L. Lovasz has shown that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. Inspired by this result we prove that classical Ky Fan's theorem admits an oriented m...
We give an intuitive combinatorial proof of Ky Fan's covering lemma based on the Borsuk-Ulam theorem. We then show how this approach can be generalized to Ky Fan's covering lemma for several linear orders....
For a simple graph, we introduce a notion of the star sequence and prove that the star sequence and the frequently sequences of a graph are inverses of each other from a combinatorial point of view. As a consequence, we express the general first Zagr...
The aim of this paper is to give a corrected bijective proof of Vershik's relations for the Kostka numbers. Our proof uses insertion and reverse insertion algorithms, as in the combinatorial proof of the Pieri rule....
Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids are used within the segmentation framework to encode a hierarchy of partitions. The different graph models used within the irregular pyramid...
When factorizing binary matrices, we often have to make a choice between using expensive combinatorial methods that retain the discrete nature of the data and using continuous methods that can be more efficient but destroy the discrete structure. Alt...
Introduction Metaverse is the next evolution of digital technologies. It includes 3D virtualization and will transform digital technologies in the next 5-10 years. Elements of a Metaverse are considered very important related to industry 4.0. Metaverse will ha…
Introduction Metaverse is the next evolution of digital technologies. It includes 3D virtualization and will transform digital technologies in the next 5-10 years. Elements of a Metaverse are considered very important related to industry 4.0. Metaverse will ha…
We consider a connected negative definite plumbing graph, and we assume that the associated plumbed 3-manifold is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg-Witten invariant of this manifold. The first one i...
In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $φ_k(n)$ and $cφ_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of...
This paper proves a deformation circle pattern theorem, which gives a complete description of those circle patterns with interstices in terms of the combinatorial type, the exterior intersections angles and the conformal structures of interstices. As...
Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and S...
We define a natural equivalence relation on collections of cards from the card game SET, and enumerate some of the equivalence classes, vastly generalizing the standard game. On this basis, we describe several alternative games for the SET deck, incl...
Given $n \geq 3$, a combinatorial object called a \textit{ pedigree } is defined using $3$-element subsets from $[n]$ obeying certain conditions. The convex hull of pedigrees is called the pedigree polytope for $n$. Pedigrees are in $1-1$ corresponde...
We give a direct combinatorial proof that the modular envelope of the cyclic operad Ass is the modular operad of (the homeomorphism classes of) 2D compact surfaces with boundary with marked points....
In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the mos...
A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ on the vertex set of a $2d$-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid....
Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight t...
This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of...
Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of...