Lattice Of Partitions Research Articles

Two Classes of Posets with Real-Rooted Chain Polynomials

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of posets, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to finite Coxeter groups, are shown to have this property. The first result generalizes one of Brenti and Welker. As a special case, the descent enumerator of permutations of the set $\{1, 2,\dots,n\}$ which have ascents at specified positions is shown to be real-rooted, hence log-concave and unimodal, and a good estimate for the location of the peak is deduced.

Partition Rank and Partition Lattices

Partition Rank and Partition Lattices

Noncrossing Partition Lattices from Planar Configurations

AbstractThe lattice of noncrossing partitions is well-known for its wide variety of combinatorial appearances and properties. For example, the lattice is rank-symmetric and enumerated by the Catalan numbers. In this article, we introduce a large family of new noncrossing partition lattices with both of these properties, each parametrized by a configuration of n points in the plane.

Arrow relations in lattices of integer partitions

We give a complete characterisation of the single and double arrow relations of the standard context K(Ln) of the lattice Ln of partitions of any positive integer n under the dominance order, thereby addressing an open question of Ganter, 2020/2022.

On the metrization of the infinite partition lattice

In this paper, we equip the partition lattice of the positive integers with a metric structure, and study the properties of this metric. We show, that this new space is complete, compact and separable. We also investigate how continuous functions look like in this structure.

Composition closed premodel structures and the Kreweras lattice

We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization systems so that the intervals detect these composition closed premodel structures. In the case that the lattice in question is a finite total order, this natural order retrieves the Kreweras lattice of noncrossing partitions as a refinement of the Tamari lattice, and model structures can be identified with certain tricolored trees.

AROUND THE ERDÖS–GALLAI CRITERION

By an (integer) partition we mean a non-increasing sequence \(\lambda=(\lambda_1, \lambda_2, \dots)\) of non-negative integers that contains a finite number of non-zero components. A partition \(\lambda\) is said to be graphic if there exists a graph \(G\) such that \(\lambda = \mathrm{dpt}\,G\), where we denote by \(\mathrm{dpt}\,G\) the degree partition of \(G\) composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.

From trees to barcodes and back again II: Combinatorial and probabilistic aspects of a topological inverse problem

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees. As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on n+1 leaf nodes fall into (2n−1)!! distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2−n. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.

Chain enumeration, partition lattices and polynomials with only real roots

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type \(B\) analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen-Macaulay posets. As a result, new families of convex polytopes whose barycentric subdivisions have real-rooted \(f\)-polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10Keywords: Chain polynomial, geometric lattice, partition lattice, real-rooted polynomial, flag \(h\)-vector, convex polytope, barycentric subdivision

On the homology of the noncrossing partition lattice and the Milnor fibre

Let L be the noncrossing partition lattice associated to a finite Coxeter group W. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of L. We define a multiplicative structure on the Whitney homology of L in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of W, the Milnor fibre of the discriminant of W, the hyperplane complement of W and the Artin group of type W, respectively. We also tabulate some computational results on the integral homology of the Milnor fibres.

Lattices of t‐structures and thick subcategories for discrete cluster categories

We classify t-structures and thick subcategories in any discrete cluster category C ( Z ) $\mathcal {C}(\mathcal {Z})$ of Dynkin type A $A$ , and show that the set of all t-structures on C ( Z ) $\mathcal {C}(\mathcal {Z})$ is a lattice under inclusion of aisles, with meet given by their intersection. We show that both the lattice of t-structures on C ( Z ) $\mathcal {C}(\mathcal {Z})$ obtained in this way and the lattice of thick subcategories of C ( Z ) $\mathcal {C}(\mathcal {Z})$ are intimately related to the lattice of non-crossing partitions of type A $A$ . In particular, the lattice of equivalence classes of non-degenerate t-structures on such a category is isomorphic to the lattice of non-crossing partitions of a finite linearly ordered set.

Partition Lattice with Limited Block Sizes

Partition Lattice with Limited Block Sizes

Components in Meandric Systems and the Infinite Noodle

Abstract We investigate here the asymptotic behaviour of a large, typical meandric system. More precisely, we show the quenched local convergence of a random uniform meandric system $\boldsymbol {M}_n$ on $2n$ points, as $n \rightarrow \infty $, towards the infinite noodle introduced by Curien et al. [3]. As a consequence, denoting by $cc( \boldsymbol {M}_n)$ the number of connected components of $\boldsymbol {M}_n$, we prove the convergence in probability of $cc(\boldsymbol {M}_n)/n$ to some constant $\kappa $, answering a question raised independently by Goulden–Nica–Puder [8] and Kargin [12]. This result also provides information on the asymptotic geometry of the Hasse diagram of the lattice of non-crossing partitions. Finally, we obtain expressions of the constant $\kappa $ as infinite sums over meanders, which allows us to compute upper and lower approximations of $\kappa $.

Maximal Chains in Bond Lattices

Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice. The elements of $BL(G)$ are the set partitions whose blocks induce connected subgraphs of $G$.
 In this article, we consider graphs $G$ whose bond lattice consists only of noncrossing partitions. We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices. We then look at the enumeration of the maximal chains in the bond lattices of triangulation graphs. Stanley's map from maximal chains in the noncrossing partition lattice to parking functions was our motivation. We find the restriction of his map to the bond lattice of certain subgraphs of triangulation graphs. Finally, we show the number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions.

Noncommutative symmetric functions and Lagrange inversion II: Noncrossing partitions and the Farahat-Higman algebra

We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing partitions. As a consequence, we obtain a quasi-symmetric version of the Farahat-Higman algebra.

The Minor Order of Homomorphisms via Natural Dualities

We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms.

Hochschild lattices and shuffle lattices

In his study of a Hochschild complex arising in connection with the free loop fibration, S. Saneblidze defined the freehedron, a certain polytope constructed via a truncation process from the hypercube. It was recently conjectured by F. Chapoton and proven by C. Combe that a certain orientation of the 1-skeleton of the freehedron carries a lattice structure. The resulting lattice was dubbed the Hochschild lattice and it is interval constructable and extremal. These properties allow for the definition of three associated structures: the Galois graph, the canonical join complex and the core label order. In this article, we study and characterize these structures. We exhibit an isomorphism from the core label order of the Hochschild lattice to a particular shuffle lattice of C. Greene. We also uncover an enumerative connection between the core label order of the Hochschild lattice, a certain order extension of its poset of irreducibles and the freehedron. These connections nicely parallel the situation surrounding the better-known Tamari lattices, noncrossing partition lattices and associahedra.

Lower bound for the number of 4-element generating sets of direct products of two neighboring partition lattices

Lower bound for the number of 4-element generating sets of direct products of two neighboring partition lattices

Notes on integer partitions

Some observations concerning the lattices of integer partitions are presented. We determine the size of the standard contexts, discuss a recursive construction and show that the lattices have unbounded breadth.

Four-generated direct powers of partition lattices and authentication

For an integer $n\geq 5$, H. Strietz (1975) and L. Zadori (1986) proved that the lattice Part$(n)$ of all partitions of $\{1,2,\dots,n\}$ is four-generated. Developing L. Zadori's particularly elegant construction further, we prove that even the $k$-th direct power Part$(n)^k$ of Part$(n)$ is four-generated for many but only finitely many exponents $k$. E.g., Part$(n)^k$ is four-generated for every $k\leq 3\cdot 10^{89}$, and it has a four element generating set that is not an antichain for every $k\leq 1.4\cdot 10^{34}$. In connection with these results, we outline a protocol how to use these lattices in authentication and secret key cryptography.