Solution of Erdős Problem 633
We classify triangles that can be tiled only into a square number of congruent triangles, settling Erdős Problem 633.
math.CO↗
arXiv subjects
Yan X. Zhang
Publications and source records attributed to Yan X. Zhang.
Discrete Envy-free Division of Necklaces and Maps
We study the discrete variation of the classical cake-cutting problem where n players divide a 1-dimensional cake with exactly (n-1) cuts, replacing the continuous, infinitely divisible "cake" with a necklace of discrete, indivisible "beads." We focus specifically on envy-free divisions, exploring different constraints on player-preferences. We show we usually cannot guarantee an envy-free division and consider situations where we can obtain an envy-free division for relatively small. We also prove a 2-dimensional result with a grid of indivisible objects. This may be viewed as a way to divide a state with indivisible districts among a set of constituents, producing somewhat gerrymandered regions that form an envy-free division of the state.
math.CO↗
Convex Neural Codes in Dimension 1
Neural codes are collections of binary strings motivated by patterns of neural activity. In this paper, we study algorithmic and enumerative aspects of convex neural codes in dimension 1 (i.e. on a line or a circle). We use the theory of consecutive-ones matrices to obtain some structural and algorithmic results; we use generating functions to obtain enumerative results.
math.CO↗
Four Variations on Graded Posets
We explore the enumeration of some natural classes of graded posets, including all graded posets, (2+2)- and (3+1)-avoiding graded posets, (2+2)-avoiding graded posets, and (3+1)-avoiding graded posets. We obtain enumerative and structural theorems for all of them. Along the way, we discuss a situation when we can switch between enumeration of labeled and unlabeled objects with ease, generalize a result of Postnikov and Stanley from the theory of hyperplane arrangements, answer a question posed by Stanley, and see an old result of Klarner in a new light.
math.CO↗
Structural Theory of 2-d Adinkras
Adinkras are combinatorial objects developed to study 1-dimensional supersymmetry representations. Recently, 2-d Adinkras have been developed to study 2-dimensional supersymmetry. In this paper, we classify all 2-d Adinkras, confirming a conjecture of T. Hübsch. Along the way, we obtain other structural results, including a simple characterization of Hübsch's even-split doubly even code.
hep-th↗
Separating hyperplanes of edge polytopes
Let $G$ be a finite connected simple graph with $d$ vertices and let $\Pc_G \subset \RR^d$ be the edge polytope of $G$. We call $\Pc_G$ \emph{decomposable} if $\Pc_G$ decomposes into integral polytopes $\Pc_{G^+}$ and $\Pc_{G^-}$ via a hyperplane. In this paper, we explore various aspects of decomposition of $\Pc_G$: we give an algorithm deciding the decomposability of $\Pc_G$, we prove that $\Pc_G$ is normal if and only if both $\Pc_{G^+}$ and $\Pc_{G^-}$ are normal, and we also study how a condition on the toric ideal of $\Pc_G$ (namely, the ideal being generated by quadratic binomials) behaves under decomposition.
math.CO↗