Combinatorial Perspective Research Articles

A consortium of Hordeum vulgare and gut microbiota against non-alcoholic fatty liver disease via data-driven analysis

Despite many recent studies on non-alcoholic fatty liver disease (NAFLD) therapeutics, the optimal treatment has yet to be determined. In this unfinished project, we combined secondary metabolites (SMs) from the gut microbiota (GM) and Hordeum vulgare (HV) to investigate their combinatorial effects via network pharmacology (NP). Additionally, we analyzed GM or barley – signalling pathways – targets – metabolites (GBSTMs) in combinatorial perspectives (HV, and GM). A total of 31 key targets were analysed via a protein-protein interaction (PPI) network, and JUN was identified as the uppermost target in NAFLD. On a bubble plot, we revealed that apelin signalling pathway, which had the lowest enrichment factor antagonize NAFLD. Holistically, we scrutinized GBSTM to identify key components (GM, signalling pathways, targets, and metabolites) associated with the Apelin signalling pathway. Consequently, we found that the primary GMs (Eubacterium limosum, Eggerthella sp. SDG-2, Alistipes indistinctus YIT 12060, Odoribacter laneus YIT 12061, Paraprevotella clara YIT 11840, Paraprevotella xylaniphila YIT 11841) to ameliorate NAFLD. The molecular docking test (MDT) suggested that tryptanthrin-JUN is an agonist, conversely, dihydroglycitein-HDAC5, 1,3-diphenylpropan-2-ol-NOS1, and (10[(Acetyloxy)methyl]-9-anthryl)methyl acetate-NOS2, which are antagonistic conformers in the apelin signalling pathway. Overall, these results suggest that combination therapy could be an effective strategy for treating NAFLD.

Couplings and Matchings: Combinatorial notes on Strassen’s theorem

Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall’s marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory, in the sense that these results can easily be derived from each other. Remarkably, this equivalence extends to Strassen’s theorem, a celebrated result on couplings of probability measures. In this paper the equivalence between Strassen’s theorem and Hall’s theorem is investigated from a combinatorial perspective. A novel combinatorial lemma will be introduced that can be used to deduce both Hall’s theorem and a finite version of Strassen’s theorem, providing a simple proof of their equivalence.

Multislant matrices and Jacobi–Trudi determinants over finite fields

The problem of counting the Fq-valued points of a variety has been well-studied from algebro-geometric, topological, and combinatorial perspectives. We explore a combinatorially flavored version of this problem studied by Anzis et al. [1], which is similar to work of Kontsevich [7], Elkies [4], and Haglund [5].Anzis et al. considered the question: what is the probability that the determinant of a Jacobi-Trudi matrix vanishes if the variables are chosen uniformly at random from a finite field? They gave a formula for various partitions such as hooks, staircases, and rectangles. We give a formula for partitions whose parts form an arithmetic progression, verifying and generalizing one of their conjectures. More generally, we compute the probability of the determinant vanishing for a class of matrices (“multislant matrices”) made of Toeplitz blocks with certain properties.We furthermore show that the determinant of a skew Jacobi-Trudi matrix is equidistributed across the finite field if the skew partition is a ribbon.

Discrete bulk reconstruction

According to the AdS/CFT correspondence, the geometries of certain spacetimes are fully determined by quantum states that live on their boundaries — indeed, by the von Neumann entropies of portions of those boundary states. This work investigates to what extent the geometries can be reconstructed from the entropies in polynomial time. Bouland, Fefferman, and Vazirani (2019) argued that the AdS/CFT map can be exponentially complex if one wants to reconstruct regions such as the interiors of black holes. Our main result provides a sort of converse: we show that, in the special case of a single 1D boundary divided into N “atomic regions”, if the input data consists of a list of entropies of contiguous boundary regions, and if the entropies satisfy a single inequality called Strong Subadditivity, then we can construct a graph model for the bulk in linear time. Moreover, the bulk graph is planar, it has O(N2) vertices (the information-theoretic minimum), and it’s “universal”, with only the edge weights depending on the specific entropies in question. From a combinatorial perspective, our problem boils down to an “inverse” of the famous min-cut problem: rather than being given a graph and asked to find a min-cut, here we’re given the values of min-cuts separating various sets of vertices, and need to find a weighted undirected graph consistent with those values. Our solution to this problem relies on the notion of a “bulkless” graph, which might be of independent interest for AdS/CFT. We also make initial progress on the case of multiple 1D boundaries — where the boundaries could be connected via wormholes — including an upper bound of O(N4) vertices whenever an embeddable bulk graph exists (thus putting the problem into the complexity class NP).

Zonotopal Algebras, Orbit Harmonics, and Donaldson–Thomas Invariants of Symmetric Quivers

Abstract We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras, respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph $G_{Q,\gamma }$ constructed from a symmetric quiver $Q$ with enough loops and a dimension vector $\gamma $. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson–Thomas (DT) invariants as the Hilbert series of the space of $S_{\gamma }$-invariants of the Postnikov–Shapiro slim subgraph space attached to $G_{Q,\gamma }$. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical DT invariants as the number of $S_{\gamma }$-orbits under the permutation action on the set of break divisors on $G$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest.

Tensor diagrams and cluster combinatorics at punctures

Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SLk-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k=2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the language of tensor diagrams, extending work of Fomin and the second author, and we formulate skein-algebraic tools for calculating in these cluster algebras. We analyze the finite mutation type examples in detail.

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.

Combinatorial perspective on ideas, concepts, and policy change

ABSTRACT The evolution of environmental policy is increasingly driven by the emergence and interaction of policy concepts and the frameworks, knowledge, and ideas they employ. We argue that policy actors’ creative use of policy concepts often leads to their combinatorial development. In this policy process, the concepts interact and shape each other’s policy relevance and future development under the influence of ideas, knowledge, and political factors. We formulate this new research approach using ideational and policy process theories. In particular, we explicate internal and contextual combinatory elements that enable policy change resulting from mutual development of policy concepts. For an empirical demonstration, we analyze the conceptual innovation, rise, and ramifications of nutrient recycling as a new segment of Finnish environmental policy.

Combinatorial Perspectives on the Crank and Mex Partition Statistics

Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistic, the mex, and Frobenius symbols, with a focus on combinatorial proof techniques. One highlight is a generating function expression for the number of partitions with a bounded crank that does not include an alternating sum. This leads to a combinatorial interpretation involving types of Durfee rectangles. A recurring combinatorial technique uses sign reversing involutions on certain triples of partitions to establish a result of Fine and other identities.

Dynamics of majority rule on hypergraphs.

A broad range of dynamical systems involve multibody interactions, or group interactions, which may not be encoded in traditional graphical structures. In this work, we focus on a canonical example from opinion dynamics, namely the majority rule, and we investigate the possibility to represent and analyze the system by means of hypergraphs. We explore the formation of consensus, and we restrict our attention to interaction groups of size 3 in order to simplify our analysis from a combinatorial perspective. We propose different types of hypergraph models, incorporating modular structure or mean-field heterogeneity, and we recast the dynamics in terms of Fokker-Planck equations, which allows us to predict the transient dynamics toward consensus. Numerical simulations show a very good agreement between the stochastic dynamics and theoretical predictions for large population sizes.

Combinatorial perspectives on Dollo-k characters in phylogenetics

Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial perspective on the notion of Dollo-k characters, i.e. traits that are gained at most once and lost precisely k times throughout evolution. We first introduce an algorithm based on the notion of spanning subtrees for finding a Dollo-k labelling for a given character and a given tree in linear time. We then compare persistent characters (consisting of the union of Dollo-0 and Dollo-1 characters) and general Dollo-k characters. While it is known that there is a strong connection between Fitch parsimony and persistent characters, we show that Dollo parsimony and Fitch parsimony are in general very different. Moreover, while it is known that there is a direct relationship between the number of persistent characters and the Sackin index of a tree, a popular index of tree balance, we show that this relationship does not generalize to Dollo-k characters. In fact, determining the number of Dollo-k characters for a given tree is much more involved than counting persistent characters, and we end this manuscript by introducing a recursive approach for the former. This approach leads to a polynomial time algorithm for counting the number of Dollo-k characters, and both this algorithm as well as the algorithm for computing Dollo-k labellings are publicly available in the Babel package for BEAST 2.

Towards a more algebraic footing for quantum field theory

The predictions of the standard model of particle physics are highly successful in spite of the fact that several parts of the underlying quantum field theoretical framework are analytically problematic. Indeed, it has long been suggested, by Einstein, Schrödinger and others, that analytic problems in the formulation of fundamental laws could be overcome by reformulating these laws without reliance on analytic methods namely, for example, algebraically. With this in mind, we focus here on the analytic ill-definedness of the quantum field theoretic Fourier and Legendre transforms of the generating series of Feynman graphs, including the path integral. To this end, we develop here purely algebraic and combinatorial formulations of the Fourier and Legendre transforms, employing rings of formal power series. These are all-purpose transform methods, i.e. their applicability is not restricted to quantum field theory. When applied in quantum field theory to the generating functionals of Feynman graphs, the new transforms are well-defined and thereby help explain the robustness and success of the predictions of perturbative quantum field theory in spite of analytic difficulties. Technically, we overcome here the problem of the possible divergence of the various generating series of Feynman graphs by constructing Fourier and Legendre transforms of formal power series that operate in a well-defined way on the coefficients of the power series irrespective of whether or not these series converge. Our new methods could, therefore, provide new algebraic and combinatorial perspectives on quantum field theoretic structures that are conventionally thought of as analytic in nature, such as the occurrence of anomalies from the path integral measure. In comparison, the use of formal power series in QFT by Bogolubov, Hepp, Parasiuk and Zimmermann concerned a different kind of divergencies, namely the UV divergencies of loop integrals and their renormalization.

Irreducible components of exotic Springer fibres II : The exotic Robinson–Schensted algorithm

Kato's exotic nilpotent cone was introduced as a substitute for the ordinary nilpotent cone of type C with cleaner properties. The geometric Robinson-Schensted correspondence is obtained by parametrizing the irreducible components of the Steinberg variety (the conormal variety for the action of a semisimple group on two copies of its flag variety); in type A the bijection coincides with the classical Robinson-Schensted algorithm for the symmetric group. Here we give a combinatorial description of the bijection obtained by using the exotic nilpotent cone instead of ordinary type C nilpotent cone in the geometric Robinson-Schensted correspondence; we refer this as the "exotic Robinson-Schensted bijection". This is interesting from a combinatorial perspective, and not a naive extension of the type A Robinson-Schensted bijection.

(Re)constructing Code Loops

The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops’ twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multiplication in Parker’s loop from a mere fragment of its twisted cocycle. We also give relatively large subspaces of the Golay code over which Parker’s loop splits as a direct product.

Computational Hardness of Collective Coin-Tossing Protocols.

Ben-Or and Linial, in a seminal work, introduced the full information model to study collective coin-tossing protocols. Collective coin-tossing is an elegant functionality providing uncluttered access to the primary bottlenecks to achieve security in a specific adversarial model. Additionally, the research outcomes for this versatile functionality has direct consequences on diverse topics in mathematics and computer science. This survey summarizes the current state-of-the-art of coin-tossing protocols in the full information model and recent advances in this field. In particular, it elaborates on a new proof technique that identifies the minimum insecurity incurred by any coin-tossing protocol and, simultaneously, constructs the coin-tossing protocol achieving that insecurity bound. The combinatorial perspective into this new proof-technique yields new coin-tossing protocols that are more secure than well-known existing coin-tossing protocols, leading to new isoperimetric inequalities over product spaces. Furthermore, this proof-technique’s algebraic reimagination resolves several long-standing fundamental hardness-of-computation problems in cryptography. This survey presents one representative application of each of these two perspectives.

Syndrome-Coupled Rate-Compatible Error-Correcting Codes: Theory and Application

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances, and we discuss decoding algorithms and correctable patterns of errors and erasures. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes, generalizing a recent construction of capacity-achieving rate-compatible polar codes. Applications of rate-compatible codes to data storage are considered. We design two-level rate-compatible codes based on Bose-Chaudhuri-Hocquenghem (BCH) and low-density parity-check (LDPC) codes which are two popular codes widely used in the data storage industry, and then we evaluate the performance of these codes in multi-level cell (MLC) flash memories. We also examine code performance on binary and $q$ -ary symmetric channels. Finally, we briefly discuss two variations of our main construction and their relative performance.

Geometric realizations of cyclic actions on surfaces

Let [Formula: see text] denote the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text], and let [Formula: see text] be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on [Formula: see text] that realizes [Formula: see text] as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of [Formula: see text]. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation [Formula: see text].

Permutation Symmetric Hypergraph States and Multipartite Quantum Entanglement

We demonstrate that a quantum hypergraph state is k-separable if and only if the hypergraph has k-connected components. The permutation symmetric states remains invariant under any permutation. We introduce permutation symmetric states generated by hypergraphs and describe their combinatorial structures. This combinatorial perspective insists us to investigate multi-partite entanglement of permutation symmetric hypergraph states. Using generalised concurrence we measure entanglement up to ten qubits. A number of examples of these states are discussed.

Simplicial structures in higher Auslander–Reiten theory

We develop a novel combinatorial perspective on the higher Auslander algebras of type A, a family of algebras arising in the context of Iyama's higher Auslander–Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg–MacLane spaces and higher-dimensional versions of Waldhausen's S•-construction in algebraic K-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to representations with values in stable ∞-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type A quivers developed by Groth and Šťovíček. Our simplicial point of view then naturally leads to an interplay between slice mutation, horn filling conditions, and the higher Segal conditions of Dyckerhoff and Kapranov. In this context, we provide a classification of higher Segal objects with values in any abelian category or stable ∞-category.

Deep Learning and AlphaGo

When the Information Processing Society of Japan (IPSJ) declared the end of the challenge to the Japanese Shogi Association in 2015, the belief was that it would take more than 20 years for computers to catch up with human professional Go players, as Go is far more complex than shogi from a combinatorial perspective. However, AlphaGo, whose paper was just published the same year, beat the Go world champion in all three games in 2017. In this paper, we describe the artificial intelligence methods adopted by AlphaGo, especially deep learning, and consider their relationship with neuroscience.