A connected graph is 2K2-free if it does not contain a pair of independent edges as an induced subgraph. In this paper, we present the structural characterization of minimal vertex separator and show that there are polynomial number of minimal vertex separators in 2K2-free graphs. Further, using the enumeration we show that finding minimum connected vertex separator in 2K2-free graphs is polynomial time solvable. We highlight that finding minimum connected vertex separator is NP-complete in Chordality 5 graphs, which is a super graph class of 2K2-free graphs. Other study includes, enumeration of all distinct maximal independent sets and testing 2K2-free graphs. Also, we present an polynomial time algorithm for feedback vertex set problem in the subclass of 2K2-free graphs.
The notions of symmetry and anonymity in strategic games have been formalized in different ways in the literature. We propose a combinatorial framework to analyze these notions, using group actions. Then, the same framework is used to define partial symmetries in payoff matrices. With this purpose, we introduce the notion of the role a player plays with respect to another one, and combinatorial relations between roles are studied. Building on them, we define relations directly between players, which provide yet another characterization of structural symmetries in the payoff matrices of strateg…
Close connections between various notions of entropy and the apparatus of category theory have been observed already in the 1980s and more vigorously developed in the past ten years. The starting point of the paper is the recent categorical understanding of structural Ramsey degrees, which then leads to a way to compute entropy of an object in a small category not as a measure of statistical, but as a measure of its combinatorial complexity. The new entropy function we propose, the Ramsey entropy, is a real-valued invariant of an object in an arbitrary small category. We require no additional …
The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian numbers have intimate relationships to the set.
We construct of the main object of the Partite Lemma as the colimit over a certain diagram. This gives a purely category theoretic take on the Partite Lemma and establishes the canonicity of the object. Additionally, the categorical point of view allows us to unify the direct Partite Lemma in the Nešetřil--Rödl Theorem with the dual Paritite Lemma by Solecki.
In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first introduced by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more patterns of length 3, and we obtain a number of well-known combinatorial sequences as a result. Along the way, we find bijections between specific sets of pattern-avoiding parking functions and a number of combinatorial objects such as partitions of polygons and trees with certain restrictions.
A natural partial ordering exists on the set of all weighted games and, more broadly, on all linear games. We describe several properties of the partially ordered sets formed by these games and utilize this perspective to enumerate proper linear games with one generator. We introduce a geometric approach to weighted voting by considering the convex polytope of all possible realizations of a weighted game and connect this geometric perspective to the weighted games poset in several ways. In particular, we prove that generic vertical lines in $C_n$, the union of all weighted $n$-player polytopes…
From the matrix point of view, we use the recursion to discuss four combinatorial numbers in terms of the integer lattice paths, this is different from Andrá's method (Andra). We give four tables and matrices, and their relations, and give their matrix decompositions. By their relations, we give the explicit formulas of the special combinatorial numbers.
White noise is a fundamental and fairly well understood stochastic process that conforms the conceptual basis for many other processes, as well as for the modeling of time series. Here we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to give new interesting insights both for modelling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that part…
The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into "mutually op…
In view of Tucker's lemma (an equivalent combinatorial version of the Borsuk- Ulam theorem), the present authors (2013) introduced the kth altermatic number of a graph G as a tight lower bound for the chromatic number of G. In this note, we present a purely combinatorial proof for this result.
The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one $0 \rightarrow 1$ transition followed by at most one $1 \rightarrow 0$ transition in the tree, and analyze these characters under different aspects. First, we illustrate the connection between persistent characters and Maximum Parsimony, where we characterize p…
Adinkras are combinatorial objects developed to study 1-dimensional supersymmetry representations. Recently, 2-d Adinkras have been developed to study 2-dimensional supersymmetry. In this paper, we classify all 2-d Adinkras, confirming a conjecture of T. Hübsch. Along the way, we obtain other structural results, including a simple characterization of Hübsch's even-split doubly even code.
In this paper we present new results for the combinatorics of web diagrams and web worlds. These are discrete objects that arise in the physics of calculating scattering amplitudes in non-abelian gauge theories. Web-colouring and web-mixing matrices (collectively known as web matrices) are indexed by ordered pairs of web-diagrams and contain information relating the number of colourings of the first web diagram that will produce the second diagram. We introduce the black diamond product on power series and show how it determines the web-colouring matrix of disjoint web worlds. Furthermore, we …
This paper presents an offering of some of the myriad connections between Combinatorics and Probability, directed in particular toward combinatorialists. The choice of material was dictated by the author's own interests, tastes and familiarity, as well as by a desire to present results with either complete proofs or well developed sketches of proofs, and to ensure that the arguments are rather accessible to combinatorialists. The first several sections collect some concepts and rudimentary results from probability theory that are needed to understand the rest of the paper.
We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type $\tilde A_{n-1}$, i.e. corresponding to the affine Lie algebra $\hat{\mathfrak{sl}}_n$. Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion's theorem and then apply it to our polyhedron to prove the fo…
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain model-theoretic structures. In 2005, Nešetřil initiated a systematic study of the so-called Ramsey classes of finite structures. This thesis is a contribution to the programme; we find Ramsey expansions of the primitive 3-constrained classes from Cherlin's catalogue of metrically homogeneous graphs. A key ingradient is an explicit combinatorial algorithm to fill-in…
Let $R$ be a ring and let $\mathcal C$ be a small class of right $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let $\mathcal V (\mathcal C)$ denote a set of representatives of isomorphism classes in $\mathcal C$ and, for any module $M$ in $\mathcal C$, let $[M]$ denote the unique element in $\mathcal V (\mathcal C)$ isomorphic to $M$. Then $\mathcal V (\mathcal C)$ is a reduced commutative semigroup with operation defined by $[M] + [N] = [M \oplus N]$, and this semigroup carries all information about direct-sum decompositions of modules in $\mathcal C…
We study using combinatorial methods the structural coefficients of the formal homogeneous universal enveloping algebra Uh(sl2) of the special linear algebra sl2 over a field of characteristic zero. We provide explicit formulae for the product of generic elements in Uh(sl2), and construct combinatorial objects giving flesh to these formulae.
We begin by explaining how arguments used by R. Wilson to give an elementary proof of the $\mathbb F_p$ case for the Ax-Katz Theorem can also be used to prove the following generalization of the Chevalley-Warning and Ax-Katz Theorems for $\mathbb F_p$, where we allow varying prime power moduli. Given any box $\mathcal B=\mathcal I_1\times\ldots\times\mathcal I_n$, with each $\mathcal I_j\subseteq\mathbb Z$ a complete system of residues modulo $p$, and a collection of nonzero polynomials $f_1,\ldots,f_s\in \mathbb Z[X_1,\ldots,X_n]$, then the set of common zeros inside the box, $$V=\{\mathbf a\…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical v…
