Related papers: The mex statistic on combinatorial structures
As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces:…
Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate…
We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free…
We demonstrate that statistics for several types of set partitions are described by generating functions which appear in the theory of integrable equations.
Algorithmic statistics considers the following problem: given a binary string $x$ (e.g., some experimental data), find a "good" explanation of this data. It uses algorithmic information theory to define formally what is a good explanation.…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set $A$ containing $\Omega(|A|^{3/2})$ three-term arithmetic progressions.
Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the…
We try to bring to light some combinatorial structure underlying formal proofs in logic. We do this through the study of the Craig Interpolation Theorem which is properly a statement about the structure of formal derivations. We show that…
We introduce in this section an Algebraic and Combinatorial approach to the theory of Numbers. The approach rests on the observation that numbers can be identified with familiar combinatorial objects namely rooted trees, which we shall here…
We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these…
We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.
We will generalize the concept of aggregation function for mathematical structures as a certain function between quantales. In fact, these functions turn to be exactly the lax morphism of quantales. This provides a global framework for the…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…