Aritro Pathak
We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d, and the set of partitions where at least one part appears d or more times.
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On Glaisher's Partition Theorem
本文基于Glaisher定理，扩展了Euler的划分定理至m=3情况，引入新的划分函数并证明了相应的恒等式，同时给出了Glaisher乘积的新级数形式。
In this paper, we proof Mizukawa-Yamada's $X-Y=C$ and $m$-tuple version of this.
Extending the notion of $r$-(class) regular partitions, we define $(r_{1},...,r_{m})$-class regular partitions. A partition identity is presented and described by making use of the Glaisher correspondence.
For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not divisible by $l$ and occurring less than $k$ times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case $l=2$, we discuss the possibility …
Some consequences of Glaisher's map and a generalization of Sylvester's theorem
本文研究Glaisher映射相关的k-正则分拆特殊情形，通过引入推广自共轭分拆的新分拆类，推广了Sylvester定理，并为若干Rogers-Ramanujan恒等式提供了新的组合解释。
Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective proofs are presented.
In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of Glaisher's bijection, the map may be widely useful in other partition combinatorics. A previous conjecture concerning iterations of Glaisher's bijection is given a counterexample.
The original Beck conjecture, now a theorem due to Andrews, states that the difference in the number of parts in all partitions into odd parts and the number of parts in all strict partitions is equal to the number of partitions whose set of even parts has one element, and also to the number of partitions with exactly one part repeated. This is a companion identity to Euler's identity. The theorem has been generalized by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity to Franklin's ide…
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…
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressoud's bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
In a recent paper, Andrews and Merca investigated the number of even parts in all partitions of $n$ into distinct parts, which arise naturally from the Euler-Glaisher bijective proof. They obtained new combinatorial interpretations for this number by using generating functions. We obtain a new direct combinatorial proof in this note.
Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of $n$ with odd parts equals the number of compositions of $n+1$ with parts greater than one, as both numbers equal the Fibonacci number $F_n$. Recently, Sills provid…
Partition theorems and the Chinese remainder theorem
本文利用中国剩余定理证明了一个涵盖欧拉、麦克马洪等所有现有划分定理的综合性定理，并基于特殊完全剩余系推广了欧拉定理，还建立了划分函数间的同余联系。
Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, versus the set of partitions of $n$ where the set of repeated parts has exactly $j$ elements. These two sets of partitions turn out to be equinumerous, and this naturally encloses Euler's theorem and Andrews' theorem as two special cases. We give two proofs, one using generating function, and the other is a direct bijection that builds on Glaisher's bijection
In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modu…
A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a simple tool, including the existence of a related statistic that realizes every possible Ramanujan-type congruence for the partition function. To further exhibit its research utility, we give an easy generalization of a theorem of Andrews, Dixit and Yee on the mock theta functions. Throughout, we state a number of observations and questions that can motivate…
Generalizations of Euler's Theorem to $k$-regular partitions
本文通过提出与$k$-正则分拆集合$B_k(n)$等势的新分拆集合$E_k(n)$，推广了Glaisher关于$k$-互异分拆与$k$-正则分拆等势的结论及欧拉分拆定理。
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。
A combinatorial proof of a partition perimeter inequality
本文通过将划分周长与组合相联系，给出了 Amdeberhan-Andrews-Ballantine 定理的组合证明，并表明 Huang 的组合定理可细化 Fu-Tang 的周长定理。
In this paper, we consider various theorems of P.A. MacMahon and M.V. Subbarao. For a non-negative integer $n$, MacMahon proved that the number of partitions of $n$ wherein parts have multiplicity greater than 1 is equal to the number of partitions of $n$ in which odd parts are congruent to 3 modulo 6. We give a new bijective proof for this theorem and its generalization, which consequently provides a new proof of Andrews' extension of the theorem. We also generalize Subbarao's finitization of Andrews' extension. This generalization is based on Glaisher's extension of Euler's mapping for odd-d…
Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k
本文将 Andrews 和 Merca 关于分拆中偶数部分数的研究推广至部分重数严格小于 k 时可被 k 整除的部分数，利用 Glaisher 双射扩展给出组合证明，并提供生成函数验证多种幂递推关系。
In a recent paper, we generalized a partition identity stated by Siladić in his study of the level one standard module of type $A_2^{(2)}$. The proof used weighted words with an arbitrary number of primary colors and all the secondary colors obtained from these primary colors, and a brand new variant of the bijection of Bressoud for Schur's partition identity. In this paper, the first of two, we analyze this variant of Bressoud's algorithm in the framework of statistical mechanics, where an integer partition is viewed as an amount of energy shared, according to certain properties, between seve…
A generalization of conjugation of integer partitions
本文对整数划分的共轭进行推广，构造了参数为 $s$ 的对合，证明了两个统计量的联合对称性，并给出了双变量生成函数。核心贡献是提供了新的自然双射，其中 $s=1$ 时退化为经典共轭。
We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a previous paper by the author , enable us to state and prove new generalizations of two of these results.
Motivated by an error-correcting generalization of Bachet's weights problem, we define and classify relaxed complete partitions. We show that these partitions enjoy a succinct description in terms of lattice points in polyhedra, with adjustments in the error being commensurate with translations in the defining hyperplanes. Our main result is that the enumeration of the minimal such partitions (those with fewest possible parts) is achieved via Brion's formula. This generalizes work of Park on classifying complete partitions and that of Rødseth on enumerating minimal complete partitions.
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.
Given a set of positive integers A = {a_1,...,a_n}, we study the number p_A (t) of nonnegative integer solutions (m_1,...,m_n) to m_1 a_1 + ... m_n a_n = t. We derive an explicit formula for the polynomial part of p_A.
On Cayley algorithm for double partition
针对双分拆问题中 Cayley 原始算法因条件不满足而失效的情况，本文提出其修改版本，以完成向标量分拆之和的归约。
Optimal transport for some symmetric, multidimensional integer partitions
本文推广Hohloch关于整数划分与对称划分间最优传输映射的结果至多维情形，利用Kantorovich形式研究高维划分与对称划分的匹配。
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.
The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (pa…
On a specific problem of partition of graph
本文研究图顶点的2-划分问题，证明当图可划分为若干“小”图时，任意割的跨边数$e(S,S^{c})$均存在良好下界。
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the {\em Shaped Partition Polytope} defined as the convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially ma…
A set partition technique that is useful for identifying wires in cables can be recast in the language of 0--1 matrices, thereby resolving an open problem stated by R.~L. Graham in Volume 1 of this journal. The proof involves a construction of 0--1 matrices having row and column sums without gaps.
A Finite-Bound Partition Equinumerosity Result Generalizing a Solution of a Problem Posed by Andrews and Deutsch
本研究对Andrews、Deutsch2016年提出、后经Smoot和Yang2018年推广的分拆等计数结论做有限界拓展，并给出对应双射证明。
Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we g…
In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of th…
For $n$ and $k$ integers we introduce the notion of some partition of $n$ being able to generate another partition of $n$. We solve the problem of finding the minimum size partition for which the set of partitions this partition can generate contains all size-$k$ partitions of $n$. We describe how this result can be applied to solving a class of combinatorial optimization problems.
Alon, Bohman, Holzman and Kleitman proved that any partition of a $d$-dimensional discrete box into proper sub-boxes must consist of at least $2^d$ sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a $d$-dimensional box of odd size, and they asked whether the trivial construction consisting of $3^d$ boxes is best possible. We show that approximately $2.93^d$ boxes are enough, and consider some natural generalisations.
We present an affirmative answer to Stanley's zrank problem, namely, the zrank and rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs depend on the number of zero entries. Similar to notion of the jrank of a skew partition, we give a characterization of the rank in terms of the Giambelli type matrices of the corresponding skew Schur functions. We also show that the sign of the determinant of a factorial Cauchy matrix is uniquely determined by the number of its zero entries, which implies the nonsingularit…
We solve a problem conjectured by Cheriyan, giving sharp bounds for incidence of certain edge partitions of the connected graph on $n$-vertices. We briefly discuss the history of the problem and relation to node connectivity of strongly regular graphs. We show that the bound cannot be made sharper.
We introduce the problem of partitioning 2D regions (usually convex regions) into mutually congruent pieces ('tiles').
A bijection between symmetric plane partitions and quasi transpose complementary plane partitions
本文提出一种双射，将对称平面划分与准转置互补平面划分联系起来，并解决了一个35年的开放问题。
Wilf's Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as \emph{Wilf partitions}. Using $f(n)$ to denote the number of Wilf partitions, we establish lead-order asymptotics for $\ln{f(n)}$.
We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $μ$ one can force the strong polarized relation with respect to the pair $μ^+,μ$.
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem …
The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition p…
Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction. This shows for the first time that some matrices have exponential-sized minimal ob…
Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant o…
In the present paper, we study the limit sets of the almost periodic functions $f(x)$. It is interesting that the values $r=\inf|f(x)|$ and $R=\sup|f(x)|$ may be expressed in the exact form. We show that the ring $r\leq |z|\leq R$ is the limit set of the almost periodic function $f(x)$ (under some natural conditions on $f$). The exact expression for $r$ coincides with the well known partition problem formula and gives a new analytical method of solving the corresponding partition problem. Several interesting examples are considered. For instance, in the case of the five numbers, the well-known…
On cone partitions for the min-cut and max-cut problems with non-negative edges
本文研究了非负边权图的最小割和最大割问题，通过构建正交限的凸锥划分并定义邻接准则，揭示了两者锥划分图的顶点度数均为指数级且直径为2，但团数分别为线性和指数级。
We describe some open problems related to homology representations of subposets of the partition lattice, beginning with questions first raised in Stanley's work on group actions on posets.
Given a sparse undirected graph G with weights on the edges, a k-plex partition of G is a partition of its set of nodes such that each component is a k-plex. A subset of nodes S is a k-plex if the degree of every node in the associated induced subgraph is at least |S|-k. The maximum edge-weight k-plex partitioning (Max-EkPP) problem is to find a k-plex partition with maximum total weight, where the partition's weight is the sum of the weights on the edges in the solution. When k=1, all components in the partition are cliques and the problem becomes the well-known maximum edge-weight clique par…
We consider the problem of finding a minimum common partition of two strings (MCSP). The problem has its application in genome comparison. MCSP problem is proved to be NP-hard. In this paper, we develop an Integer Programming (IP) formulation for the problem and implement it. The experimental results are compared with the previous state-of-the-art algorithms and are found to be promising.
Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characteri…
On the complexity of Sandwich Problems for $M$-partitions
本文研究了$M$-划分问题的复杂性，通过结构分类和高效算法，对矩阵划分和沙丁鱼问题进行了P与NP完全的分类。
The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part. This problem was introduced by Gerber and Kobler [European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors, but its parameterized complexity remains open until now. It is known that the Satisfactory Partition problem, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We enhance our understa…
In the k-partition problem (k-PP), one is given an edge-weighted undirected graph, and one must partition the node set into at most k subsets, in order to minimise (or maximise) the total weight of the edges that have their end-nodes in the same cluster. Various hierarchical variants of this problem have been studied in the context of data mining. We consider a 'two-level' variant that arises in mobile wireless communications. We show that an exact algorithm based on intelligent preprocessing, cutting planes and symmetry-breaking is capable of solving small- and medium-size instances to proven…
Complexity Dichotomies for the Maximum Weighted Digraph Partition Problem
本文提出了最大加权有向图划分（MWDP）问题，并证明了其在任意、有向定向及对称有向图上的三个复杂性二分法，同时展示了该理论在二元行动polymatrix博弈及图论问题中的应用。
We study the optimization version of the set partition problem (where the difference between the partition sums are minimized), which has numerous applications in decision theory literature. While the set partitioning problem is NP-hard and requires exponential complexity to solve (i.e., intractable); we formulate a weaker version of this NP-hard problem, where the goal is to find a locally optimal solution. We show that our proposed algorithms can find a locally optimal solution in near linear time. Our algorithms require neither positive nor integer elements in the input set, hence, they are…
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