A group of students in 7-9 grades are inventing combinatorial impartial games. The games are played on graphs, piles, and grids. We found winning positions, optimal strategies, and other interesting facts about the games....
We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic numbers, that the...
We give a combinatorial criterion for determining which Bestvina--Brady group is isomorphic to a right-angled Artin group....
Right-angled Artin groups and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uni...
The celebrated Foata combinatorial model for Hermite polynomials, and his seminal and beautiful proof of the Mehler formula, are straightened to deal with two sexes rather than one, with the exclusion of same-sex relationships (both marital and non...
Given a linear equation $\mathcal{L}$, a set $A$ of integers is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. This notion incorporates many central topics in combinatorial number theory such as sum-free and...
Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials $G_π$ indexed by permutations in the basis of stable Grothendieck polynomials $G_λ$ index...
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved...
This paper classifies and constructs explicitly all the irreducible representations of affine Hecke algebras of rank two root systems. The methods used to obtain this classification are primarily combinatorial and are, for the most part, an applica...
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors have shown that Patience Sorting provides an algorithmic description for permutatio...
Despite having been introduced in 1962 by C.L. Mallows, the combinatorial algorithm Patience Sorting is only now beginning to receive significant attention due to such recent deep results as the Baik-Deift-Johansson Theorem that connect it to field...
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors extended Patience Sorting to a full bijection between the symmetric group and certa...
We classify combinations of isolated singularities that can occur on complex cubic threefolds generalizing analogous results for cubic surfaces due to Schläfli and Bruce--Wall. In addition, we provide concise combinatorial description of the possibl...
In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that can arise re...
From self-assembly and protein folding to combinatorial metamaterials, a key challenge in material design is finding the right combination of interacting building blocks that yield targeted properties. Such structures are fiendishly difficult to find...
We provide a complete and explicit characterization of the exposed extreme rays of the cone of sums of nonnegative circuit (SONC) polynomials. The criterion we derive is purely combinatorial and depends only on the existence of certain circuits withi...
Scalable quantum computers hold the promise to solve hard computational problems, such as prime factorization, combinatorial optimization, simulation of many-body physics, and quantum chemistry. While being key to understanding many real-world phenom...
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approa...
We characterize folding patterns, the combinatorial options of folding minimal alcove-to-alcove galleries in affine Coxeter complexes positively with respect to Weyl chamber orientations of the Coxeter complex, by drawing a connection to the Bruhat m...
We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs wi...