One of our result is that 5 measurable sets in $R^8$ always admit an equipartition by 2 hyperplanes. This is an instance of a general equipartition problem (formulated by B. Gr{\" u}nbaum and H. Hadwiger) which can be reduced to the question of (non)existence of a $W_k$-equivariant map where $W_k$ is the group of symmetries of a $k$-cube. We show that the computation of relevant cohomology/bordism obstruction classes often reduces to the question of enumerating the classes of immersed curves in $\mathbb{R}^2$ with a prescribed type and number of intersections with the coordinate axes, which in turn leads to a problem of enumerating classes of cyclic signed $AB$-words.
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …
We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D_8.
Let F be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and pro…
The partitioning of space by hyperplanes in the context of discrete classification problem is considered. We obtain some relations for the number of partitions and establish a recurrence relation for the maximal number of partitions of R^n by m hyperplanes. We rederive an explicit formula for the number of components into which the space can be partitioned by m hyperplanes.
We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky's theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $σ$-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.
We give an interpretation of the coefficients of the two variable refinement $D_{\Sh_n}(q,t)$ of the distance enumerator of the Shi hyperplane arrangement $\Sh_n$ in $n$ dimensions. This two variable refinement was defined by Stanley \cite{stan-rota} for the general $r$-extended Shi hyperplane arrangements. We give an interpretation when $r=1$. We define three natural three-dimensional partitions of the number $(n+1)^{n-1}$. The first arises from parking functions of length $n$, the second from special posets on $n$ vertices defined by Athanasiadis and the third from spanning trees on $n+1$ ve…
We study nested partitions of $R^d$ obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among $k$ sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any $t$ measures there is a path formed only by horizontal and vertical segments using at most $t-1$ turns that splits them by half simultaneously, and optimal mass-partitioning res…
Extending the notion of $r$-(class) regular partitions, we define $(r_{1},...,r_{m})$-class regular partitions. A partition identity is presented and described by making use of the Glaisher correspondence.
In this paper we study the topology of the strata, indexed by number partitions $λ$, in the natural stratification of the space of monic hyperbolic polynomials of degree $n$. We prove stabilization theorems for removing an independent block or an independent relation in $λ$. We also prove contractibility of the one-point compactifications of the strata indexed by a large class of number partitions, including $λ=(k^m,1^r)$, for $m\geq 2$. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition $λ$, o…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.
We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2 and j=4 2^k+1 which for k=0 reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. This is an example of a genuine combinatorial geometric result which involves Z_4-torsion in an essential way and cannot be obtained by the application of either Stiefel…
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$ and $s$. Classical notions such as Dyck paths and parking functions are naturally generalized by considering the set of partitions included in a given triangular partition. One of our striking results is that the restriction of the Young lattice to triangular partition has a planar Hasse diagram, wi…
The discriminantal arrangement is the space of configurations of $n$ hyperplanes in generic position in a $k$ dimensional space (see \cite{MS}). Differently from the case $k=1$ in which it corresponds to the well known braid arrangement, the discriminantal arrangement in the case $k>1$ has a combinatorics which depends from the choice of the original $n$ hyperplanes. It is known that this combinatorics is constant in an open Zariski set $\mathcal{Z}$, but to assess wether or not $n$ fixed hyperplanes in generic position belongs to $\mathcal{Z}$ proved to be a nontrivial problem. Even to simply…
We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.
We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.
We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropica…
An arrangement of k-semilines in the Euclidean (projective) plane or on the 2-sphere is called a k-fan if all semilines start from the same point. A k-fan is an $α$-partition for a probability measure $μ$ if $μ(σ_i)=α_i$ for each $i=1,...,k$ where $\{σ_i\}_{i=1}^k$ are conical sectors associated with the k-fan and $α= (α_1,... ,α_k)$. The set of all $α= (α_1,... ,α_m)$ such that for any collection of probability measures $μ_1,... ,μ_m$ there exists a common $α$-partition by a k-fan is denoted by ${\cal A}_{m,k}$. We prove, as a central result of this paper, that ${\cal A}_{3,2} = \{(s,t)\in \m…
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions…
The Grünbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the "admissible" triples $(d,j,k)$ such that for any $j$ masses in $\mathbb{R}^d$ there are $k$ hyperplanes that cut each of the masses into $2^k$ equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition $dk\ge j(2^k-1)$ is also sufficient. We develop a "join scheme" for this problem, such that non-existence of an $G_k$-equivariant map between spheres $(S^d)^{*k} \rightarrow S(W_k\oplus U_k^{\oplus j})$ t…
Integer compositions, integer partitions, Fibonacci numbers, and generalizations of these have recently been shown to be interconnected via two-toned tilings of horizontal grids. In this article, we present refinements of two-toned tilings, describe functions which analyze them, and apply these to generalizations of integer compositions and partitions which interpolate between the two.
In this paper we extend the block combinatorics partition theorems of Hindman and Milliken in the setting of the recursive system of the block Schreier families (B^xi) consisting of families defined for every countable ordinal xi. Results contain (a) a block partition Ramsey theorem for every countable ordinal xi (Hindman's theorem corresponding to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topolo…
For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not divisible by $l$ and occurring less than $k$ times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case $l=2$, we discuss the possibility …
We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions. We provide closed forms for the generating functions of these…
In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation.
In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.
For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs in \cite{BW} using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.
We study oscillations in the eigenfunctions for a fractional Schrödinger operator on the real line. An argument in the spirit of Courant's nodal domain theorem applies to an associated local problem in the upper half plane and provides a bound on the number of nodal domains for the extensions of the eigenfunctions. Using the combinatorial properties of noncrossing partitions, we turn the nodal domain bound into an estimate for the number of sign changes in the eigenfunctions. We discuss applications in the periodic setting and the Steklov problem on planar domains.
We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles ρ_{R_i} in the homology of the proper part \bar{L…
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.
We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In particular, we examine the correspondence between the wavefunctions of the XXZ type, Felderhof type and the boson type integrable models and symmetric polynomials such as the Schur, Grothendieck and symplectic Schur functions. We also give a brief report of our work on generalizing the correspondence between the Felderhof models and factorial Schur and symplectic …
The partition number $π(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojević, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.
We generalize the Yao-Yao partition theorem by showing that for any smooth measure in $R^d$ there exist equipartitions using $(t+1)2^{d-1}$ convex regions such that every hyperplane misses the interior of at least $t$ regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into $n$ regions. We also present a simple proof of a Borsuk-Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao--Yao partition theorem and the cen…
We present a study of three families of Kronecker coefficients, which we describe in terms of reduced Kronecker coefficients. This study is grounded on the generating function of the coefficients, proved by a bijection between two combinatorial objects. This study includes the connection between plane partitions and these three families of reduced Kronecker coefficients, providing us their combinatorial interpretation. As an application, we verify that the saturation hypothesis holds for our three families of reduced Kronecker coefficients. The study also includes other interpretation in terms…
In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of interlacing polynomials. Many of the open questions on stability and unimodality of polynomials pertain to the enumeration of faces of cell complexes. In this paper, we relate the theory of interlacing polynomials to the shellability of cell complexes. We first derive a sufficient condition for stability of the $h$-polynomial of a subdivision of a shellable comp…
Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of $d+1$ measures in $\mathbb{R}^d$ by parallel hyperplanes and of $d+2$ measures in $\mathbb{R}^d$ by concentric spheres. For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concent…
Sometimes we need the approximate value of the partition number in a simple and efficient way. There are already several formulae to calculate the partition number p(n). But they are either inconvenient for most people (not majored in math) who do not want do write programs, or unsatisfying in accuracy. By bringing in two parameters in the Hardy-Ramanujan's Asymptotic formula and fitting the data of the two parameters by least square method, iteration method and some other special designed methods, several revised elementary estimation formulae with high accuracy for p(n) are obtained. With th…
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 normalized volumes of the UPGMA cones for small n. Keywords: phylogenetic trees, polyhedral combinatorics, partition lat…
We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…
The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $λ$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irr…
We give the fermionic character formulas for the spaces of coinvariants obtained from level $k$ integrable representations of $\hat{\mathfrak sl}_2$. We establish the functional realization of the spaces dual to the coinvariant spaces. We parameterize functions in the dual spaces by rigged partitions, and prove the recursion relations for the sets of rigged partitions.
We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a polynomial in $q$ that specializes as $d_{n,r}(1) = n^r$ and $d_{n,r}(0) = B(r)$, the $r$th Bell number. We compute $d_{n,r}(q)$ in two ways. The first is purely combinatorial. We show that $d_{n,r}(q) = \sum_λf^λ(q) m_r^λ$, where $f^λ(q)$ is the $q$-hook number and $m_r^λ$ is the number of $r$-vacillating tableaux. Using a Schensted bijection, we write this …
Many natural counting problems arise in connection with the normal form of braids--and seem to have never been considered so far. Here we solve some of them by analysing the normality condition in terms of the associated permutations, their descents and the corresponding partitions. A number of different induction schemes appear in that framework.
We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps.
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the characteristic polynomial of the Shi threshold and the Catalan threshold arrangements was an open problem in Stanley's list of problems in [1]. Seunghyun Seo solved both the problems by clever arguments using the finite field method in [3,4]. However, in his paper, he left open the problem of computing the characteristic polynomial of a broader class of thr…
