Lattice Of Noncrossing Partitions Research Articles

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.

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.

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 $.

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.

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.

The Noncrossing Bond Poset of a Graph

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$.
 Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.

The Whitney duals of a graded poset

We introduce the notion of a Whitney dual of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge labelings which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly construct a Whitney dual using a technique involving quotient posets.As applications of our main theorem, we show that geometric lattices and the lattice of noncrossing partitions all have Whitney duals. Our technique gives a combinatorial description of the Whitney dual of the partition lattice in terms of a poset of increasing forests. More generally we give combinatorial descriptions of Whitney duals of geometric lattices in terms of NBC sets. We also provide a combinatorial description of a Whitney dual of the noncrossing partition lattice in terms of collections of decorated Dyck paths.Finally, we show that a graded poset with a Whitney labeling admits a local action of the 0-Hecke algebra of type A on its set of maximal chains. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of the poset. The existence of this action implies, using a result of McNamara, that when the maximal intervals of the constructed Whitney duals are bowtie-free, they are also snellable. In the case where these maximal intervals are lattices, they are supersolvable.

The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders). In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner. In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups. Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.

Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of Möbius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions.

Oriented Flip Graphs and Noncrossing Tree Partitions

Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.

Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

On the Sperner property for the absolute order on complex reflection groups

Two partial orders on a reflection group, the codimension order and the prefix order, are together called the absolute order when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type $D_n$, for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice $NC_W$, a certain maximal interval in the absolute order, but not for the entire poset, except in the case of the symmetric group. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.

The antipode of the noncrossing partition lattice

We prove the cancellation-free formula for the antipode of the noncrossing partition lattice in the reduced incidence Hopf algebra of posets due to Einziger. The proof is based on a map from chains in the noncrossing partition lattice to noncrossing hypertrees and expressing the alternating sum over these fibers as an Euler characteristic.

Combinatorics of Exceptional Sequences in Type A

Exceptional sequences are certain sequences of quiver representations. We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions.

Noncrossing partitions and Milnor fibers

For a finite real reflection group $W$ we use non-crossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$-discriminant $\Delta_W$ and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of $\Delta_W$.

Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions

Given a tree embedded in a disk, we introduce a simplicial complex of noncrossing geodesics supported by the tree, which we call the noncrossing complex. The facets of the noncrossing complex have the structure of an oriented flip graph. Special cases of these oriented flip graphs include the Tamari lattice, type A Cambrian lattices, Stokes posets of quadrangulations, and oriented exchange graphs of quivers mutation-equivalent to a type A Dynkin quiver. We prove that the oriented flip graph is a polygonal, congruence-uniform lattice. To do so, we express the oriented flip graph as a lattice quotient of a lattice of biclosed sets.The facets of the noncrossing complex have an alternate ordering known as the shard intersection order. We prove that this shard intersection order is isomorphic to a lattice of noncrossing tree partitions, which generalizes the classical lattice of noncrossing set partitions. The oriented flip graph inherits a cyclic action from its congruence-uniform lattice structure. On noncrossing tree partitions, this cyclic action generalizes the classical Kreweras complementation on noncrossing set partitions.

Generalized Non-Crossing Partitions and Buildings

For any finite Coxeter group $W$ of rank $n$ we show that the order complex of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give a new proof of the fact that the non-crossing partition lattice in type $A_n$ is supersolvable for all $n$. Moreover, we show that in case $B_n$, this is only the case if $n<4$. We also obtain a lower bound on the radius of the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the radius is $\binom{n}{2}$. A Corrigendum for this paper was added on May 17, 2018.

Undesired Parking Spaces and Contractible Pieces of the Noncrossing Partition Link

There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. The first author and his coauthors conjectured that various collections of simplices of the noncrossing partition link (determined by the undesired parking spaces in the corresponding parking functions) form contractible subcomplexes. In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.

Two Posets of Noncrossing Set Partitions

In this manuscript we consider a subposet of the lattice of noncrossing set partitions of an n-element set under refinement order. This subposet is induced by those noncrossing set partitions, which do not contain the block {n−1,n}, or which do not contain the singleton block {n} whenever 1 and n−1 are in the same block. We prove that the resulting poset is in fact a supersolvable lattice, and we give a combinatorial proof for the value of its Möbius function between least and greatest element by using Blass and Sagan's theory of NBB bases.As a corollary we prove a conjecture by Bruce, Dougherty, Hlavacek, Kudo and Nicolas about the homotopy type of a certain subposet of our poset, which comes from parking functions with undesired positions.