'Congruent Partitions' of Polygons - a Short Introduction

Decompositions of a rectangle into non-congruent rectangles of equal area

In this paper, we deal with a simple geometric problem: Is it possible to partition a rectangle into $k$ non-congruent rectangles of equal area? This problem is motivated by the so-called `Mondrian art problem' that asks a similar question for dissections with rectangles of integer sides. Here, we generalize the Mondrian problem by allowing rectangles of real sides. In this case, we show that the minimum value of $k$ for a rectangle to have a `perfect Mondrian partition' (that is, with non-congruent equal-area rectangles) is seven. Moreover, we prove that such a partition is unique (up to symm…

Sequentially congruent partitions and related bijections

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and parti…

Colored partitions of a convex polygon by noncrossing diagonals

For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions.

序列一致划分与平方数划分

Sequentially congruent partitions and partitions into squares

本文通过构造双射证明序列一致划分与平方数划分计数相等，并推广至k次幂划分。

球面由角度全等五边形进行边到边镶嵌

Edge-to-edge Tilings of the Sphere by Angle Congruent Pentagons

研究球面由全等五边形镶嵌时，仅通过角度信息确定镶嵌方式及简化过程对镶嵌特征的影响。

球面的侧向等边曲边三角形镶嵌

Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles

该研究分类了球面侧向等边曲边三角形镶嵌，发现其本质是直线三角形镶嵌的变形。

Cutting Mutually Congruent Pieces from Convex Regions

What is the shape of the 2D convex region P from which, when 2 mutually congruent convex pieces with maximum possible area are cut out, the highest fraction of the area of P is left over? When P is restricted to the set of all possible triangular shapes, our computational search yields an approximate upper bound of 5.6% on the area wasted when any triangle is given its best (most area utilizing) partition into 2 convex pieces. We then produce evidence for the general convex region which wastes the most area for its best convex 2-partition not being a triangle and briefly discuss some further g…

关于由凸多边形的不相交对角线划分构造花带图案

On the construction of frieze patterns from partitions of convex polygons by nonintersecting diagonals

本文初等地展示了如何从凸m边形的不相交对角线划分构造宽度为m-3的花带图案。

Congruences for 9-regular partitions modulo 3

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4 and 5. A collection of conjectures includes two congruences modulo higher powers of 2 and a large family of "congruences with exceptions" for these and other regular partitions mod 3.

Proper partitions of a polygon and k-Catalan numbers

Let P be a polygon whose vertices have been colored (labeled) cyclically with the numbers 1,2,...,c. Motivated by conjectures of Propp, we are led to consider partitions of P into k-gons which are proper in the sense that each k-gon contains all c colors on its vertices. Counting the number of proper partitions involves a generalization of the k-Catalan numbers. We also show that in certain cases, any proper partition can be obtained from another by a sequence of moves called flips.

彩色合成、Dyck 路径与多边形剖分之间的双射

Bijections between colored compositions, Dyck paths, and polygon partitions

该研究通过识别并装饰合成、Dyck 路径与凸多边形剖分中的构建块，构造了保持部分且支持彩色情形的双射，从而统一刻画三类组合对象。

A New Companion to Capparelli's Identities

We discuss a new companion to Capparelli's identities. Capparelli's identities for m=1,2 state that the number of partitions of $n$ into distinct parts not congruent to m, -m modulo $6$ is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not con…

Efficient covering of convex domains by congruent discs

In this paper, we consider the problem of covering a plane region with unit discs. We present an improved upper bound and the first nontrivial lower bound on the number of discs needed for such a covering, depending on the area and perimeter of the region. We provide algorithms for efficient covering of convex polygonal regions using unit discs. We show that the computational complexity of the algorithms is pseudo-polynomial in the size of the input and the output. We also show that these algorithms provide a constant factor approximation of the optimal covering of the region.

Berkovich与Uncu的分拆定理的双射证明

A bijective proof of a partition theorem of Berkovich and Uncu

针对Berkovich与Uncu2016年提出的分拆定理，研究者构建了由2-模图等要素组成的显式双射，还借助AxiomProver在Lean中形式化验证了该主定理的证明，解决了其提出的组合证明问题。

Beck-type identities: new combinatorial proofs and a theorem for parts congruent to $t$ mod $r$

Let $\mathcal O_r(n)$ be the set of $r$-regular partitions of $n$, $\mathcal D_r(n)$ the set of partitions of $n$ with parts repeated at most $r-1$ times, $\mathcal O_{1,r}(n)$ the set of partitions with exactly one part (possibly repeated) divisible by $r$, and let $\mathcal D_{1,r}(n)$ be the set of partitions in which exactly one part appears at least $r$ times. If $E_{r, t}(n)$ is the excess in the number of parts congruent to $t \pmod r$ in all partitions in $\mathcal O_r(n)$ over the number of different parts appearing at least $t$ times in all partitions in $\mathcal D_r(n)$, then $E_{r…

Partitioning orthogonal polygons into at most 8-vertex pieces, with application to an art gallery theorem

We prove that every simply connected orthogonal polygon of $n$ vertices can be partitioned into $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ mobile guards are sufficient to control the interior of an $n$-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only required at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answer…

'Fair' Partitions of Polygons - an Introduction

We address the question: Given a positive integer $N$, can any 2D convex polygonal region be partitioned into $N$ convex pieces such that all pieces have the same area and same perimeter? The answer to this question is easily `yes' for $N$=2. We prove the answer to be `yes' for $N$=4 and also discuss higher powers of 2.

从锥体和半圆出发的非交叉划分

Noncrossing Partitions From Cones and Semicircles

研究凸多边形边上的点配置生成的非交叉划分格，探讨其三种变体及与卡特兰数的关联。

A Double Bounded Version of Schur's Partition Theorem

Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts congruent 0,1 (mod 3) and another on the parts congruent 2 (mod 3), and by keeping track of the number of parts in each of the residue classes (mod 3). Despite the long history of Schur's theorem, our result is new, and extends earlier work of Andrews, Alladi-G…

$q$-Series congruences involving statistical mechanics partition functions in regime III and IV of Baxter's solution of the hard-hexagon model

For each $s\in\{2,4\}$, the generating function of $R_s(n)$, the number of partitions of $n$ into odd parts or congruent to $0$, $\pm s\pmod {10}$, arises naturally in regime III of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. For each $s\in\{1,3\}$, the generating function of $R^*_s(n)$, the number of partitions of $n$ into parts not congruent to $0$, $\pm s\pmod {10}$ and $10-2s \pmod {20}$, arises naturally in regime IV of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. In this paper, we investigate the parity of $R_s(n)$ and $R^*…

非交叉划分与围栏配置

Noncrossing Partitions From Hull Configurations

本文研究了围栏配置所确定的非交叉划分格，证明其为布尔子偏格的并集，并在特定条件下具有对称链分解。

关于Mao提出的Beck分拆统计量模8猜想的一个证明

A proof of a conjecture of Mao on Beck's partition statistics modulo 8

该研究在Beck分拆统计量的已有同余与恒等式研究基础上，证明了模8的NT统计量与模4的M_ω统计量的恒等式，证实了Mao提出的相关猜想。

Partitions of unity in $\mathrm{SL}(2,\mathbb Z)$, negative continued fractions, and dissections of polygons

We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array} \right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots \left( \begin{array}{cc} a_1&-1 1&0 \end{array} \right) $ is either the identity matrix $\mathrm Id$, its negative $-\mathrm Id$, or square root of $-\mathrm Id$. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.

Congruences for the $k$ dots bracelet partition functions

By finding the congruent relations between the generating function of the 5 dots bracelet partitions and that of the 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime $p$, we study the arithmetic properties modulo $p$ of the $k$ dots bracelet partitions.

MacMahon Partition Analysis: A discrete approach to broken stick problems

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for $n\geq k\geq 3$, the probability that a $k$-gon cannot be formed from a stick broken into $n$ parts is given by $n!$ over a product of linear combinations of partial sums of generalized F…

Races among products

We will revisit a 1987 question of Rabbi Ehrenpreis. Among many things, we will provide an elementary injective proof that P_1(L,y,n)>=P_2(L,y,n) for any L,n>0 and any odd y>1 . Here, P_1(L,y,n) denotes the number of partitions of n into parts congruent to 1, y+2, or 2y mod 2(y+1) with the largest part not exceeding 2(y+1)L-2 and P_2(L,y,n) denotes the number of partitions of n into parts congruent to 2, y, or 2y+1 mod 2(y+1) with the largest part not exceeding 2(y+1)L-1.

Subdivision of complexes of k-Trees

Consider the poset of partitions of {1,...(n-1)k+1} with block sizes congruent to 1 modulo k. We prove that its order complex is a subdivision of the complex of k-trees, thereby answering a question posed by Feichtner. The result is obtained by an ad-hoc generalization of concepts from the theory of nested set complexes to non-lattices.

On the Number of 2-Hooks and 3-Hooks of Integer Partitions

Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a \bmod{b}$. For $t\in \{2, 3\}$, arithmetic progressions $r_1 \bmod{m_1}$ and $r_2 \bmod{m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males, and Ono using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of $t$-cores and $t$-quotients.

配对分拆定理I

The Partition Pairing Theorems I

本文引入分拆的配对理论，定义配对指数、配对秩、配对宽度等统计量，证明相关等分布、同余式等结论，关联超分拆与弗罗贝尼乌斯表示，还研究负配对秩的单配对分拆并得到相关恒等式与行列式结果。

Classification and enumeration of lattice polygons in a disc

In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of given area. Since then, this problem and its analogues have been studied by many authors, including $\mathrm{B\acute{a}r\acute{a}ny}$, Lagarias, Pach, Santos, Ziegler and Zong. Recently, Zong proposed two computer programs to prove Hadwiger's covering conjecture and Borsuk's partition problem, respectively, based on enumeration of the convex lattice polytopes contained in certain balls. For this purpose, similar to $\mathrm{B\acute{a}r\acute{a}ny}$ and Pach's work on volume and Liu and Zong's work on cardin…

Cellular Structures determined by Polygons and Trees

The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and incorporate the results into properties of the real moduli space of Riemann spheres. Connections are discussed with relation to classic combinatorial problems as well as to other sciences.

关于凸多边形加权部分三角剖分的研究

On weighted partial triangulations of convex polygons

本文研究了如何对凸多边形进行加权部分三角剖分的采样问题，提出了一种高效的精确采样方法，解决了传统马尔可夫链方法存在的混合时间不优问题。

螺旋、井字棋划分与深对角映射

Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps

本研究推广五角星映射提出深对角映射$T_k$，证明其保持α、β型$k$-螺旋不变，推导$T_3$有理公式与四个代数不变量，揭示其轨道的井字棋网格划分结构，完善了映射的几何推广理论。

一类Tverberg型的正多边形划分

Regular Polygonal Partitions of a Tverberg Type

本文研究Tverberg定理不适用的少点集划分问题，提出正多边形/正多胞体形的P(r,d)划分，证明平面及高维空间中对应点数的划分可行性与紧界，并给出素数幂下的拓扑推广及相关衍生结论。