf 46(4):349–395, 2001]. It relies on the assumption that the lattice is infinitely supremum distributive, and it fails for pictorial objects forming a non-distributive lattice. In this paper we give a more general theory of geodesic operations, that can be applied to non-distributive lattices; it is compatible with the previous theory when the lattice is infinitely supremum distributive. We study one particular form of geodesic operator, the impulsive geodesic dilation, which gives good results for images with values in a bundle lattice (e.g., the lattice of labels and the reference lattice). We also briefly discuss geodesy on the lattice of partitions.

Chains in the Noncrossing Partition Lattice

We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e., without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the $m$-divisible noncrossing partition lattice.

Congruences on equivalence algebras

A certain class of atomic, semimodular, semisimple partition lattices is studied. It is shown that this class is precisely the class of congruence lattices of equivalence algebras.

Partition lattices in Milton Babbitt's Woodwind Quartet (1953)

This article explores how lattices arising from the interaction of pitch-class partitions can provide insight into small-scale and large-scale analytical relationships in an early 12-tone work by Milton Babbitt. We begin with a review of the partition-lattice analytical model, as formulated by Richard Kurth. Individual passages from Milton Babbitt's Woodwind Quartet (1953) are then explored by first identifying the pitch-class partitions that govern the pitch structure of each section, and then constructing lattices from partition interactions. Lattice interactions of the trichordal array partitions used in the Quartet reveal how recurring hexachords and hexachordal partitions govern the distribution of pitch materials. In addition, one hexachordal partition that is not readily apparent from the arrays emerges from partition lattice interactions as a background unifying structure.

Non-crossing partition lattices in finite real reflection groups

For a finite real reflection group W W with Coxeter element γ \gamma we give a case-free proof that the closed interval, [ I , γ ] [I, \gamma ] , forms a lattice in the partial order on W W induced by reflection length. Key to this is the construction of an isomorphic lattice of spherical simplicial complexes. We also prove that the greatest element in this latter lattice embeds in the type W W simplicial generalised associahedron, and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere.

Algebraic Structure of Lattices of SK Partitions

Algebraic Structure of Lattices of SK Partitions

Strict partitions and discrete dynamical systems

We prove that the set of partitions with distinct parts of a given positive integer under dominance ordering can be considered as a configuration space of a discrete dynamical model with two transition rules and with the initial configuration being the singleton partition. This allows us to characterize its lattice structure, fixed point, and longest chains as well as their length, using Chip Firing Game theory. Finally, we study the recursive structure of infinite extension of the lattice of strict partitions.

Convex partitions of a finite lattice

The purpose of this paper is to introduce the lattice of convex partitions for a lattice L. Then we will show some properties of this lattice. Finally, we will show that if the convex partition lattice of L is finite and modular if and only if L is a finite chain.

Plethysm for Wreath Products and Homology of Sub-Posets of Dowling Lattices

We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal.

Complexes of trees and nested set complexes

We exhibit an identity of abstract simplicial complexes between the well-studied complex of trees Tn and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of k-trees by Hanlon, and we propose yet another generalization, more natural in the context of nested set complexes.

Shellability of noncrossing partition lattices

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type D n and those of exceptional type and rank at least three.

Grothendieck Bialgebras, Partition Lattices, and Symmetric Functions in Noncommutative Variables

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.

Noncrossing Partitions in Surprising Locations

Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser known example: the noncrossing partition lattice. The focus of the article is a gentle introduction to the lattice itself in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a building block for a contractible space acted on by a braid group. Since this article is aimed primarily at nonspecialists, each area is briefly introduced along the way.

Noncrossing Partitions in Surprising Locations

1. Introduction.Certain mathematical structures make a habit of reoccuring in the most diverselist of settings. Some obvious examples exhibiting this intrusive type of behaviorinclude the Fibonacci numbers, the Catalan numbers, the quaternions, and themodular group. In this article, the focus is on a lesser known example: the non-crossing partition lattice. The focus of the article is a gentle introduction to thelattice itself in three of its many guises: as a way to encode parking functions, asa key part of the foundations of noncommutative probability, and as a buildingblock for a contractible space acted on by a braid group. Since this article is aimedprimarily at nonspecialists, each area is briefly introduced along the way.The noncrossing partition lattice is a relative newcomer to the mathematicalworld. First defined and studied by Germain Kreweras in 1972 [33], it caught theimagination of combinatorialists beginning in the 1980s [20], [21], [22], [23], [29],[37], [39], [40], [45], and has come to be regarded as one of the standard objectsin the field. In recent years it has also played a role in areas as diverse as low-dimensional topology and geometric group theory [9], [12], [13], [31], [32] as wellas the noncommutative version of probability [2], [3], [35], [41], [42], [43], [49],[50]. Due no doubt to its recent vintage, it is less well-known to the mathematicalcommunity at large than perhaps it deserves to be, but hopefully this short paperwill help to remedy this state of affairs.2. A motivating example.Before launching into a discussion of the noncrossing partition lattice itself, wequickly consider a motivating example: the Catalan numbers. The Catalan num-bers are a favorite pastime of many amateur (and professional) mathematicians. Inaddition, they also have a connection with the noncrossing partition lattice (Theo-rem 3.1).Example 2.1(Catalan numbers). The Catalan numbers are the numbers C