Noncrossing Partitions Research Articles

Weyl Group $${\varvec{q}}$$ q -Kreweras Numbers and Cyclic Sieving

Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A, B, C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A, B, C, D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

Weak Z-Property of the Absolute Order on Groups Generated by Sets Closed Under Taking Inverses

The Bruhat order is a well-studied partial order on Coxeter groups and Schubert varieties. Deodhar provided several characterizations of the Bruhat order, including the so-called “Z-property.” Another partial order on Coxeter groups that is relevant in combinatorics is the absolute order, which extends the notion of the classical noncrossing partitions to Coxeter groups. In this note, we prove a result that implies that the absolute order satisfies a weaker version of the Z-property.

Electroid varieties and a compactification of the space of electrical networks

We construct a compactification of the space of circular planar electrical networks studied by Curtis–Ingerman–Morrow [4] and Colin de Verdière–Gitler–Vertigan [3], using cactus networks. We embed this compactification as a linear slice of the totally nonnegative Grassmannian, and relate Kenyon and Wilson's grove measurements to Postnikov's boundary measurements. Intersections of the slice with the positroid stratification leads to a class of electroid varieties, indexed by matchings. The uncrossing partial order on matchings arising from electrical networks is shown to be dual to a subposet of affine Bruhat order. The analogues of matroids in this setting are certain distinguished collections of non-crossing partitions.

On the graded dual numbers, arcs, and non-crossing partitions of the integers

We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the bounded derived category, and of the perfect complexes, in terms of non-crossing partitions. We also make some comments on the symmetries of these lattices, exceptional collections, and the analogous problem for the unbounded derived category.

New refined enumerations of set partitions related to sorting

ABSTRACTIn this paper, we consider a statistic defined on the sequential representation of a finite set partition which tracks the number of steps required, i.e. pushes, to put the sequence in weakly increasing order. A generating function is computed for the distribution of the statistic that records the number of pushes considered jointly with some other combinatorial parameters on the set of partitions of for each fixed n. Some further attention is paid to the distribution for the number of pushes by itself. Generating function formulas are also found for the comparable distribution on the set of non-crossing and non-nesting partitions of . As a consequence of our results, one obtains new polynomial generalizations of the Stirling, Bell and Catalan numbers.

SO(N) Lattice Gauge Theory, planar and beyond

AbstractLattice gauge theories have been studied in the physics literature as discrete approximations to quantum Yang‐Mills theory for a long time. Primary statistics of interest in these models are expectations of the so‐called Wilson loop variables. In this article we continue the program initiated by Chatterjee [3] to understand Wilson loop expectations in lattice gauge theories in a certain limit through gauge‐string duality. The objective in this paper is to better understand the underlying combinatorics in the strong coupling regime by giving a more geometric picture of string trajectories involving correspondence to objects such as decorated trees and noncrossing partitions. Using connections with free probability theory, we provide an elaborate description of loop expectations in the planar setting, which provides certain insights into structures of higher‐dimensional trajectories as well. Exploiting this, we construct an example showing that in any dimension, the Wilson loop area law lower bound does not hold in full generality. © 2018 Wiley Periodicals, Inc.

New results on production matrices for geometric graphs

We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. For instance, a formula for the number of connected geometric graphs with given root degree, drawn on a set of n points in convex position in the plane, is presented. Further, we find the characteristic polynomials and we provide a characterization of the eigenvectors of the production matrices.

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.

Δ-Cumulants in terms of moments

The [Formula: see text]-convolution of real probability measures, introduced by Bożejko, generalizes both free and Boolean convolutions. It is linearized by the [Formula: see text]-cumulants, and Yoshida gave a combinatorial formula for moments in terms of [Formula: see text]-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the [Formula: see text]-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida’s formula involving Schröder trees.

Restricted growth function patterns and statistics

A restricted growth function (RGF) of length n is a sequence w=w1w2…wn of positive integers such that w1=1 and wi≤1+max⁡{w1,…,wi−1} for i≥2. RGFs are of interest because they are in natural bijection with set partitions of {1,2,…,n}. An RGF w avoids another RGF v if there is no subword of w which standardizes to v. We study the generating functions ∑w∈Rn(v)qst(w) where Rn(v) is the set of RGFs of length n which avoid v and st(w) is any of the four fundamental statistics on RGFs defined by Wachs and White. These generating functions exhibit interesting connections with multiset permutations, integer partitions, and two-colored Motzkin paths, as well as noncrossing and nonnesting set partitions.

Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n)

We consider closed meandric systems, and their equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions $NC(n)$. In this equivalent description, the number of components of a random meandric system of order $n$ translates into the distance between two partitions in $NC(n)$. We focus on a class of couples $(\pi,\rho)\in NC(n)^2$ -- namely the ones where $\pi$ is conditioned to be an interval partition -- for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a non-trivial class of meanders (i.e. connected meandric systems), which we call "meanders with shallow top", and which can be explicitly enumerated. Moreover, the expected number of components for a random "meandric system with shallow top", is asymptotically $(9n+28)/27$. Our calculations concerning expected number of components are related to the idea of taking the derivative at $t=1$ in a semigroup for the operation $\boxplus$ of free probability (but the underlying considerations are presented in a self-contained way, and can be followed without assuming a free probability background). Let $c_{n}'$ denote the expected number of components of a general, unconditioned, meandric system of order $n$. A variation of the methods used in the shallow-top case allows us to prove that $\mathrm{lim\ inf}_{n\to\infty}c_{n}'/n\geq0.17$. We also note that, by a direct elementary argument, one has $\mathrm{lim\ sup}_{n\to\infty}c_{n}'/n\leq0.5$. These bounds support the conjecture that $c_{n}'$ follows a regime of "constant times $n$" (where numerical experiments suggest that the constant should be $\approx0.23$).

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 Partitions Coming From Undesired Parking Spaces

Considere las particiones sin cruces de un conjunto de n elementos que no usan el bloque {n - 1, n}, ni usan a la vez el bloque {n} y un bloque que contenga a 1 y n - 1. En este artículo estudiamos el subposet del retículo de particiones sin cruces inducido por estos elementos. Probamos que este retículo es supersoluble, y por lo tanto es lexicogríaficamente descascarable. También damos un modelo combinatorio de las bases NBB de este retículo y derivamos una fórmula explicita para el valor de su función de Möbius entre el elemento mínimo y el máximo. Este trabajo es motivado por un artículo reciente de M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, e I. Nicolas en el cual introducen un subposet del retículo de particiones sin cruces que es determinado por funciones de parqueo con ciertas entradas prohibidas. En particular, ellos conjeturan que el poset resultante siempre tiene un complejo de orden contráctil. En este artículo probamos esta conjetura, sumergiendo su poset en el nuestro y mostrando que esta inmersión hereda la descascarabilidad lexicográfica.

Counting water cells in bargraphs of compositions and set partitions

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

Set partitions and parity successions

By a (parity) succession within a sequence w = w1w2 … , we will mean an index i such that wi ≡ wi + 1 (mod2). In this paper, we address the problem of counting successions in set partitions, represented sequentially as restricted growth functions. Among our results, we find explicit formulas for the relevant generating functions and for the number of parity-al­ternating set partitions, i.e., those having no successions. We also compute a formula for the total number of successions within all partitions of a fixed length and number of blocks, and a combinatorial proof of this result is provided. Finally, we consider the problem of counting successions in non-crossing partitions, i.e., those having no occurrence of the pattern 1212, and determine, with the aid of programming, formulas for the generating functions.

The classification of tensor categories of two-colored noncrossing partitions

Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These so called categories of partitions are exactly the tensor categories being used in the theory of Banica–Speicher quantum groups (also called easy quantum groups). In our approach, we additionally allow a coloring of the points. This serves as the basis for the introduction of unitary Banica–Speicher quantum groups, which is done in a separate article. The present article however is purely combinatorial. We find all categories of two-colored noncrossing partitions. For doing so, we extract certain parameters with values in the natural numbers specifying the colorization of the categories on a global as well as on a local level. It turns out that there are ten series of categories, each indexed by one or two parameters from the natural numbers, plus two additional categories. This is just the beginning of the classification of categories of two-colored partitions and we point out open problems at the end of the article.

Simple dual braids, noncrossing partitions and Mikado braids of type Dn

We show that the simple elements of the dual Garside structure of an Artin group of type $D_n$ are Mikado braids, giving a positive answer to a conjecture of Digne and the second author. To this end, we use an embedding of the Artin group of type $D_n$ in a suitable quotient of an Artin group of type $B_n$ noticed by Allcock, of which we give a simple algebraic proof here. This allows one to give a characterization of the Mikado braids of type $D_n$ in terms of those of type $B_n$ and also to describe them topologically. Using this topological representation and Athanasiadis and Reiner's model for noncrossing partitions of type $D_n$ which can be used to represent the simple elements, we deduce the above mentioned conjecture.

K-divisible random variables in free probability

We introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not divisible by k.First, we consider the combinatorial convolution ⁎ in the lattices NC of non-crossing partitions and NCk of k-divisible non-crossing partitions and show that convolving k times with the zeta function in NC is equivalent to convolving once with the zeta function in NCk. Furthermore, when x is k-divisible, we derive a formula for the free cumulants of xk in terms of the free cumulants of x, involving k-divisible non-crossing partitions.Second, we prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (on a and s) of degree k−2 on s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results.Next, we show that free multiplicative convolution between a measure concentrated in the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed.Finally, we concentrate on free additive powers of k-symmetric distributions and prove that μ⊞t is a well defined k-symmetric probability measure, for all t>1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ→μk. We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.

Operator-valued Jacobi parameters and examples of operator-valued distributions

In the setting of distributions taking values in a C⁎-algebra B, we define generalized Jacobi parameters and study distributions they generate. These include numerous known examples and one new family, of B-valued free binomial distributions, for which we are able to compute free convolution powers. Moreover, we develop a convenient combinatorial method for calculating the joint distributions of B-free random variables with Jacobi parameters, utilizing two-color non-crossing partitions. This leads to several new explicit examples of free convolution computations in the operator-valued setting. Additionally, we obtain a counting algorithm for the number of two-color non-crossing pairings of relative finite depth, using only free probabilistic techniques. Finally, we show that the class of distributions with Jacobi parameters is not closed under free convolution.