Combinatorial Objects Research Articles

GENERALIZED PASCAL’S PYRAMIDS AND THEIR APPLICATIONS IN STOCHASTIC PROCESSES

This work studies combinatorial objects of pyramidal structure. It proposes fundamental relations for the elements of generalized Pascal’s triangle and generalized Pascal’s pyramid. Among the important special cases under consideration, there are generalized Stirling’s numbers and generalizations of trinomial coefficients and their applications while constructing discrete models of a number of stochastic processes.

Solving Paths Search Problems in Complex Graphs

The construction of models of various systems is associated with the enumeration of the values of the parameters of the elements of the structure and taking into account all the characteristics of operation and interaction of components to find a certain set of solutions that determine the configuration of the system. Such tasks belong to enumeration type tasks and imply that some of the next solutions from this set are obtained from the previous solution in a certain order. It is known that any problem of the enumeration type is solved only by methods of exhaustive search, and other methods for their enumeration do not exist yet. The paper presents a new method of searching paths in a graph – the method of node-graph transformation. The proposed method, unlike the existing ones, allows one to search all directed simple paths in an oriented graph of arbitrary structure much faster. In the known graph search methods (Breadth First Search and Depth First Search), the object of the search is a path. The total number of such paths in the graph determines the size of the search space. The main idea of the node-graph transformation method is to significantly reduce the size of the search space by enlarging the search objects. The enlargement of enumeration objects is performed by clustering paths into combinatorial objects that concentrate some set of paths of the same length according to certain rules. These combinatorial objects are called node-graphs. A node-graph refers to center-peripheral combinatorial objects, and specific node-graph transformation operations have been developed to enumerate all paths in the graph, which allow finding new paths based on previous paths. The method can be used as a basic toolkit to reduce the dimensionality of the search space for solutions to NP-complete problems while maintaining the universality and accuracy of the full search.

Charmed roots and the Kroweras complement

AbstractAlthough both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition of charmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.

GENERALIZED PASCAL’S PYRAMIDS AND COMBINATORIAL INFORMATION RETRIEVAL PROBLEMS

The work studies combinatorial objects of pyramidal structure. It proposes fundamental relations for generalizations of Pascal’s triangle and pyramid. It considers problems of finding new ways of constructing information relevance indices and offers a way of constructing such an index with the help of Pascal’s generalized pyramids. Using structures of a type of Pascal’s generalized pyramids with weight factors allow construction of an index that reflects a degree of relevance of keywords, as well as makes it possible to allocate subsets of keywords with specific properties. The offered methods of the combinatorial analysis of hierarchical structures in information retrieval problems can be used in creating and analyzing knowledge databases.

Quiver tails and brane webs

A new type of quiver theories, denoted twin quivers, was recently introduced for studying 5d SCFTs engineered by webs of 5-branes ending on 7-branes. Twin quivers provide an alternative perspective on various aspects of such webs, including Hanany-Witten moves and the s-rule. More ambitiously, they can be regarded as a first step towards the construction of combinatorial objects, generalizing brane tilings, encoding the corresponding BPS quivers. This paper continues the investigation of twin quivers, focusing on their non-uniqueness, which stems from the multiplicity of toric phases for a given toric Calabi-Yau 3-fold. We find that the different twin quivers are necessary for describing what we call quiver tails, which in turn correspond to certain sub-configurations in the webs. More generally, the multiplicity of twin quivers captures the roots of the Higgs branch in the extended Coulomb branch of 5d theories.

Cyclotomic Generating Functions

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

On generalized Sidon spaces

Sidon spaces have been introduced by Bachoc, Serra and Zémor as the q-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of r-Sidon spaces, as an extension of Sidon spaces, which may be seen as the q-analogue of Br-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and r-Sidon spaces, providing some upper and lower bounds on the possible dimension of their r-span and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of r-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of Br-sets constructed by means of them.

Surfaceology for colored Yukawa theory

Arkani-Hamed and collaborators have recently shown that scattering amplitudes for colored theories can be expressed as integrals over combinatorial objects simply constructed from surfaces decorated by kinematic data. In this paper we extend the curve integral formalism to theories with colored fermionic matter and present a compact formula for the all-loop, all-genus, all-multiplicity amplitude integrand of a colored Yukawa theory. The curve integral formalism makes certain properties of the amplitudes manifest and repackages non-trivial numerators into a single combinatorial object. We also present an efficient formula for L-loop integrated amplitudes in terms of a sum over 2L combinatorial determinants.

Pattern Avoidance in Weak Ascent Sequences

In this paper, we study pattern avoidance in weak ascent sequences, giving some results for patterns of length 3. This is an analogous study to one given by Duncan and Steingr\'imsson (2011) for ascent sequences. More precisely, we provide systematically the generating functions for the number of weak ascent sequences avoiding the patterns $001, 011, 012, 021$, and $102$. Additionally, we establish bijective connections between pattern-avoiding weak ascent sequences and other combinatorial objects, such as compositions, upper triangular 01-matrices, and plane trees.

Parameter estimation for Gibbs distributions

A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We consider sampling problems coming from Gibbs distributions , which are families of probability distributions over a discrete space \(\Omega\) with probability mass function of the form \(\mu^{\Omega}_{\beta}(\omega)\propto e^{\beta H(\omega)}\) for \(\beta\) in an interval \([\beta_{\min},\beta_{\max}]\) and \(H(\omega)\in\{0\}\cup[1,n]\) . Two important parameters are the partition function , which is the normalization factor \(Z(\beta)=\sum_{\omega\in\Omega}e^{\beta H(\omega)}\) , and the vector of preimage counts \(c_{x}=|H^{-1}(x)|\) . We develop black-box sampling algorithms to estimate the counts using roughly \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions, where \(q=\log\frac{Z(\beta_{\max})}{Z(\beta_{\min})}\) (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we estimate all values of the partition function using \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate \(Z(\beta_{\max})\) and which uses a slightly larger amount of samples. We show matching lower bounds, demonstrating this complexity is optimal as a function of \(n\) and \(q\) up to logarithmic terms.

Plactic-like monoids arising from meets and joins of stalactic and taiga congruences

We study the four plactic-like monoids that arise by taking the meets and joins of stalactic and taiga congruences. We obtain the combinatorial objects associated with the meet monoids, establishing Robinson–Schensted-like correspondences and giving extraction and iterative insertion algorithms for these objects. We then obtain results on the sizes of classes of words equal in plactic-like monoids, show that some of these monoids are syntactic, and characterise their equational theories.

Combinatorics of semi-toric degenerations of Schubert varieties in type C

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand–Tsetlin polytopes, Kiritchenko–Smirnov–Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand–Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko–Smirnov–Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.

Moduli spaces of ℤ/kℤ-constellations over 𝔸2

Let [Formula: see text] be a representation of a finite abelian group and let [Formula: see text] be the space of generic stability conditions on the set of [Formula: see text]-constellations. We provide a combinatorial description of all the chambers [Formula: see text] and prove that there are [Formula: see text] of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric [Formula: see text]-constellations, it is enough to build all simple chambers. We also prove that there are [Formula: see text] simple chambers. Finally, we provide an explicit formula for the tautological bundles [Formula: see text] over the moduli spaces [Formula: see text] for all chambers [Formula: see text] which only depends upon the chamber stair which is a combinatorial object attached to the chamber [Formula: see text].

On a variant of Flory model

We consider a one-dimensional variant of a recently introduced settlement planning problem in which houses can be built on finite portions of the rectangular integer lattice subject to certain requirements on the amount of insolation they receive. In our model, each house occupies a unit square on a 1×n strip, with the restriction that at least one of the neighboring squares must be free. We are interested mostly in situations in which no further building is possible, i.e. in maximal configurations of houses in the strip. We reinterpret the problem as a problem of restricted packing of vertices in a path graph and then apply the transfer matrix method in order to compute the bivariate generating functions for the sequences enumerating all maximal configurations of a given length with respect to the number of houses. This allows us to determine the asymptotic behavior of the enumerating sequences and to compute some interesting statistics. Along the way, we establish close connections between our maximal configurations and several other types of combinatorial objects, including restricted permutations and walks on certain small oriented graphs. We then generalize our results in several directions by considering multi-story houses, by varying the insolation restrictions, and, finally, by considering strips of width 2 and 3. At the end we comment on several possible directions of future research.

Combinatorics of essential sets for positroids

Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows the study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic intervals. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.

Ptolemy diagrams and torsion pairs in m-cluster categories of type D

In this paper, we give a complete classification of torsion pairs in m-cluster categories of type D when m is odd, denoted by C D n m , via a bijection to combinatorial objects called Ptolemy diagrams of type D. As applications, we classify m-rigid subcategories of C D n m , which gives Jacquet-Malo’s classification of m-cluster tilting subcategories of C D n m . When m = 1, we generalizes the work of Holm, Jørgensen and Rubey [ 14 ] for the classification of torsion pairs in cluster categories of type Dn . Communicated by Steffen Oppermann

Cocharge and Skewing Formulas for Δ-Springer Modules and the Delta Conjecture

Abstract We prove that $\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta $-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $\Delta $-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^{2}$ coefficients of $\omega \Delta ^{\prime}_{e_{k}}e_{n}$.

Central Limit Theorems via Analytic Combinatorics in Several Variables

The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of analytic combinatorics in several variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. In this work we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large class of combinatorial generating functions. To illustrate the techniques involved, we also establish an explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem stated as a conjecture in recent work of Chung et al.), integer compositions, and n-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in an arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate LU-factorization. It is our hope that this work provides readers a blueprint to apply the powerful tools of ACSV in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields.

Non-disjoint strong external difference families can have any number of sets

AbstractStrong external difference families (SEDFs) are much-studied combinatorial objects motivated by an information security application. A well-known conjecture states that only one abelian SEDF with more than 2 sets exists. We show that if the disjointness condition is replaced by non-disjointness, then abelian SEDFs can be constructed with more than 2 sets (indeed any number of sets). We demonstrate that the non-disjoint analogue has striking differences to, and connections with, the classical SEDF and arises naturally via another coding application.

Density results for the parity of (4k,k)-singular overpartitions

The (k,i)-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6k. In this paper we consider the parity of the number C‾k,i(n) of (k,i)-singular overpartitions of n. In particular, we give a sufficient condition on even values of k so that the values of C‾4k,k(n) are almost always even. Furthermore, we show that for odd values of k≤23, k≠19, certain subsequences of C‾4k,k(n) are almost always even.