Maya Mohsin Ahmed
Magic labelings of graphs are studied in great detail by Stanley and Stewart. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.
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The n'th Birkhoff polytope is the set of all doubly stochastic n-by-n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n up to 8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function counting the integer points in the dilated polytope. One reason to be interested in this counting function is that the leading term of the Ehrhart polynomial is--up to a trivial factor--the volume of…
We compute the set of facets of the polytope which is the convex hull of the Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$ we found $2$ orbits of facets which contradicts previous results published in \cite{birkhoff}. For the group $\mathsf{H}_4$ we found $1063$ orbits of facets which provides a counterexample to the conjecture of \cite{birkhoff}.
The Birkhoff polytope, defined to be the convex hull of $n\times n$ permutation matrices, is a well studied polytope in the context of the Ehrhart theory. This polytope is known to have many desirable properties, such as the Gorenstein property and existence of regular, unimodular triangulations. In this paper, we study analogues of the Birkhoff polytope for finite irreducible Coxeter groups of other types. We focus on a type-$B$ Birkhoff polytope $\mathcal{BB}(n)$ arising from signed permutation matrices and prove that it and its dual polytope are reflexive, and hence Gorenstein, and also pos…
Type B c-Birkhoff polytopes are order polytopes
本文证明 B 型 c-Birkhoff 多胞体与堆偏序集的序多胞体单模等价，从而回答了此前关于 A 型结果的推广问题。
$c$-Birkhoff polytopes
本文针对对称群的每个Coxeter元$c$定义了$c$-Birkhoff多面体，证明其与最长置换$c$-排序词的堆偏序集的序多面体幺模等价，既肯定回答了Davis和Sagan的问题，也得出其归一化体积为A型$c$-Cambrian格最长链数的结论。
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics. In this paper we study combinatorial types L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face with combinatorial type L. By a result of Billera and Sarangarajan, a combinatorial type L of a d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is…
Restricted Birkhoff polytopes and Ehrhart period collapse
本研究证明带最长递增子序列约束的受限Birkhoff多胞体的Ehrhart拟多项式为真实多项式，通过与Gelfand-Tsetlin多胞体的类Robinson-Schensted-Knuth对应分段线性双射实现，揭示多胞体格点计数性质。
We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.
The n'th Birkhoff polytope $B_n$ is the set of all doubly stochastic $n \times n$ matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A long-standing open problem is the determination of the relative volume of $B_n$. In [arXiv:math.CO/0202267] we introduced a method of calculating this volume and used it to compute $\vol B_9$. This note is an update on our progress: with the same program (but much longer computing time), we have now derived $\vol B_{10}$.
Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of …
We present a change of basis that may allow more efficient calculation of the volumes of Birkhoff polytopes using a slicing method. We construct the basis from a special set of square matrices. We explain how to construct this basis easily for any Birkhoff polytope, and give examples of its use. We also discuss possible directions for future work.
A new class of magic positive Ehrhart polynomials of reflexive polytopes
本文构造了一类新的具有魔正Ehrhart多项式的自反多面体，证明Stasheff多面体满足该性质，并部分证明圈的对称边多面体对偶的魔正性。
Consider the Birkhoff polytope of n by n doubly-stochastic matrices. As the Birkhoff-von Neumann theorem famously states, its vertex set coincides with the set of all n by n permutation matrices. Here we seek a higher-dimensional analog of this basic fact. Namely, consider the polytope which consists of all tristochastic arrays of order n. These are n by n by n arrays with nonnegative entries in which every line sums to 1. What can be said about its vertex set? It is well-known that an order-n Latin square may be viewed as a tristochastic array where every line contains n-1 zeros and a single …
Cliques and independent subgroups of the Birkhoff polytope graph
研究Birkhoff多面体图的团和独立子群结构，证明了关于Sym(n)中3循环排列子集的最大大小的Erdős-Ko-Rado型定理。
The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices in $\mathbb{R}^{n\times n}$. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group $G\leq \operatorname{GL}(d,\mathbb{R})$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.
We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.
Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes
本文证明了伪对称单纯反射多胞体的极对偶的Ehrhart多项式具有魔幻正性，并进一步证明伪对称光滑Fano多胞体的魔幻系数严格为正，从而得出其极对偶Ehrhart正且$h^*$-多项式实根且$\u03b3$-正，同时解决了Konoike关于循环对称边多胞体的问题。
Luck and magic for Pitman-Stanley polytopes and parking functions
本文研究Pitman-Stanley多面体的Ehrhart理论，证明其Ehrhart多项式具有强正性现象，进而推导出其h*-多项式实根性。通过解读系数于magic基中，关联到修改停车协议中的幸运汽车数。
On the magic positivity of Ehrhart polynomials of dilated polytopes
本文研究多项式的魔力正性，证明了正实系数多项式经充分放大后必为魔力正，并构造了任意给定维度下不满足此性质的放大多胞体，最后探讨了使Ehrhart多项式魔力正所需的放大程度。
We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric piec…
The set of bistochastic or doubly stochastic N by N matrices form a convex set called Birkhoff's polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff's polytope. For N=3 we present fairly complete results. For N=4 partial results are obtained. An interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der Waerden matrix for N=3, while this is not the case for N=4.
Normality of $k$-Matching Polytopes of Bipartite Graphs
本文证明二部图的 $k$-匹配多胞体是正规的，并给出其 $H$-表示，推广了 Birkhoff 多胞体正规性与 Birkhoff-Von Neumann 定理。
Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph $K_{n,n}$ with labels coming from $F_2^d$ , that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size …
Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces
本文研究了一类特殊的鲁棒Hadamard矩阵，其投影到二维子空间形成Hadamard矩阵。证明了当存在skew Hadamard矩阵时，此类矩阵存在。对于偶数维度n≤20，展示了Birkhoff多面体中任意射线上的双随机矩阵都是unistochastic的，并通过鲁棒Hadamard矩阵构造了具有相同纠缠度的正交基，从而在产品基和最大纠缠基之间进行插值。
Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
本文提出一种利用常数项和有限域精确计算 Birkhoff 多胞形 Ehrhart 级数的方法，通过根过滤器、对称函数和广义 Todd 系数实现高效计算，并推广到 World Cup 问题，给出直至阶数 12 的完整级数。
Ehrhart polynomials of cyclic polytopes as averages of zonotope Ehrhart polynomials
本文证明循环多胞体的 Ehrhart 多项式可表示为格带形多胞体 Ehrhart 多项式的平均值，从而推出其魔幻正性、$h^\ast$-多项式实根性及 $h^\ast$-向量对数凹与单峰性。
In the chapter "Magic with a Matrix" in \emph{Hexaflexagons and Other Mathematical Diversions} (1988), Martin Gardner describes a delightful "party trick" to fill the squares of a $d$-by-$d$ chessboard with nonnegative integers such that the sum of the numbers covered by any placement of $d$ nonthreatening rooks is a given number $N$. We consider such chessboards from a geometric perspective which gives rise to a family of lattice polytopes. The polyhedral structure of these Gardner polytopes explains the underlying trick and enables us to count such chessboards for given $N$ in three differen…
Lattice slices, Ehrhart polynomials, and magic positivity of generalized parking-function polytopes
研究广义停车函数多面体，证明其格切片性质并得出格点数递归，推导埃尔哈特多项式公式，在特殊情况有显式形式与生成函数，还对双参数族多面体的神奇正性分类，回答相关问题并扩展前人工作。
Birkhoff-von Neumann Graphs that are PM-compact
本文研究了Birkhoff-von Neumann图和PM-紧图的交集，给出了匹配覆盖图中同时满足BvN和PMc条件的完整刻画，从而使得相关决策问题处于P复杂度类中。
We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.
Magic Positivity for the Ehrhart Polynomials of Partial Permutohedra
研究部分排列多面体\(\mathcal{P}(m,n)\)的埃尔哈特多项式的神奇正性。通过证明，当\(n \ge m - 1\)时，除\((m,n)=(2,1)\)外该多项式是神奇正的；对于\(n<m - 1\)，讨论了\(n = 1,2,3\)时的情况并给出反例。
On the $f$-vectors of flow polytopes for the complete graph
本文利用图双射与子图枚举结果，给出了完全图流多面体及$CRY_n$多面体$f$-向量的生成函数与显式公式，并将其扩展至任意非负净流场景，恢复了Tesler多面体的$f$-向量。
Reducing the NP-problems to the convex/linear analysis on the Birkhoff polytope.
Geometric and Combinatorial Properties of the Alternating Sign Matrix Polytope
本文研究交替符号矩阵多面体 $ASM_n$ 的面，结合初等流网格与双向图刻画其对称性、2-level 性质、顶点/边/面数界、面结构及低维组合类型。
The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities
本文证明了m=2 amplituhedron的魔力数猜想，即其每个铺砌的基数均为组合数${n-2 \choose k}$，并给出了铺砌的组合必要条件、体积公式及Parke-Taylor恒等式的新变体。
We use the results of AG/0406290 to discuss the counting formulas of network flow polytopes and magic squares, i.e. the formula for the corresponding Ehrhart polynomial in terms of residues. We also discuss a description of the big cells using the theory of non broken circuit bases.
Compact graphs and quantum automorphisms
提出量子自同构群在态上的基本魔法酉矩阵，定义介于经典自同构多面体与分数自同构多面体之间的双随机矩阵闭凸集，证明量子紧图等价于经典紧图。
Monotonicity in Quadratically Regularized Linear Programs
针对二次正则化线性规划，研究正则化参数松弛时解支撑集的单调性，证明其等价于多面体的几何性质，验证低维最优传输问题成立但高维不成立，并将几何视角应用于Erdős问题。
In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to b…
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