Lattice Of Partitions Research Articles

The Splitting Process in Free Probability Theory

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Möbius calculus over the lattice of non-crossing partitions. In this work, we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed-point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work.

A natural extension of the Young partition lattice

Abstract Recently Andrews introduced the concept of signed partition: a signed partition is a finite sequence of integers ak, . . . , a1, a−1, . . . , a−l such that ak ≥ ... ≥ a1 > 0 > a−1 ≥ ... ≥ a−l. So far the signed partitions have been studied from an arithmetical point of view. In this paper we first generalize the concept of signed partition and we next use such a generalization to introduce a partial order on the set of all the signed partitions. Furthermore, we show that this order has many remarkable properties and that it generalizes the classical order on the Young lattice.

The role of residue and quotient tables in the theory of k-Schur functions

Recently, residue and quotient tables were defined by Fishel and the author, and were used to describe strong covers in the lattice of k-bounded partitions. In this paper, we prove (and, in some cases, conjecture) that residue and quotient tables can be used to describe many other results in the theory of k-bounded partitions and k-Schur functions, including k-conjugates, weak horizontal and vertical strips, and the Murnaghan–Nakayama rule. Evidence is presented for the claim that one of the most important open questions in the theory of k-Schur functions, a general rule that would describe their product, can be also concisely stated in terms of residue tables.

A Note on Statistical Averages for Oscillating Tableaux

Oscillating tableaux are certain walks in Young's lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits. We show that the average weight of all oscillating tableaux of shape $\lambda$ and length $|\lambda|+2n$ (where $|\lambda|$ denotes the size of $\lambda$ and $n \in \mathbb{N}$) has a surprisingly simple formula: it is a quadratic polynomial in $|\lambda|$ and $n$. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm of Propp and Roby may lead to a more conceptual proof of this result and reveal a hidden symmetry in the set of perfect matchings.

Dwyer-Kan homotopy theory of enriched categories

We study the Fadell-Husseini index of the configuration space F(R^d,n) with respect to different subgroups of the symmetric group S_n. For p prime and d>0, we completely determine Index_{Z/p}(F(R^d,p);F_p) and partially describe Index{(Z/p)^k}(F(R^d,p^k);F_p). In this process we obtain results of independent interest, including: (1) an extended equivariant Goresky-MacPherson formula, (2) a complete description of the top homology of the partition lattice Pi_p as an F_p[Z_p]-module, and (3) a generalized Dold theorem for elementary abelian groups. The results on the Fadell-Husseini index yield a new proof of the Nandakumar & Ramana Rao conjecture for a prime. For n=p^k a prime power, we compute the Lusternik-Schnirelmann category cat(F(R^d,n)/S_n)=(d-1)(n-1). Moreover, we extend coincidence results related to the Borsuk-Ulam theorem, as obtained by Cohen & Connett, Cohen & Lusk, and Karasev & Volovikov.

A combinatorial model for exceptional sequences in type A

Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya’s work) to classify exceptional sequences of representations of $Q$, the linearly ordered quiver with $n$ vertices. We also show how to use variations of this model to classify $c$-matrices of $Q$, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of $c$-matrices, we also give an interpretation of $c$-matrix mutation in terms of our noncrossing trees with directed edges. Les suites exceptionnelles sont certaines suites ordonnées de représentations de carquois. Nous utilisons des arbres aux arêtes étiquetés et aux sommets dans le bord d’un disque (expansion sur le travail de T. Araya) pour classifier les suites exceptionnelles de représentations du carquois linéairement ordonné à $n$ sommets. Nous exploitons des variations de ce modèle pour classifier les $c$-matrices dudit carquois, pour interpréter les suites exceptionnelles comme des extensions linéaires, et pour donner une bijection élémentaire entre les suites exceptionnelles et certaines chaînes dans le réseau des partitions sans croisement. Dans le cas des $c$-matrices, nous donnons également une interprétation de la mutation des $c$-matrices en termes des arbres sans croisement aux arêtes orientés.

On non-conjugate Coxeter elements in well-generated reflection groups

Given an irreducible well-generated complex reflection group $W$ with Coxeter number $h$, we call a Coxeter element any regular element (in the sense of Springer) of order $h$ in $W$; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in $W$ under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element $c$ is a Coxeter element if and only if there exists a simple system $S$ of reflections such that $c$ is the product of the generators in $S$. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of $W$ associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of $W$ on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order. Étant donnés un groupe de réflexion complexe $W$, irréductible et bien engendré, et $h$ son nombre de Coxeter, nous appelons élément de Coxeter un élément régulier (au sens de Springer) d’ordre $h$; ceci est une extension de la notion la plus habituelle d’élément de Coxeter. Nous montrons que l’ensemble de ces éléments de Coxeter forme une seule orbite sous l’action des automorphismes de réflexion de $W$. Pour les groupes de Coxeter et de Shephard, ceci implique qu’un élément $c$ est un élément de Coxeter si et seulement s’il existe un système simple $S$ de réflexions tel que $c$ soit le produit des générateurs dans $S$. Nous déduisons de cette propriété plusieurs autres résultats. En particulier, nous obtenons que tous les treillis de partitions non-croisées de $W$, associés à différents éléments de Coxeter, sont isomorphes. Nous montrons également qu’il existe une action simplement transitive du groupe de Galois du corps de définition de $W$ sur l’ensemble des classes de conjugaison d’éléments de Coxeter. Enfin, nous étendons plusieurs de ces propriétés au cas des éléments réguliers d’ordre quelconque.

The Toric h-Vector of a Cubical Complex in Terms of Noncrossing Partition Statistics

This paper introduces a new and simple statistic on noncrossing partitions that expresses each coordinate of the toric h-vector of a cubical complex, written in the basis of the Adin h-vector entries, as the total weight of all noncrossing partitions. The same model may also be used to obtain a very simple combinatorial interpretation of the contribution of a cubical shelling component to the toric h-vector. In thismodel, a strengthening of the symmetry expressed by the Dehn-Sommerville equations may be derived from the self-duality of the noncrossing partition lattice, exhibited by the involution of Simion and Ullman.

EL-shellability and noncrossing partitions associated with well-generated complex reflection groups

In this article we prove that the lattice of noncrossing partitions is EL-shellable when associated with the well-generated complex reflection group of type G(d,d,n), for d,n≥3, or with the exceptional well-generated complex reflection groups which are no real reflection groups. This result was previously established for the real reflection groups and it can be extended to the well-generated complex reflection group of type G(d,1,n), for d,n≥3, as well as to three exceptional groups, namely G25,G26 and G32, using a braid group argument. We thus conclude that the lattice of noncrossing partitions of any well-generated complex reflection group is EL-shellable. Using this result and a construction by Armstrong and Thomas, we conclude further that the poset of m-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. Finally, we derive results on the Möbius function of these posets previously conjectured by Armstrong, Krattenthaler and Tomie.

A combinatorial proof of strict unimodality for [formula omitted]-binomial coefficients

I. Pak and G. Panova recently proved that the q-binomial coefficient m+nmq is a strictly unimodal polynomial in q for m,n≥8, via the representation theory of the symmetric group. We give a direct combinatorial proof of their result by characterizing when a product of chains is strictly unimodal and then applying O’Hara’s structure theorem for the partition lattice L(m,n). In fact, we prove a stronger result: if m,n≥8d, and 2d≤r≤mn/2, then the rth rank of L(m,n) has at least d more elements than the next lower rank.

Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices

Inequalities for convex functions on the lattice of partitions of a set partially ordered by refinement lead to multivariate generalizations of inequalities of Cauchy and Rogers-Holder and to eigenvalue inequalities needed in the theory of population dynamics in Markovian environments: If A is an n× n nonnegative matrix, n > 1, D is an n× n diagonal matrix with positive diagonal elements, r(·) is the spectral radius of a square matrix, r(A) > 0, and x ∈ [1,∞), then rx−1(A)r(DxA) ≥ rx(DA). When A is irreducible and ATA is irreducible and x > 1, then equality holds if and only if all elements of D are equal. Conversely, when x > 1 and rx−1(A)r(DxA) = rx(DA) if and only if all elements of D are equal, then A is irreducible and ATA is irreducible.

Counting Dyck Paths by Area and Rank

The set of Dyck paths of length 2n inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: area (the area under the path) and rank (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting (q, t)-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: One refines an identity of Carlitz and Riordan; the other refines the notion of γ-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane’s correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the n-cycle (12· · ·n) in the absolute order on the symmetric group.

On maximal chains in the non-crossing partition lattice

A weak order on the set of maximal chains of the non-crossing partition lattice is introduced and studied. A 0-Hecke algebra action is used to compute the radius of the graph on these chains in which two chains are adjacent if they differ in exactly one element.

Reflection arrangements and ribbon representations

Ehrenborg and Jung (2011) recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a Specht module. Their work unifies that of Calderbank, Hanlon, Robinson (1986), and Wachs (1996). By focusing on the underlying geometry, we strengthen and extend these results from type A to all real reflection groups and the complex reflection groups known as Shephard groups.

Tropical decomposition of Young’s partition lattice

Young’s partition lattice L(m,n) consists of integer partitions having m parts where each part is at most n. Using methods from complex algebraic geometry, R. Stanley proved that this poset is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that it has a symmetric chain decomposition, which is a stronger property. Despite many efforts, this conjecture has only been proved for min(m,n)≤4. In this paper, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we find that the resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of “sufficiently generic” partitions.

A Poset Fiber Theorem for Doubly Cohen-Macaulay Posets and Its Applications

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen- Macaulayness of a poset. Applications to complexes of injective words are also included.

The h, k-equal Partition Lattice is EL-shellable when h $\geq$ k

It has been shown that the h, k-equal partition lattice \(\tilde \Pi_n^{h, k}\) is EL-shellable when h < k. We produce an EL-shelling for \(\tilde \Pi_n^{h, k}\) when n ≥ h ≥ k ≥ 2 and observe that, in this shelling, there are no weakly decreasing chains. This shows that \(\tilde \Pi_n^{h, k}\) is contractible for such values of h and k, which can also be seen by the fact that \(\tilde \Pi_n^{h, k}\) is noncomplemented.

Chain lengths in the dominance lattice

We construct the largest union of two or three chains in the lattice of partitions of n under the dominance order. This construction provides a framework for finding the largest union of four or more chains. We also see how these chain lengths relate to a theorem of Gansner and Saks and the problem of finding the largest antichain.

On the Shard Intersection Order of a Coxeter Group

Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the case of the symmetric group, we provide combinatorial models for shard intersections of all classical types and use this understanding to prove that the shard intersection order is EL-shellable. Further, inspired by work of Simion and Ullman on the lattice of noncrossing partitions, we show that the shard intersection order on the symmetric group admits a symmetric Boolean decomposition, i.e., a partition into disjoint Boolean algebras whose middle ranks coincide with the middle rank of the poset. Our decomposition also yields a new symmetric Boolean decomposition of the noncrossing partition lattice.

Polyhedral combinatorics of UPGMA cones

Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the spherical volumes of the UPGMA cones for small n.