Noncrossing Partitions Research Articles

On maximal dihedral reflection subgroups and generalized noncrossing partitions

In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group W W , every pair t , t ′ t,t’ of distinct reflections lie in a unique maximal dihedral reflection subgroup of W W . Our proof only relies on the combinatorics of words, in particular we do not use root systems at all. As an application, we deduce a new proof of a recent result of Delucchi-Paolini-Salvetti, stating that the poset [ 1 , c ] T [1,c]_T of generalized noncrossing partitions in any Coxeter group of rank 3 3 is a lattice. We achieve this by showing the more general statement that any interval of length 3 3 in the absolute order on an arbitrary Coxeter group is a lattice. This implies that the interval group attached to any interval [ 1 , w ] T [1,w]_T where w w is an element of an arbitrary Coxeter group with ℓ T ( w ) = 3 \ell _T(w)=3 is a quasi-Garside group.

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.

Fluctuation Moments for Regular Functions of Wigner Matrices

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.

Combinatorial aspects of weighted free Poisson random variables

This paper will be devoted to the study of weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted (deformed) free Fock space together with the vacuum expectation. We shall provide a combinatorial moment formula of non-commutative Poisson random variables. This formula gives us a very nice combinatorial interpretation to two parameters of weights. One can see that the deformation treated in this paper interpolates free and boolean Poisson random variables, their distributions and moments, and yields some conditionally free Poisson distribution by taking limit of the parameter.

Generating series and matrix models for meandric systems with one shallow side

In this article, we investigate meandric systems having one shallow side: the arch configuration on that side has depth at most two. This class of meandric systems was introduced and extensively examined by I. P. Goulden, A. Nica, and D. Puder [Int. Math. Res. Not. IMRN 2020 (2020), 983–1034]. Shallow arch configurations are in bijection with the set of interval partitions. We study meandric systems by using moment-cumulant transforms for non-crossing and interval partitions, corresponding to the notions of free and Boolean independence, respectively, in non-commutative probability. We obtain formulas for the generating series of different classes of meandric systems with one shallow side by explicitly enumerating the simpler, irreducible objects. In addition, we propose random matrix models for the corresponding meandric polynomials, which can be described in the language of quantum information theory, in particular that of quantum channels.

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.

Enumeration of non-crossing partitions according to subwords with repeated letters

In this paper, we enumerate members of the set NCn of non-crossing partitions of length n according to the number of occurrences of several infinite families of subword patterns each containing repeated letters. As a consequence of our results, we obtain explicit generating function formulas counting the members of NCn for n ? 0 according to all subword patterns of length three containing a repeated letter. In particular, we find extensions of the subword equivalences 211 ? 221, 1211 ? 1121 and 112 ? 122 over NCn.

Navigating the realm of noncommutative probability: Historical perspectives and future directionsstorical perspectives and future directions

This review article provides a comprehensive overview of the fascinating field of noncommutative probability theory, tracing its evolution from its inception in the early 1980s by Romanian-American mathematician Dan Voiculescu to its current state of prominence in mathematics. Through a meticulous examination of seminal works and recent advancements, we explore the key concepts, methodologies, and significant developments in this field, emphasizing the combinatorial aspects of noncommutative probability spaces, including non-crossing partitions and linked partitions. This exploration encompasses various aspects, including analytical methods, operator algebras, random matrices, and combinatorial structures. Additionally, it concludes with the current understanding and potential directions for future research.

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.

The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces

The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.

Analogues of Poisson-type limit theorems in discrete bm-Fock spaces

In this paper, we present analogues of the Poisson limit distribution for the noncommutative bm-independence, which is associated with several positive symmetric cones. We construct related discrete Fock spaces with creation, annihilation and conservation operators, and prove Poisson type limit theorems for them. Properties of the positive cones, in particular the volume characteristic property they enjoy, and the combinatorics of labeled noncrossing partitions, play crucial role in these considerations.

A combinatorial basis for the fermionic diagonal coinvariant ring

Let \(\Theta_n = (\theta_1, \dots, \theta_n)\) and \(\Xi_n = (\xi_1, \dots, \xi_n)\) be two lists of \(n\) variables and consider the diagonal action of \(\mathfrak{S}_n\) on the exterior algebra \(\wedge \{ \Theta_n, \Xi_n \}\) generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring \(FDR_n\) obtained from \(\wedge \{ \Theta_n, \Xi_n \}\) by modding out by the \(\mathfrak{S}_n\)-invariants with vanishing constant term. The author and Rhoades gave a basis for the maximal degree components of this ring where the action of \(\mathfrak{S}_n\) could be interpreted combinatorially via noncrossing set partitions. This paper will do similarly for the entire ring, although the combinatorial interpretation will be limited to the action of \(\mathfrak{S}_{n-1} \subset \mathfrak{S}_n\). The basis will be indexed by a certain class of noncrossing partitions.Mathematics Subject Classifications: 05E10, 05E18, 20C30Keywords: Skein relation, coinvariant algebra, noncrossing set partition, cyclic sieving

Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron

The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher's result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions for contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumerate bipartite planar trees and forests, and we apply our results to enumerate various families of permutations: for example, bipartite planar trees are in bijection with separable permutations. It is postulated by Livia Ferro, Tomasz Łukowski and Robert Moerman (2020) that contracted Grassmannian forests are in bijection with boundary strata of the momentum amplituhedron, an object encoding the tree-level S-matrix of maximally supersymmetric Yang-Mills theory. With this assumption, our results give a rank generating function for the boundary strata of the momentum amplituhedron, and imply that the Euler characteristic of the momentum amplituhedron is \(1\).Mathematics Subject Classifications: 05A05, 05A15, 05C10Keywords: Generating functions, permutations, planar forests

Counting occurrences of subword patterns in non-crossing partitions

A permutation pattern in which all letters within an occurrence are required to be adjacent is known as a subword. In this paper, we consider the distribution of several infinite families of subword patterns on the set of non-crossing partitions of size n, denoted by NCn, and derive formulas for the generating functions of these distributions on NCn. As special cases of our results, we obtain formulas for the generating functions enumerating the members of NCn according to the number of occurrences of any subword of length three with distinct letters. Simple expressions for the total number of occurrences of a pattern over all members of NCn are also deduced. Some connections are made with the related problem of counting Dyck paths according to the number of occurrences of certain types of strings. Further, for the subwords 12⋯m and 213⋯m where m ≥ 3, we consider the joint distribution with the descents statistic and make use of the kernel method to establish the results in these cases.

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.

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

We consider the group (G,⁎) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where “⁎” denotes the convolution operation. We introduce a larger group (G˜,⁎) of unitized functions from the same incidence algebra, which satisfy a weaker semi-multiplicativity condition. The natural action of G˜ on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of G˜ in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bożejko and Wysoczanski.It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that G˜ can also be identified as group of characters of a Hopf algebra T, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion G⊆G˜ turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.

Some Properties of the Parking Function Poset

In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with two well-known polytopes : the associahedron and the permutohedron.

A shuffle algebra point of view on operator-valued probability theory

We extend the shuffle algebra perspective on scalar-valued non-commutative probability theory to the operator-valued case. Given an operator-valued probability space with an algebra B acting on it (on the left and on the right), we associate operators in the operad of multilinear maps on B to the operator-valued distribution and free cumulants of a random variable. These operators define a representation of a PRO of non-crossing partitions. Using concepts from higher category theory, specifically 2-monoidal categories, we define a notion of unshuffle Hopf algebra on an underlying PRO. We introduce a PRO of words insertions and show that both the latter and the PRO of non-crossing partitions are unshuffle Hopf algebras. The two relate by mean of a map of unshuffle bialgebra (in a 2-monoidal sense) which we call the splitting map. Ultimately, we obtain a left half-shuffle fixed point equation corresponding to free moment-cumulant relations in a shuffle algebra of bicollection homomorphisms on the PRO of words insertions. Right half-shuffle and shuffle laws are interpreted in the framework of boolean and monotone non-commutative probability theory, respectively.

A web basis of invariant polynomials from noncrossing partitions

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.