We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.
In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of th…
We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence $\{a_n\}_{n\geq 0}$ is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence $\{a_{n+1}/a_{n}\}_{n\geq 0}$ is log-concave. Furthermore, we prove that if a sequence $\{a_n\}_{n\geq k}$ is ratio log-concave, then the sequence $…
We establish a combinatorial connection between the sequence $(i_{n,k})$ counting the involutions on $n$ letters with $k$ descents and the sequence $(a_{n,k})$ enumerating the semistandard Young tableaux on $n$ cells with $k$ symbols. This allows us to show that the sequences $(i_{n,k})$ are not log-concave for some values of $n$, hence answering a conjecture due to F. Brenti.
We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász's colorful Helly theorem, Bárány's colorful Carathéodory's theorem, and the colorful Tverberg theorem.
We show quantitative versions of classic results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of Bárány, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions.
Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand-Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or ``hives'') on a ``convex part'' of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applicat…
A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes using independent random variables. The principle outlines conditions under which the total variation distance between the true joint distribution and the approximation should be small, and was successfully exploited by Pittel in the cases of integer partitions and set partitions. We provide sufficient conditions for this principle to be true in a general context, valid for certain discrete probability di…
Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\Rscr$ of these faces forms a convex polygon. The question is: given a function $σ$ on the boundary edges of $G$, does there exist a concave function $f$ on $\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=σ(e)$ for each boundary edge $e=(u,v)$? The functions $σ$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\Rscr$ is a triangle, $C$ turns out to be exactly the cone of boundary …
Classes of set functions along with a choice of ground set are a bedrock to determine and develop corresponding variants of greedy algorithms to obtain efficient solutions for combinatorial optimization problems. The class of approximate constrained submodular optimization has seen huge advances at the intersection of good computational efficiency, versatility and approximation guarantees while exact solutions for unconstrained submodular optimization are NP-hard. What is an alternative to situations when submodularity does not hold? Can efficient and globally exact solutions be obtained? We i…
We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph. Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences. The above findings are a natural backd…
This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.
We study combinatorial and asymptotic properties of the rank of strongly unimodal sequences. We find a generating function for the rank enumeration function, and give a new combinatorial interpretation of the ospt-function introduced by Andrews, Chan, and Kim. We conjecture that the enumeration function for the number of unimodal sequences of a fixed size and varying rank is log-concave, and prove an asymptotic result in support of this conjecture. Finally, we determine the asymptotic behavior of the rank for strongly unimodal sequences, and prove that its values (when appropriately renormaliz…
Studying the virtual Euler characteristic of the moduli space of curves, Harer and Zagier compute the generating function $C_g(z)$ of unicellular maps of genus $g$. They furthermore identify coefficients, $κ^{\star}_{g}(n)$, which fully determine the series $C_g(z)$. The main result of this paper is a combinatorial interpretation of $κ^{\star}_{g}(n)$. We show that these enumerate a class of unicellular maps, which correspond $1$-to-$2^{2g}$ to a specific type of trees, referred to as O-trees. O-trees are a variant of the C-decorated trees introduced by Chapuy, Féray and Fusy. We exhaustively …
Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for the integer lattice grow polynomially in each fixed dimension, their variants with boxes as witness sets are uniformly bounded. We prove several colorful and fractional variations on this theorem. W…
The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz, which provides some information about the polynomial map $P|_{\X_1\times...\times\X_n}$ when only incomplete information about the polynomial $P(X_1,...c,X_n)$ is given. In a very general working frame, the grid points $x\in\X_1\times\...b\times\X_n$ which do not vanish under an algebraic solution -- a certain describing polynomial $P(X_1,...c,X_n)$ -- correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that th…
We study the lower tail behavior of the least singular value of an $n\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix. This setting has previously been considered in a series of influential works by Tao and Vu, most notably in …
Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial {of a complete graph} can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independe…
We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories. 1. We show a quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental theorem which has been used within diverse areas of mathematics. The Perron-Frobenius theorem states that every irreducible nonnegative matrix $R$ has a largest positive eigenvalue $r$, and every other eigenvalue $λ$ of $R$ is such that $\text{Re}λ<r$ and $|λ|\leq r$. We capture the notion of irreducibility through the widely studied notion of edge expansion $ϕ$ …
We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of $q$-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a co…
