Related papers: Combinatorics of positional colored compositions
We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erd\H{o}s and Szekeres. In our setting we have $n$ points in general position in the plane,…
In this paper we study the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization…
Compositionality is a widely discussed property of natural languages, although its exact definition has been elusive. We focus on the proposal that compositionality can be assessed by measuring meaning-form correlation. We analyze…
We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the…
The purpose of this paper is twofold. Firstly, we generalize the notion of characteristic polynomials of hyperplane and toric arrangements to those of certain abelian Lie group arrangements. Secondly, we give two interpretations for the…
We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this…
Bright, saturated structural colors in birds have inspired synthesis of self-assembled, disordered arrays of assembled nanoparticles with varied particle spacings and refractive indices. However, predicting colors of assembled…
This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for…
We compute the motivic nearby cycles of functions obtained by composition of two functions with distinct sets of variables with a two variable function
We determine all composition-closed equational classes of Boolean functions. These classes provide a natural generalization of clones and iterative algebras: they are closed under composition, permutation and identification…
We study the enumeration of bargraphs with respect to some corner statistics. We find generating functions for the number of bargraphs that tracks the corner statistics of interest, the number of cells, and the number of columns. The…
Coherence and correlations represent two related properties of a compound system. The system can be, for instance, the polarization of a photon, which forms part of a polarization-entangled two-photon state, or the spatial shape of a…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
Finite graphs that have a common chromatic polynomial have the same number of regular $n$-colorings. A natural question is whether there exists a natural bijection between regular $n$-colorings. We address this question using a functorial…
The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a…
A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer…
In physics, timbre is a complex phenomenon, like color. Musical timbres are given by the superposition of sinusoidal signals, corresponding to longitudinal acoustic waves. Colors are produced by the superposition of transverse…
Mobiles are a particular class of decorated plane trees which serve as codings for planar maps. Here we address the question of enumerating mobiles in their most general flavor, in correspondence with planar Eulerian (i.e., bicolored) maps.…