In this note, we give a new necessary condition for the existence of non-trivial partitions of a finite vector space. Precisely, we prove that, if V is a finite vector space over a field of order q, then the number of the subspaces of minimum dimension t of a non-trivial partition of V is greater than or equal to q+t. Moreover, we give some extensions of a well known Beutelspacher-Heden's result on existence of T-partitions.
A vector space partition $\mathcal{P}$ of the projective space $\operatorname{PG}(v-1,q)$ is a set of subspaces in $\operatorname{PG}(v-1,q)$ which partitions the set of points. We say that a vector space partition $\mathcal{P}$ has type $(v-1)^{m_{v-1}} \dots 2^{m_2}1^{m_1}$ if precisely $m_i$ of its elements have dimension $i$, where $1\le i\le v-1$. Here we determine all possible types of vector space partitions in $\operatorname{PG}(7,2)$.
Based on discrete truncated powers, the beautiful Popoviciu's formulation for restricted integer partition function is generalized. An explicit formulation for two dimensional multivariate truncated power functions is presented. Therefore, a simplified explicit formulation for two dimensional vector partition functions is given. Moreover, the generalized Frobenius problem is also discussed.
A {\it vector space partition} is here a collection $\mathcal P$ of subspaces of a finite vector space $V(n,q)$, of dimension $n$ over a finite field with $q$ elements, with the property that every non zero vector is contained in a unique member of $\mathcal P$. Vector space partitions relates to finite projective planes, design theory and error correcting codes. In the first part of the talk I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the talk contains a survey of known results on the type of a vector space partition, more…
Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t,q) that contains m_d spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V(2t,q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.
We will show that in a space of dimension $m$, any family of $2^{m-1}$ distinct Hadamard vectors (where you can choose x or -x but not both) can be partitioned into Hadamard matrices if and only if $m=2^n$ for some n. We will solve this problem with a simple algorithm for assigning the vectors to the Hadamard matrices.
We prove identities generating higher dimensional vector partitions. We derive theorems for integer lattice points in the 2D first quadrant, then generalize the approach to find 3D and $n$-space lattice point vector region extensions. We also state combinatorial identities for Visible Point Vectors in 2D up to 5D and $n$D first hyperquadrant and hyperpyramid lattices. 2D and 3D theorems for vector partitions with binary components are also derived.
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 this model, 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…
A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring dimension in a vector space partition. To this end, we introduce the notion of $q^r$-divisible sets of $k$-subspaces in $\mathbb{F}_q^v$. By geometric arguments we obtain non-existence results for these objects, which then imply the improved result of Heden.
A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$.
Let $d, r \in \N$, $\|\cdot\|$ any norm on $\R^d$ and $B$ denote the unit ball with respect to this norm. We show that any sequence $v_1,v_2,...$ of vectors in $B$ can be partitioned into $r$ subsequences $V_1, ..., V_r$ in a balanced manner with respect to the partial sums: For all $n \in \N$, $\ell \le r$, we have $\|\sum_{i \le k, v_i \in V_\ell} v_i - \tfrac 1r \sum_{i \le k} v_i\| \le 2.0005 d$. A similar bound holds for partitioning sequences of vector sets. Both results extend an earlier one of Bárány and Grinberg (1981) to partitions in arbitrarily many classes.
We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age indivisible relational structure which is not weakly indivisible.
We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an algorithm to compute the numerators in their partial fraction decomposition "formally," i.e., as a formal expression in the parameter. We also analyze the type of generalized rational functions that appear during the execution of an algorithm based on iterated partial fraction decomposition and explain how to handle these objects.
Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope contains a finite set of lattice points corresponding to nonnegative integral linear combinations. The partition polytope associated to an integral $a$ is a rational convex polytope, and any rational convex polytope can be realized canonically as a pa…
A case-free proof is given that the entries of the $h$-vector of the cluster complex $Δ(Φ)$, associated by S. Fomin and A. Zelevinsky to a finite root system $Φ$, count elements of the lattice $\nc$ of noncrossing partitions of corresponding type by rank. Similar interpretations for the $h$-vector of the positive part of $Δ(Φ)$ are provided. The proof utilizes the appearance of the complex $Δ(Φ)$ in the context of the lattice $\nc$, in recent work of two of the authors, as well as an explicit shelling of $Δ(Φ)$.
Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf s},{\bf D})$, i.e., a number of integer solutions to a linear system ${\bf x} \ge 0, {\bf D x} = {\bf s}$. It is shown that $W({\bf s},{\bf D})$ can be expressed through the vector Bernoulli polynomials of higher order.
A result of Rosenthal says that for every $q>1$ and $n \in \mathbb{N}$ there is $N \in \mathbb{N}$ such that every sequence of $N$ distinct positive numbers contains, after a suitable translation and possible multiplication by $-1$, a subsequence $a_1,\ldots,a_n$ that is either $q$-increasing (that is, $a_{i+1}>qa_i$ for all $i$) or $1/q$-decreasing ($a_{i+1}<a_i/q$ for all $i$). One of our main theorems extends this result to vector sequences. This theorem is then used to prove the universality theorem for Tverberg partitions which says that, for every $d$ and $r$, every long enough sequence …
The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a finite vector space is the vector space analogue of a set partition. This paper develops the formulas for the number of direct-sum decompositions that are the q-analogs of the formulas for: (1) the number of set partitions with a given number partition signature; (2) the number of set partitions of an n-element set with m blocks (the Stirling numbers of the sec…
This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in a subsequent paper.
Ferrers graphs and tables of partitions are treated as vectors. Matrix operations are used for simple proofs of identities concerning partitions. Interpreting partitions as vectors gives a possibility to generalize partitions on negative numbers. Partitions are then tabulated into lattices and some properties of these lattices are studied. There appears a new identity counting Ferrers graphs packed consecutively into isoscele form. The lattices form the base for tabulating combinatorial identities.
We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating function of the number of generalized lattice paths and a new version of Chaundy-Bullard identity for the vector partition function.
We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $Π_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of spheres of dimension $(n-2)$. We extend on this result to show that $Π_{n,s}$ is edge-lexicographic shellable. We then use this edge-labeling to find a recursive expression for the number of spheres, and show that when $s=1$ the number of spheres is equal to the number of complete non-ambiguous trees, first defined in 2014 by Aval, Boussicault, Bouvel and Silim…
We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A\_r$, $B\_r$, $C\_r$, $D\_r$. We therefore get efficient {\tt Maple} programs that compute for these Lie algebras: the multiplicity of a weight in an irreducible finite-dimensional representation; the decomposition coefficients of the tensor product of two irreducible finite-dimensional representations. These programs can also calculate associated Ehrhart quasipolynomials.
Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake \& Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be use…
We consider the {\em vector partition problem}, where $n$ agents, each with a $d$-dimensional attribute vector, are to be partitioned into $p$ parts so as to minimize cost which is a given function on the sums of attribute vectors in each part. The problem has applications in a variety of areas including clustering, logistics and health care. We consider the complexity and parameterized complexity of the problem under various assumptions on the natural parameters $p,d,a,t$ of the problem where $a$ is the maximum absolute value of any attribute and $t$ is the number of agent types, and raise so…
We study the Kronecker coefficients $g_{λ, μ, ν}$ via a formula that was described by Mishna, Rosas, and Sundaram, in which the coefficients are expressed as a signed sum of vector partition function evaluations. In particular, we use this formula to determine formulas to evaluate, bound, and estimate $g_{λ, μ, ν}$ in terms of the lengths of the partitions $λ, μ$, and $ν$. We describe a computational tool to compute Kronecker coefficients $g_{λ, μ, ν}$ with $\ell(μ) \leq 2,\ \ell(ν) \leq 4,\ \ell(λ) \leq 8$. We present a set of new vanishing conditions for the Kronecker coefficients by relatin…
If a pure simplicial complex is partitionable, then its $h$-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex $Δ$, we construct a complex $Γ\supseteq Δ$ of the same dimension such that both $Γ$ and the relative complex $(Γ,Δ)$ are partitionable. This allows us to rewrite the $h$-vector of any pure simplicial complex as the difference of two $h$-vectors of partitionable complexes, giving an analogous interpretation of the $h$-vector of a non-partitionable complex. By contrast, for a given complex $Δ$ it is not always possibl…
Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective proofs are presented.
This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the number of partitions on $n$. The analogy with convex polytopes indicates that the linear inequalities satisfied by $fG$ may be both interesting and accessible. Such would provide inequalities both sharp and subtle on the combinatorial structure of $G$. These may be related to Ramsey theory.
Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions o…
We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new `direct' proof of Rado's result.
We revisit the following problem (along with its higher dimensional variant): Given a set $S$ of $n$ points inside an axis-parallel rectangle $U$ in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in $U$ but contains no points of $S$. (I) We present an algorithm that finds a large empty box amidst $n$ points in $[0,1]^d$: a box whose volume is at least $\frac{\log{d}}{4(n + \log{d})}$ can be computed in $O(n+d \log{d})$ time. (II) To better analyze the above approach, we introduce the concepts of perfect vector sets and properly overlapping partitions, in connectio…
We prove a lower bound expansion on the probability that a random $\pm 1$ matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form $e_i \pm e_j$, which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form $e_i \pm e_j \pm e_k \pm e_\ell$, which corresponds to the partition 1111. The fifth…
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplect…
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb{N}$ is finitely colored, there must be some $\overset{\rightarrow}{x}$ with entries from $\mathbb{N}$ such that all entries of $A \overset{\rightarrow}{x}$ are in the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition r…
We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb N$ is finitely colored, there must be some $\vec{x}$ with entries from $\mathbb N$ such that all entries of $A\vec{x}$ are in some color class. In [6] and [8], the notion of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite image partition regular matrices. It was shown that centrally image partition regular matrices are closed under diagonal sum. In the present paper, we show that the diagonal sum of two matrices, one …
We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension.
Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in $\mathbb{R}^n$ obtained from the partitions of the fixed positive integer $n$. These distributions arise naturally when considering equally-likely random permutations on the set of $n$ letters. For one of the distributions, the expectation vector and covariance matrix is derived. For the other distribution, conjectures for several elements of the expectation vector are provided.
We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver $…
In this paper we consider the sequence whose n^{th} term is the number of h-vectors of length n. We show that the n^{th} term of this sequence is bounded above by the n^{th} Fibonacci number and bounded below by the number if integer partitions of n into distinct parts. Further we show embedded sequences that directly relate to integer partitions.
In this paper we find exact formulas for the numbers of partitions and compositions of an element into $m$ parts over a finite field, i.e. we find the number of nonzero solutions of the equation $x_1+x_2+...+x_m=z$ over a finite field when the order does not matter and when it does, respectively. We also give an application of our results in the study of polynomials of prescribed ranges over finite fields.
Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the {\em Shaped Partition Polytope} defined as the convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially ma…
Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure.
In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some surjection having the poset as domain. We then obtain combinatorial characterisations of such notions in terms of blocks, without reference to surjection. Finally, we give a further, equivalent definition …
Let $\mathbf a=(a_1,\ldots,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.
