Combinatorial Aspects Research Articles

Weighted PBW degenerations and tropical flag varieties

We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type [Formula: see text]. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.

Tomographic reconstruction of 2-convex polyominoes using dual Horn clauses

Among the very many research interests of Maurice Nivat, a special role, according to the produced literature, was played by the study of the algorithmic and combinatorial aspects of connected finite discrete sets of points called polyominoes. In particular, he addressed the problem of a faithful reconstruction of some subclasses of them by imposing convexity constraints. The present study fits in this research line, and relies on a well known algorithm that Maurice Nivat and co-authors defined 1996 for the reconstruction of hv-convex polyominoes by orthogonal projections in polynomial time. Here, we consider a hierarchy on this class of polyominoes, and we continue a longstanding research on the reconstruction of its first levels by specializing the above mentioned result. The algorithm we propose bases on the possibility of characterizing the elements of the second level of the hierarchy, by a logic formula belonging to Dual-Horn and so polynomially solvable. Some related open problems are also presented.

Bernstein operators and super-Schur functions: combinatorial aspects

The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.

Two cryptomorphic formalizations of projective incidence geometry

Incidence geometry is a well-established theory which captures the very basic properties of all geometries in terms of points belonging to lines, planes, etc. Moreover, projective incidence geometry leads to a simple framework where many properties can be studied. In this article, we consider two very different but complementary mathematical approaches formalizing this theory within the Coq proof assistant. The first one consists of the usual and synthetic geometric axiom system often encountered in the literature. The second one is more original and relies on combinatorial aspects through the notion of rank which is based on the matroid structure of incidence geometry. This paper mainly contributes to the field by proving the equivalence between these two approaches in both 2D and 3D. This result allows us to study the further automation of many proofs of projective geometry theorems. We give an overview of techniques that will be heavily used in the equivalence proof and are generic enough to be reused later in yet-to-be-written proofs. Finally, we discuss the possibilities of future automation that can be envisaged using the rank notion.

PBW Degeneration: Algebra, Geometry, and Combinatorics

We review the recent progress in the theory of Poincare–Birkhoff–Witt degenerations of irreducible representations of simple Lie algebras. We describe algebraic, geometric, and combinatorial aspects of the theory.

Combinatorial Aspects of the Quantized Universal Enveloping Algebra of $$\mathfrak {sl}_{n+1}$$ sl n + 1

Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon $$\mathscr {U}_h(\mathfrak {sl}_2)$$ , the quantized universal enveloping algebra of the Lie algebra $$\mathfrak {sl}_2$$ . In this paper, combinatorial structure in $$\mathscr {U}_h(\mathfrak {sl}_2)$$ is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case $$n=1$$ . We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s $$sR$$ -matrix, but also for the arguably mysterious ribbon elements of $$\mathscr {U}_h(\mathfrak {sl}_2)$$ . Finally, we extend these techniques to the higher-dimensional algebras $$\mathscr {U}_h(\mathfrak {sl}_{n+1})$$ . While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.

Combinatorial aspects of classical resolution of singularities

We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka’s maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.

Tverberg’s theorem is 50 years old: A survey

This survey presents an overview of the advances around Tverberg’s theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg’s theorem and its applications. The survey contains several open problems and conjectures.

Robust allocation of operating rooms: A cutting plane approach to handle lognormal case durations

The problem of allocating operating rooms (OR) to surgical cases is a challenging task, involving both combinatorial aspects and uncertainty handling. We formulate this problem as a parallel machines scheduling problem, in which job durations follow a lognormal distribution, and a fixed assignment of jobs to machines must be computed. We propose a cutting-plane approach to solve the robust counterpart of this optimization problem. To this end, we develop an algorithm based on fixed-point iterations that identifies worst-case scenarios and generates cut inequalities. The main result of this article uses Hilbert’s projective geometry to prove the convergence of this procedure under mild conditions. We also propose two exact solution methods for a similar problem, but with a polyhedral uncertainty set, for which only approximation approaches were known. Our model can be extended to balance the load over several planning periods in a rolling horizon. We present extensive numerical experiments for instances based on real data from a major hospital in Berlin. In particular, we find that: (i) our approach performs well compared to a previous model that ignored the distribution of case durations; (ii) compared to an alternative stochastic programming approach, robust optimization yields solutions that are more robust against uncertainty, at a small price in terms of average cost; (iii) the longest expected processing time first (LEPT) heuristic performs well and efficiently protects against extreme scenarios, but only if a good prediction model for the durations is available. Finally, we draw a number of managerial implications from these observations.

Open problems on [formula omitted]-orbit polytopes

We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.

On Partial Barycentric Subdivision

The $$l{\mathrm{th}}$$ partial barycentric subdivision is defined for a $$(d-1)$$ -dimensional simplicial complex $$\Delta $$ and studied along with its combinatorial and geometric aspects. We analyze the behavior of the f- and h-vector under the lth partial barycentric subdivision extending previous work of Brenti and Welker on the standard barycentric subdivision – the case $$l = d$$ . We discuss and provide properties of the transformation matrices sending the f- and h-vector of $$\Delta $$ to the f- and h-vector of its lth partial barycentric subdivision. We conclude with open problems.

Bayesian Poisson calculus for latent feature modeling via generalized Indian Buffet Process priors

Statistical latent feature models, such as latent factor models, are models where each observation is associated with a vector of latent features. A general problem is how to select the number/types of features, and related quantities. In Bayesian statistical machine learning, one seeks (nonparametric) models where one can learn such quantities in the presence of observed data. The Indian Buffet Process (IBP), devised by Griffiths and Ghahramani (2005), generates a (sparse) latent binary matrix with columns representing a potentially unbounded number of features and where each row corresponds to an individual or object. Its generative scheme is cast in terms of customers entering sequentially an Indian Buffet restaurant and selecting previously sampled dishes as well as new dishes. Dishes correspond to latent features shared by individuals. The IBP has been applied to a wide range of statistical problems. Recent works have demonstrated the utility of generalizations to nonbinary matrices. The purpose of this work is to describe a unified mechanism for construction, Bayesian analysis, and practical sampling of broad generalizations of the IBP that generate (sparse) matrices with general entries. An adaptation of the Poisson partition calculus is employed to handle the complexities, including combinatorial aspects, of these models. Our work reveals a spike and slab characterization, and also presents a general framework for multivariate extensions. We close by highlighting a multivariate IBP with condiments, and the role of a stable-Beta Dirichlet multivariate prior.

A Combinatorial Study on Quiver Varieties

This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating function of Poincar\'e polynomials of quiver varieties in rank 1 cases. Our main tools are cores and quotients of Young diagrams. In the second part, we give a brief survey of instanton counting in physics, where quiver varieties appear as moduli spaces of instantons, focusing on its combinatorial aspects.

Combinatorics of $\lambda$-terms: a natural approach

We consider combinatorial aspects of $\lambda$-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for $\lambda$-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral $\lambda$-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size sampler for Motzkin trees of Bodini et alli. Using the powerful theory of analytic combinatorics, we state several results concerning the asymptotic growth rate of $\lambda$-terms in neutral, normal, and head normal forms. Finally, we investigate the asymptotic density of $\lambda$-terms containing arbitrary fixed subterms showing that, inter alia, strongly normalising or typeable terms are asymptotically negligible in the set of all $\lambda$-terms.

New Genome Similarity Measures based on Conserved Gene Adjacencies.

Many important questions in molecular biology, evolution, and biomedicine can be addressed by comparative genomic approaches. One of the basic tasks when comparing genomes is the definition of measures of similarity (or dissimilarity) between two genomes, for example, to elucidate the phylogenetic relationships between species. The power of different genome comparison methods varies with the underlying formal model of a genome. The simplest models impose the strong restriction that each genome under study must contain the same genes, each in exactly one copy. More realistic models allow several copies of a gene in a genome. One speaks of gene families, and comparative genomic methods that allow this kind of input are called gene family-based. The most powerful-but also most complex-models avoid this preprocessing of the input data and instead integrate the family assignment within the comparative analysis. Such methods are called gene family-free. In this article, we study an intermediate approach between family-based and family-free genomic similarity measures. Introducing this simpler model, called gene connections, we focus on the combinatorial aspects of gene family-free genome comparison. While in most cases, the computational costs to the general family-free case are the same, we also find an instance where the gene connections model has lower complexity. Within the gene connections model, we define three variants of genomic similarity measures that have different expression powers. We give polynomial-time algorithms for two of them, while we show NP-hardness for the third, most powerful one. We also generalize the measures and algorithms to make them more robust against recent local disruptions in gene order. Our theoretical findings are supported by experimental results, proving the applicability and performance of our newly defined similarity measures.

Combinatorial aspects of exceptional sequences on (rational) surfaces

We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric surface whose torus fixpoints are either smooth or cyclic T-singularities (in the sense of Wahl) of type $\frac{1}{r^2}(1, kr - 1)$. We also show that any exceptional sequence can be transformed by mutation into an exceptional sequence which consists only of objects of rank one.

Persistence property for some classes of monomial ideals of a polynomial ring

Let [Formula: see text] be a field and [Formula: see text] be a polynomial ring in the variables [Formula: see text]. In this paper, we introduce two classes of monomial ideals of [Formula: see text], which have the following properties: (i) The (strong) persistence property of associated prime ideals. (ii) There exists a strongly superficial element. (iii) Ratliff–Rush closed. Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

Dynamics of Polynomial Diffeomorphisms of $$\mathbb {C}^2$$ C 2 : Combinatorial and Topological Aspects

The Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer. This complicated object is the Julia set of a complex Henon map $$f_{c, b}(x, y)=(x^2+c-by, x)$$ defined on $$\mathbb {C}^2$$ together with its stable and unstable manifolds, hence it is a fractal set in the real 4-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of $$\mathbb {C}^2$$ including complex Henon maps with an emphasis on the combinatorial and topological aspects of their Julia sets.

Some combinatorial aspects of S^1 fixed points of Peterson varieties

Some combinatorial aspects of S^1 fixed points of Peterson varieties

Bounds and constructions of t-spontaneous emission error designs

The combinatorial aspects of quantum codes were demonstrated in the study of decay processes of certain quantum systems used in the newly emerging field of quantum computing. Among them, the configuration of t-spontaneous emission error design (t-SEED) was proposed to correct errors caused by quantum jumps. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. In this paper the upper and lower bounds on the dimensions of 3-\((v,\,4;\,m)\) SEEDs are studied and the necessary and sufficient conditions for 3-SEEDs attaining the upper bounds are described. Several new combinatorial constructions are presented for general t-SEEDs and lots of t-SEEDs of new parameters with \(t\ge 3\) are shown to exist.