Noncrossing Partitions Research Articles

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.

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.

The rank enumeration of certain parabolic non-crossing partitions

We consider $m$-divisible non-crossing partitions of $\{1,2,\ldots,mn\}$ with the property that for some $t\leq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapoton's $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.

Coxeter Pop-Tsack Torsing

Given a finite irreducible Coxeter group W with a fixed Coxeter element c, we define the Coxeter pop-tsack torsing operator Pop T :W→W by Pop T (w)=w·π T (w) -1 , where π T (w) is the join in the noncrossing partition lattice NC(w,c) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W. In each of these types, we obtain a natural lift from W to the dual braid monoid of W. We also prove that W is coincidental if and only if it has a unique forward orbit of size h. For arbitrary W, we show that the forward orbit of c -1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.

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.

New bounds for the site percolation threshold of the hexagonal lattice

The site percolation threshold of the hexagonal lattice satisfies 0.656 246 < p c < 0.739 695. For comparison, the largest previous lower bound of 0.652 703… was established in 1981, and the smallest previous upper bound of 0.743 359 was derived in 2007. The bound is obtained by using the substitution method to compare the hexagonal lattice site model to an exactly-solved two-parameter site percolation model on the martini lattice. Computational reductions involving graph-welding, symmetry, non-crossing partitions, and network flow computations overcome challenges to establishing stochastic ordering between the models.

Counting chains in the noncrossing partition lattice via the W-Laplacian

We give an elementary, case-free, Coxeter-theoretic derivation of the formula hnn!/|W| for the number of maximal chains in the noncrossing partition lattice NC(W) of a real reflection group W. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the W-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

Random matrices with independent entries: Beyond non-crossing partitions

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, have attracted much attention. The [Formula: see text]th moment of the limit equals the number of non-crossing pair-partitions of the set [Formula: see text]. There are several extensions of this result in the literature. In this paper, we consider a unifying extension which also yields additional results. Suppose [Formula: see text] is an [Formula: see text] symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems. Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erdős–Rényi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.

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.

Traces on diagram algebras I: Free partition quantum groups, random lattice paths and random walks on trees

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica–Speicher [5] and Weber [42]. For the infinite-dimensional Temperley–Lieb algebra (corresponding to the quantum group O N + $O^+_N$ ) and the Motzkin algebra ( B N + $B^+_N$ ), the classification of extremal traces implies a classification result for well-known types of central random lattice paths. For the 2-Fuss–Catalan algebra ( H N + $H_N^+$ ), we solve the classification problem by computing the minimal or exit boundary (also known as the absolute) for central random walks on the Fibonacci tree, thereby solving a probabilistic problem of independent interest, and to our knowledge the first such result for a nonhomogeneous tree. In the course of this article, we also discuss the branching graphs for all seven examples of free partition quantum groups, compute those that were not already known, and provide new formulae for the dimensions of their irreducible representations.

Troupes, cumulants, and stack-sorting

In several cases, a sequence of free cumulants that counts certain binary plane trees corresponds to a sequence of classical cumulants that counts the decreasing versions of the same trees. Using two new operations on binary plane trees that we call insertion and decomposition, we prove that this surprising phenomenon holds for families of trees that we call troupes. We give a simple characterization of troupes, showing that they are plentiful. Troupes provide a broad framework for generalizing several of the results that are known about West's stack-sorting map s. Indeed, we give new proofs of some of the main theorems underlying techniques that have been developed recently for understanding s; these new proofs are far more conceptual than the original ones, explain how the objects called valid hook configurations arise very naturally, and generalize to the context of troupes. To illustrate these general techniques, we enumerate 2-stack-sortable and 3-stack-sortable alternating permutations of odd length and 2-stack-sortable and 3-stack-sortable permutations whose descents are all peaks.The unexpected connection between troupes and cumulants provides a powerful new tool for analyzing the stack-sorting map that hinges on free probability theory. We give numerous applications of this method. For example, we show that if σ∈Sn−1 is chosen uniformly at random and des denotes the descent statistic, then the expected value of des(s(σ))+1 is(3−∑j=0n1j!)n. Furthermore, the variance of des(s(σ))+1 is asymptotically (2+2e−e2)n. We obtain similar results concerning the expected number of descents of postorder readings of decreasing binary plane trees of various types. We also obtain improved estimates for |s(Sn)| and an improved lower bound for the degree of noninvertibility of s:Sn→Sn. The combinatorics of valid hook configurations allows us to give two novel formulas that convert from free to classical (univariate) cumulants. The first formula is given by a sum over noncrossing partitions, and the second is given by a sum over 231-avoiding valid hook configurations. We pose several conjectures and open problems.

Refined lattice path enumeration and combinatorial reciprocity

It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fu{\ss}-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps. We give exact formulas for the number of such paths with fixed height, fixed number of returns and (i) fixed number of valleys, (ii) fixed number of valleys with $x$-coordinate divisible by $m$ and (iii) fixed number of valleys with $x$-coordinate not divisible by $m$. The enumeration (ii) combinatorially realizes the $H$-triangle appearing in a recent article of Krattenthaler and the first author (Algebr. Comb. 5, 2022) in the context of certain parabolic noncrossing partitions. Through a transformation formula due to Chapoton, we give an explicit formula for the associated $F$-triangle. We realize this polynomial combinatorially by means of generalized Schr\"oder paths as well as flats in certain hyperplane arrangements. Along the way we exhibit two new combinatorial reciprocity results.

Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

Let $Q$ be an acyclic quiver and $w \geqslant 1$ be an integer. Let $\mathsf {C}_{-w}({\mathbf {k}} Q)$ be the $(-w)$-cluster category of ${\mathbf {k}} Q$. We show that there is a bijection between simple-minded collections in $\mathsf {D}^b({\mathbf {k}} Q)$ lying in a fundamental domain of $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$. This generalises the same result of Iyama–Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf {H}$ of a bounded t-structure in a Hom-finite, Krull–Schmidt, ${\mathbf {k}}$-linear saturated triangulated category $\mathsf {D}$ is functorially finite in $\mathsf {D}$ if and only if $\mathsf {H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure $\mathfrak{m}$, then its fluctuation from the EED of the principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with $\mathfrak{m}$ by the Markov--Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of $\mathfrak{m}$. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.

New production matrices for geometric graphs

We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another, simple and elegant, way of counting the number of such objects. Counting geometric graphs is then equivalent to calculating the powers of a production matrix. Applying the technique of Riordan Arrays to these production matrices, we establish new formulas for the numbers of geometric graphs as well as combinatorial identities derived from the production matrices. Further, we obtain the characteristic polynomial and the eigenvectors of such production matrices.

Geometric combinatorics of polynomials I: The case of a single polynomial

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial p with d distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally ▪ metric that is locally Euclidean away from a finite set of interior points indexed by the critical points of p, and each of its 2-cells is a metric rectangle. From this planar rectangular 2-complex one can use metric graphs known as metric cacti and metric banyans to read off several pieces of combinatorial data: a chain in the partition lattice, a cyclic factorization of a d-cycle, a real noncrossing partition (also known as a primitive d-major), and the monodromy permutations for the polynomial. This article is the first in a series.

Combinatorics of type D Exceptional Sequences

Exceptional sequences are important sequences of quiver representations in the study of representation theory of algebras. They are also closely related to the theory of cluster algebras and the combinatorics of Coxeter groups. We combinatorially classify exceptional sequences of a family of type D Dynkin quivers, and we show how our model for exceptional sequences connects to the combinatorics of type D noncrossing partitions.

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua.