Lattice Of Partitions Research Articles

A global compositional complexity measure for biological sequences: AT-rich and GC-rich genomes encode less complex proteins.

Different local regions of natural amino acid or nucleotide sequences show remarkable heterogeneity in residue composition, reflecting diversity in evolutionary history and physiochemical constraints. Compositional complexity measures are helpful for describing and understanding this variegation. Motivated by some open problems in comparative genomics and protein folding, we have developed a new 'global' compositional complexity measure, G1, which overcomes a crucial limitation of earlier methods. The 'local' measures used in previous research resemble entropy functions and are inherently dependent on an underlying probability distribution. Local measures cannot rigorously compare complexity across sequences of substantially different size, because real sequences show very irregular heterogeneity and do not have the necessary ergodicity in scaling and asymptotic properties. G1 is a member of a new class of scale-independent, distribution-independent complexity functions. For a sequence S of length L on an N-letter alphabet, G1 is derived from ratios in the integer partition lattice, P¿L,N¿ of L with N parts, where the elements of P¿L,N¿ are the state vectors of S, (n1, n2,..., nN), ranked by an order principle. We present theorems and proofs relating to the metric properties of G1 and its relationship to other state-vector-dependent compositional complexity functions, together with a fully-efficient O(L) algorithm to compute G1. The distributions of G1 were calculated for the entire sets of translated proteins encoded by extensively sequenced genomes. The results establish the existence of a clear evolutionary principle, common to bacteria, archaea and eukaryotes, that the proteins encoded by more extreme AT-rich and GC-rich genomes have generally lower compositional complexity than those of more typical organisms.

Collapsibility of Δ(Π_{𝑛})/𝒮_{𝓃} and some related CW complexes

Let Δ ( Π n ) \Delta (\Pi _n) denote the order complex of the partition lattice. The natural S n \mathcal {S}_n -action on the set [ n ] [n] induces an S n \mathcal {S}_n -action on Δ ( Π n ) \Delta (\Pi _n) . We show that the regular CW complex Δ ( Π n ) / S n \Delta (\Pi _n)/\mathcal {S}_n is collapsible. Even more, we show that Δ ( Π n ) / S n \Delta (\Pi _n)/\mathcal {S}_n is collapsible, where Π Δ \Pi _\Delta is a suitable type selection of the partition lattice. This allows us to generalize and reprove in a conceptual way several previous results regarding the multiplicity of the trivial character in the S n \mathcal {S}_n -representation on H ∗ ( Δ ( Π n ) ) H_*(\Delta (\Pi _n)) .

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r− 1,q ) for r& 3 is the only rank- r(combinatorial) geometry with (qr− 1)/(q− 1) points in which all lines have at least q+ 1 points. For r= 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.

Partitions and network reliability

The all-terminal reliability and K-terminal reliability of networks are often considered as separate problems. We use the partition lattice of the vertex set of an undirected graph to show that these reliability measures are closely related. As an application of the presented approach, a formula for the K-terminal reliability polynomial of the complete graph and Buzacott's node partition formula are derived.

Partition Lattices in Twelve-Tone Music: An Introduction

Partition Lattices in Twelve-Tone Music: An Introduction

A Class of Lattices Whose Intervals are Spherical or Contractible

We study a class of lattices called weak*complemented lattices which are shown to have the property that the order complex of any interval of the lattice is either contractible or homotopy equivalent to a sphere. The two main examples are lattices generated by intervals in a total order and the lattices of partitions of integers under dominance order. The proofs are done mainly using homotopy complementation formulas for lattices and with a method called B-labeling. We also show that a class of lattices called Greene lattices are either contractible or spherical. Lattices generated by multisets are also discussed.

Convex and extendable partitions of a lattice

Abstract In his paper, Ore studied the lattice of partitions of a set. For many current applications such as formal concept analysis where the underlying set is a lattice, it makes more sense to restrict our attention to those partitions that respect the partial order on the set in some way. This paper looks at two classes of partitions that respect the partial order in different ways. It then shows some of the lattice theoretic properties of each of these classes of lattices.

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

We study complex hyperplane arrangements whose intersection lattices, known as the Dowling lattices, are a natural generalization of the partition lattice. We give a combinatorial description of the Dowling lattice via enriched partitions to obtain an explicit EL -labeling and then find a recursion for the flag h -vector in terms of weighted derivations. When the hyperplane arrangements are real they correspond to the braid arrangements A n and B n . By applying a result due to Billera and the authors, we obtain a recursive formula for the cd-index of the lattice of regions of the braid arrangements A n and B n .

Deformation of Chains via a Local Symmetric Group Action

A symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that $(i,i+1)$ sends each maximal chain either to itself or to one differing only at rank $i$. We prove that when $S_n$ acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as $R^* S$-labellings have symmetric chain decompositions and provide $R^* S$-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.

Finite partition lattices and butler groups

We investigate a special relation between finite partition lattices and distributive lattices. This will allow us to explore the close relationship between a class of torsionfree Abelian groups (the subclass of the class of Butler groups consisting of B (1) - groups) and a category of order-preserving morphisms mapping partition lattices into distributive lattices.

Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W. We use this correspondence to find all W for which L is supersolvable. In particular, we show that the only infinite Coxeter group for which L is supersolvable is the infinite dihedral group. Also, we show how this isomorphism gives an embedding of L into the partition lattice whenever W is of type An, Bn or Dn. In addition, we give several results concerning non-broken circuit bases (NBC bases) when W is finite. We show that L is supersolvable if and only if all NBC bases are obtainable by a certain specific combinatorial procedure, and we use the lattice of parabolic subgroups to identify a natural subcollection of the collection of all NBC bases.

Homotopy of Non-Modular Partitions and the Whitehouse Module

We present a class of subposets of the partition lattice Π n with the following property: The order complex is homotopy equivalent to the order complex of Π n − 1, and the S n -module structure of the homology coincides with a recently discovered lifting of the S n − 1-action on the homology of Π n − 1. This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet P n n − 1 of the lattice of set partitions Π n , obtained by removing all elements with a unique nontrivial block. More generally, for 2 ≤ k ≤ n − 1, let Q k n denote the subposet of the partition lattice Π n obtained by removing all elements with a unique nontrivial block of size equal to k, and let P k n = ∩ k i = 2 Q i n . We show that P k n is Cohen-Macaulay, and that P k n and Q k n are both homotopy equivalent to a wedge of spheres of dimension (n − 4), with Betti number \((n - 1)!\frac{{n - k}}{k}\). The posets Q k n are neither shellable nor Cohen-Macaulay. We show that the S n -module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property.

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.

A Basis for the Top Homology of a Generalized Partition Lattice

For a fixed positive integer k, consider the collection of all affine hyperplanes in n-space given by xi − xj = m, where i, j ∈ [n], i ≠ j, and m ∈ {0, 1,..., k}. Let Ln,k be the set of all nonempty affine subspaces (including the empty space) which can be obtained by intersecting some subset of these affine hyperplanes. Now give Ln,k a lattice structure by ordering its elements by reverse inclusion. The symmetric group Gn acts naturally on Ln,k by permuting the coordinates of the space, and this action extends to an action on the top homology of Ln,k. It is easy to show by computing the character of this action that the top homology is isomorphic as an Gn-module to a direct sum of copies of the regular representation, CGn. In this paper, we construct an explicit basis for the top homology of Ln,k, where the basis elements are indexed by all labelled, rooted, (k + 1)-ary trees on n-vertices in which the root has no 0-child. This construction gives an explicit Gn-equivariant isomorphism between the top homology of Ln,k and a direct sum of copies of CGn.

On the (co)homology of the partition lattice and the free Lie algebra

We explore the well-known relationship between the representations of the symmetric group on the homology of the partition lattice and on the ln component of the free Lie algebra. We give two new combinatorial proofs of the sign twisted isomorphism between the two modules. One involves a bijection between bases and the other involves a bijection between generating sets which takes relations to relations.

The Size of the Largest Antichain in the Partition Lattice

Consider the posetΠnof partitions of ann-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Leta=12−elog(2)/4. We prove the following upper bound for the ratio of the size of the largest antichain to the size of the largest rank:d(Πn⩽)S(n,Kn)⩽c2na(logn)−a−1/4,for suitable constantc2andn>1. This upper bound exceeds the best known lower bound for the latter ratio by a multiplicative factor ofO(1).

Extensive operators in partition lattices for image sequence analysis

This paper deals with the class of extensive operators in the lattice of partitions. These operators are merging techniques. They can be used as filtering tools or as segmentation algorithms. In the first case, they are known as connected operators and, in the second case, they are region growing techniques. This paper discusses the basic elements that have to be defined to create a merging algorithm: merging order, merging criterion and region model. This analysis highlights the similarity and differences between a filtering tool, such as a connected operator, and a segmentation algorithm. Taking benefit from the filtering and segmentation viewpoints, we propose a general merging algorithm that can be used to create new connected operators (in particular self-dual operators) and efficient segmentation algorithms (robust criteria and efficient implementation).

A chromatic partition polynomial

A polynomial in two variables is defined by C n ( x , t ) = Σ π ∈ Π n x ( G π , x ) t | π | , where Π n is the lattice of partitions of the set {1, 2, …, n}, G π is a certain interval graph defined in terms of the partition gp, χ( G π, x ) is the chromatic polynomial of G π and |π| is the number of blocks in π. It is shown that C n ( x , t ) = Σ k = 0 n t k Σ i = 0 k ( n − k n − 1 ) S ( n , i ) ( x ) i , where S( n, i) is the Stirling number of the second kind and ( x) i = x( x − 1) ··· ( x − i + 1). As a special case, C n (−1, − t) = A n ( t), where A n ( t) is the nth Eulerian polynomial. Moreover, A n ( t ) = Σ π ∈ Π n a π t | π | where a π is the number of acyclic orientations of G π .

Combinatorial theory of the free product with amalgamation and operator-valued free probability theory

Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory Operator-valued stochastic processes and stochastic differential equations Bibliography.

Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, \widehat{\rm des}, is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of des on genus zero permutations and Baxter permutations. Our q-enumerative results relate the des statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.