Related papers: Simple extensions of combinatorial structures
Induction is the process by which we obtain predictive laws or theories or models of the world. We consider the structural aspect of induction. We answer the question as to whether we can find a finite and minmalistic set of operations on…
We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by…
A combinatorial theory of associative $n$-categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a combinatorial notion of homotopy with a natural…
Given a compact set K in the plane, which contains no triple of points forming a vertical and a horizontal segment, and a continuous real-valued map f on K, we give a construction of real-valued continuous maps of one variable g,h such that…
Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they…
We uncover a close relationship between combinatorial and syntactic proofs for first-order logic (without equality). Whereas syntactic proofs are formalized in a deductive proof system based on inference rules, a combinatorial proof is a…
In this paper we construct a subclass of the composite access structure introduced by Mart\'inez et al. based on schemes realizing the structure given by the set of codewords of minimal support of linear codes. This class enlarges the…
This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the…
In a previous work arXiv:physics/0611108v2, it was shown that the volume spanned by a molecular system in its conformational space can be effectively bounded by a polyhedral cone, this cone is described by means of a simple combinatorial…
Associating to each pre-order on the indices 1,...,n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n x n matrix rings. Rings within the order-convex hull of…
A family of subsets $\mathcal{F} \subseteq \mathcal{P}(\{1, 2, \ldots, n\})$ has the disparate union property if any two disjoint subfamilies $\mathcal{F}_1, \mathcal{F}_2 \subseteq \mathcal{F}$ have distinct unions $\bigcup \mathcal{F}_1…
This paper attempts a more formal approach to the legibility of text based programming languages, presenting, with proof, minimum possible ways of representing structure in text interleaved with information. This presumes that a minimalist…
In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster…
We consider composite $n$ satisfying the congruence $$n \cdot \sigma_k(n) \equiv 2 \pmod{\phi(n)},$$ and show a "flanking" structure: $14$ appears in both $S_{k-1}$ and $S_{k+1}$ whenever certain values of $n$ appear in $S_k$; and,…
We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…
We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…
For an arbitrary group, the subgroups form a lattice with order determined by set inclusion. Not every lattice is isomorphic to the subgroup lattice for a group. However, Birkhoff and Frink proved that any compactly generated lattice is…
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example,…
We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the factorizations of…