Combinatorial Aspects Research Articles

Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)

Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.

Hex-Mesh Generation and Processing: A Survey

In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post-processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.

Generalisations of Capparelli's and Primc's identities, I: Coloured Frobenius partitions and combinatorial proofs

The partition identities of Capparelli and Primc were originally discovered via representation theoretic techniques, and have since then been studied and refined combinatorially, but the question of giving a very broad generalisation remained open. In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra An−1(1).In this first paper, we focus on combinatorial aspects. We give a n2-coloured generalisation of Primc's identity by constructing a n2×n2 matrix of difference conditions, Primc's original identities corresponding to n=2 and n=3. While most coloured partition identities in the literature connect partitions with difference conditions to partitions with congruence conditions, in our case, the natural way to generalise these identities is to relate partitions with difference conditions to coloured Frobenius partitions. This gives a very simple expression for the generating function. With a particular specialisation of the colour variables, our generalisation also yields a partition identity with congruence conditions.Then, using a bijection from our new generalisation of Primc's identity, we deduce a large family of identities on (n2−1)-coloured partitions which generalise Capparelli's identity, also in terms of coloured Frobenius partitions. The particular case n=2 is Capparelli's identity and one of the cases where n=3 recovers an identity of Meurman and Primc.In the second paper, we will focus on crystal theoretic aspects. We will show that the difference conditions we defined in our n2-coloured generalisation of Primc's identity are actually energy functions for certain An−1(1) crystals. We will then use this result to retrieve the Kac-Peterson character formula and derive a new character formula as a sum of infinite products for all the irreducible highest weight An−1(1)-modules of level 1.

Causal Structure Learning: A Combinatorial Perspective

In this review, we discuss approaches for learning causal structure from data, also called causal discovery. In particular, we focus on approaches for learning directed acyclic graphs and various generalizations which allow for some variables to be unobserved in the available data. We devote special attention to two fundamental combinatorial aspects of causal structure learning. First, we discuss the structure of the search space over causal graphs. Second, we discuss the structure of equivalence classes over causal graphs, i.e., sets of graphs which represent what can be learned from observational data alone, and how these equivalence classes can be refined by adding interventional data.

A subsemigroup of the rook monoid

We define a subsemigroup \(S_n\) of the rook monoid \(R_n\) and investigate its properties. To do this, we represent the nonzero elements of \(S_n\) (which are \(n\times n\) matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that \(S_n\) consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, compute nilpotency indexes, determine Green’s relations and ideals, and come up with a minimal generating set. Furthermore, we give a necessary and sufficient condition for the jth root of a nonzero element to exist in \(S_n\), show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of \(S_n\) (some of which are familiar from previous studies); and, using rook n-diagrams, graphically interpret many of our results.

On Snapshot-Based Model Reduction Under Compatibility Conditions for a Nonlinear Flow Problem on Networks

This paper is on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The nonlinear flow problem finds broad application in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy bound and port-Hamiltonian structure. We also derive a wellposedness result for them. In the training phase, the compatibility conditions pose challenges, we face constrained data approximation problems as opposed to the unconstrained training problems in the conventional reduction methods. The training of our model order reduction consists of a principal component analysis under a compatibility constraint and, notably, yields reduced models that fulfill an optimality condition for the snapshot data. The training of our quadrature-type complexity reduction involves a semi-definite program with combinatorial aspects, which we approach by a greedy procedure. Efficient algorithmic implementations are presented. The robustness and good performance of our structure-preserving reduced models are showcased at the example of gas network simulations.

An improved penalty algorithm using model order reduction for MIPDECO problems with partial observations

This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra issues, arising from the PDE discretization. Since classical solution approaches such as the branch-and-bound framework are typically overwhelmed by such large-scale problems, this work extends an improved penalty algorithm proposed by the authors, to the time-dependent setting. The main contribution is a novel combination of an interior point method, preconditioning, and model order reduction yielding a tailored local optimization solver at the heart of the overall solution procedure. A thorough numerical investigation is carried out both for the heat equation as well as a convection-diffusion problem demonstrating the versatility of the approach.

Formalizing the Face Lattice of Polyhedra

Faces play a central role in the combinatorial and computational aspects of polyhedra. In this paper, we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under interval sublattices. We also prove a theorem due to Balinski on the $d$-connectedness of the adjacency graph of polytopes of dimension $d$.

Combinatorics of NC-probability spaces with independent constants

The boolean and monotone notions of independence lack the property of independent constants. We address this problem from a combinatorial point of view (based on cumulants defined from weights on set-partitions, in the general framework of operator-valued probability spaces). We show that if the weights are singleton inductive (SI), then all higher-order cumulants involving constants vanish, just as in the free and classical case. Our combinatorial considerations lead rather directly to mild variations of boolean and monotone probability theories which are closely related to the usual notions. The SI-boolean case is related to c-free and Fermi convolutions. We also describe some standard combinatorial aspects of the SI-boolean and cyclic-boolean lattices, such as their Möbius functions, featuring well-known combinatorial integer sequences.

Combinatorics of F-Polynomials

Abstract We use the stabilization functors to study the combinatorial aspects of the $F$-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the $F$-polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the $F$-polynomial is saturated for any rigid representation. We provide many examples and counterexamples and pose several conjectures.

On Proper Labellings of Graphs with Minimum Label Sum

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture, that may be attributed to Chartrand et al., is that almost every graph G can be turned into a locally irregular multigraph M, i.e., with no two adjacent vertices having the same degree, by replacing each of its edges by at most three parallel edges. In other words, for almost every graph G there should be a locally irregular multigraph M with the same adjacencies and having a relatively small number of edges. The 1–2–3 Conjecture, if true, would indeed imply that there is such an M with \(|E(M)| \le 3|E(G)|\). In this work, we study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. In other words, given a graph G, we are looking for a locally irregular multigraph \(M^*\) with the smallest number of edges possible that can be obtained from G by multiplying edges. This problem is actually quite different from the 1–2–3 Conjecture, as we prove that there is no absolute constant k such that \(M^*\) can always be obtained from G by replacing each edge with at most k parallel edges. We investigate several aspects of this problem, covering algorithmic and combinatorial aspects. In particular, we prove that the problem of designing proper labellings with minimum label sum is \({\mathcal {N}}{\mathcal {P}}\)-hard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that for almost every connected graph G there should be a proper labelling with label sum at most 2|E(G)|, which we verify for several classes of graphs.

A survey on t-core partitions

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

On Sombor indices of line graph of silicate carbide Si 2 C 3-I[p,q

Topological indices are numerical parameters associated with underlying topology of a molecular structure. They are correlated with several physio-chemical properties of chemical compounds. Recently, Euclidean metric based topological index has been introduced named as Sombor index. Therefore, in this article, we will discuss combinatorial aspects and compute Sombor index, average Sombor index and the reduced Sombor index of line graph of silicate carbides Si 2 C 3 -I[p, q].

Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties.

Age-Related Differences in Resting-State Functional Connectivity in Women

Cartan matrices, quasi-Cartan matrices and associated upper triangular Gram matrices control important combinatorial aspects of Lie theory and representation theory of associative algebras. We provide a graph theoretic proof of the fact that the absolute values of the coefficients of a non-negative quasi-Cartan matrix A as well as of its (minimal) symmetrizer D are bounded by 4, and that the analogous bound in case of the associated Gram matrix GˇA is 8. Moreover, we show that D (and GˇA) has at least one diagonal coefficient equal to 1. We describe some other restrictions and interrelations between the coefficients of A, D and GˇA, and the corank and other properties of A relevant in Lie theory. We apply our results to construct an algorithm by which we classify all non-negative quasi-Cartan matrices of small sizes.

On the hardness of finding normal surfaces

For many fundamental problems in computational topology, such as unknot recognition and 3-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms for these problems is normal surface theory. However, we currently have a poor understanding of the computational complexity of problems in normal surface theory: many such problems are still not known to have polynomial-time algorithms, yet proofs of \(\mathrm {NP}\)-hardness also remain scarce. We give three results that provide some insight on this front. A number of modern normal surface theoretic algorithms depend critically on the operation of finding a non-trivial normal sphere or disc in a 3-dimensional triangulation. We formulate an abstract problem that captures the algebraic and combinatorial aspects of this operation, and show that this abstract problem is \(\mathrm {NP}\)-complete. Assuming \(\mathrm {P}\ne \mathrm {NP}\), this result suggests that any polynomial-time procedure for finding a non-trivial normal sphere or disc will need to exploit some geometric or topological intuition. Another key operation, which applies to a much wider range of topological problems, involves finding a vertex normal surface of a certain type. We study two closely-related problems that can be solved using this operation. For one of these problems, we give a simple alternative solution that runs in polynomial time; for the other, we prove \(\mathrm {NP}\)-completeness.

Combinatorial aspects of Löwenstein's rule

Combinatorial aspects of Löwenstein's rule

Combinatorial Aspects of An Odd Linkage Property for General Linear Supergroups

Combinatorial Aspects of An Odd Linkage Property for General Linear Supergroups

Another approach for Leslie model via linear difference equations

We explore here an approach to study the Leslie model, describing the growth of the discrete age structured populations. Our main tool is based upon properties of some linear difference equations. Involving the analytical and combinatorial aspects of solutions of these equations, we establish new formulas for the entries of the population vector and derive their asymptotic behavior. Significant illustrative examples and applications are given.

Relationships between multiple zeta values of depths 2 and 3 and period polynomials

Some combinatorial aspects of relations between multiple zeta values of depths 2 and 3 and period polynomials are discussed.