On the Asymptotic Formula for the Number of Plane Partitions of Positive Integers

Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester

A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of consecutive integers, or, more generally, whose parts form a finite arithmetic progression. This paper reviews the relation between trapezoidal numbers, partitions, and the set of divisors of a positive integer. There is also a complete proof of a theorem of Sylvester that produces a stratification of the partitions of an integer into odd parts and partitions int…

A generalized Hardy-Ramanujan formula for the number of restricted integer partitions

We derive the asymptotic formula for $p_n(N,M)$, the number of partitions of integer $n$ with part size at most $N$ and length at most $M$. We consider both $N$ and $M$ are comparable to $\sqrt{n}$. This is an extension of the classical Hardy-Ramanujan formula and Szekeres' formula. The proof relies on the saddle point method.

Computation of q-Binomial Coefficients with the $P(n,m)$ Integer Partition Function

Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be computed in $O(n^2)$, and the q-binomial coefficient can be computed in $O(n^3)$. Using the definition of the q-binomial coefficient, some properties of the q-binomial coefficient and $P(n,m,p)$ are derived. The q-multinomial coefficient can be computed as a product of q-binomial coefficients. A formula for $Q(n,m,p)$, the number of integer partitions of $n$ into e…

Divisibility and distribution of $MEX$ related integer partitions of Andrews and Newman

Andrews and Newman introduced the minimal excludant or ``$mex$'' function for an integer partition $π$ of a positive integer $n$, $mex(π)$, as the smallest positive integer that is not a part of $π$. They defined $σmex(n)$ to be the sum of $mex(π)$ taken over all partitions $π$ of $n$. We prove infinite families of congruence and multiplicative formulas for $σmex(n)$. By restricting to the part of $π$, Andrews and Newman also introduced $moex(π)$ to be the smallest odd integer that is not a part of $π$ and $σmoex(n)$ to be the sum of $moex(π)$ taken over all partitions $π$ of $n$. In this arti…

关于含拟必需元素与禁用元素的整数分拆相关集合

On sets related to integer partitions with quasi-required elements and disallowed elements

本文研究与整数分拆相关的极小集合计算问题，并将其转化为数值半群框架求解。

关于整数划分的分形与递推方程

The Fractal and The Recurrence Equations Concerning The Integer Partitions

本文提出利用分形方法解决整数划分计数问题，并借此推导出包括五边形数定理在内的相关递推方程。

Dyson's Crank and the Mex of Integer Partitions

Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius symbols. Recently the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of the mex) by proving a number of properties that naturally relate these partition statistics.

整数分拆格中的箭头关系

Arrow Relations in Lattices of Integer Partitions

本文刻画任意正整数 n 的整数分拆格在优势序下标准背景中的单、双箭头关系，解决了 Ganter 2022 年的开放问题。

A Comparison of Integer Partitions Based on Smallest Part

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu in 2019. Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the case…

On the distribution of rank and crank statistics for integer partitions

Let $k$ be a positive integer and $m$ be an integer. Garvan's $k$-rank $N_k(m,n)$ is the number of partitions of $n$ into at least $(k-1)$ successive Durfee squares with $k$-rank equal to $m$. In this paper give some asymptotics for $N_k(m,n)$ with $|m|\ge \sqrt{n}$ as $n\rightarrow \infty$. As a corollary, we give a more complete answer for the Dyson's crank distribution conjecture. We also establish some asymptotic formulas for finite differences of $N_k(m,n)$ with respect to $m$ with $m\gg \sqrt{n}\log n$.

某些对称多维整数划分的最优传输

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。

Statistics on integer partitions arising from seaweed algebras

Using the index theory of seaweed algebras, we explore various new integer partition statistics. We find relations to some well-known varieties of integer partitions as well as a surprising periodicity result.

关于小幂次整数划分的中心极限定理

A Central Limit Theorem for Integer Partitions into Small Powers

本文研究了将整数划分为小幂次和的变体，并利用鞍点法证明了此类划分中加数数量的中心极限定理。

A note on enumerating colored integer partitions

In this short note, we give basic enumerative results on colored integer partitions.

On integer partitions corresponding to numerical semigroups

Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions.

六类可分整数分拆

Six types of separable integer partitions

本文从Andrews提出的可分整数分拆类视角出发，探讨六类分拆，扩展了该领域的相关定理研究。

A product of integer partitions

I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on partitions that is associative with a natural grading thus defining a free associative algebra on the set of integer partitions. As an outcome of the computations, certain sets of integers appear that I call difference sets and the product of the integers in a difference set is an invariant for a family of sets of partitions. The main combinatorial objects u…

带连续分拆部限制的可分整数分拆类

Separable integer partition classes with restrictions on consecutive parts

本文研究带连续分拆部限制的整数分拆，证明其属于Andrews提出的可分整数分拆类，并推导了这类分拆的生成函数。

Andrews-Merca整数分拆定理的一个双射证明

A bijection proof of Andrews-Merca integer partition theorem

本文通过构造n的分拆集合上的一般双射，证明六个相关统计量中两个三元组的等分布性，从而将Andrews和Merca关于奇偶部分之和的模2结果推广到所有整数m≥2的模m情形。

Optimal transport and integer partitions

We link the theory of optimal transportation to the theory of integer partitions. Let $\mathscr P(n)$ denote the set of integer partitions of $n \in \mathbb N$ and write partitions $π\in \mathscr P(n)$ as $(n_1, \dots, n_{k(π)})$. Using terminology from optimal transport, we characterize certain classes of partitions like symmetric partitions and those in Euler's identity $|\{ π\in \mathscr P(n) |$ all $ n_i $ distinct $ \} | = | \{ π\in \mathscr P(n) | $ all $ n_i $ odd $ \}|$. Then we sketch how optimal transport might help to understand higher dimensional partitions.

Counting pattern-avoiding integer partitions

A partition $α$ is said to contain another partition (or pattern) $μ$ if the Ferrers board for $μ$ is attainable from $α$ under removal of rows and columns. We say $α$ avoids $μ$ if it does not contain $μ$. In this paper we count the number of partitions of $n$ avoiding a fixed pattern $μ$, in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $μ$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic $p$-groups. We f…

Explicit formulas using partitions of integers for numbers defined by recursion

In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare our result with similar formulas obtained with different approaches already in the XIX century. Examples are given for Bernoulli, Euler and Fibonacci numbers.

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.

相交整数分拆：星形上界与各尺度反例

Intersecting integer partitions: star bounds and counterexamples at every scale

本文研究整数分拆的最大t-相交族，证明在n≥Ak^3时典范t-星最大，并构造了Borg猜想在各尺度的反例。

Exact Limit Theorems for Restricted Integer Partitions

For a set of positive integers $A$, let $p_A(n)$ denote the number of ways to write $n$ as a sum of integers from $A$, and let $p(n)$ denote the usual partition function. In the early 40s, Erdős extended the classical Hardy--Ramanujan formula for $p(n)$ by showing that $A$ has density $α$ if and only if $\log p_A(n) \sim \log p(αn)$. Nathanson asked if Erdős's theorem holds also with respect to $A$'s lower density, namely, whether $A$ has lower-density $α$ if and only if $\log p_A(n) / \log p(αn)$ has lower limit $1$. We answer this question negatively by constructing, for every $α> 0$, a set …

Generalization of a few results in Integer Partitions

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a …

Eventual log-concavity of $k$-rank statistics for integer partitions

Let $N_k(m,n)$ denote the number of partitions of $n$ with Garvan $k$-rank $m$. It is well-known that Andrews-Garvan-Dyson's crank and Dyson's rank are the $k$-rank for $k=1$ and $k=2$, respectively. In this paper, we prove that the sequence $\{N_k(m,n)\}_{|m|\le n-k-71}$ is log-concave for all sufficiently large $n$ and each integer $k$. In particular, we partially solve the log-concavity conjecture for Andrews-Garvan-Dyson's crank and Dyson's rank, which was independently proposed by Bringmann-Jennings-Shaffer-Mahlburg and Ji-Zang.

Distributions of Hook lengths in integer partitions

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $μ_t(n)\sim \frac{\sqrt{6n}}π-\frac{t}{2}$ and variance $σ_t^2(n)\sim \frac{(π^2-6)\sqrt{6n}}{2π^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $θ=\sqrt{2/(t-1)}.$

On Conjectures Concerning the Smallest Part and Missing Parts of Integer Partitions

For positive integers $s$ and $L \geq 3$, Berkovich and Uncu (Ann. Comb. $23$ ($2019$) $263$--$284$) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval $\{s, \ldots, L+s\}$. Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases $s=1$ and $s=2$. In the present article, we prove the conjecture for general $s$ by proving a stronger theorem. We also prove other related conjectures found in the same paper.

由二重级数生成的正加权分拆

Positive Weighted Partitions Generated by Double Series

本文研究生成函数为二重级数的加权整数分拆，通过建立 $q$-二重级数的闭式公式证明其系数非负，进而导出整数分拆间的不等式。

可分离整数划分类与Slater的列表 -- I

Separable integer partition classes and Slater's list -- I

本文利用Andrews的可分离整数划分理论，对Slater列表中的若干恒等式进行解析，提供自然的划分解释和参数化推广，并通过q-超几何变换推导出新的表达式。

On conjugates for set partitions and integer compositions

There is a familiar conjugate for integer partitions: transpose the Ferrers diagram, and a conjugate for integer compositions: transpose a Ferrers-like diagram. Here we propose a conjugate for set partitions and show that it interchanges # singletons and # adjacencies. Its restriction to noncrossing partitions cropped up in a 1972 paper of Kreweras. We also exhibit an analogous pairs of statistics interchanged by the composition conjugate.

Numerical Sets, Core Partitions, and Integer Points in Polytopes

We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results about core partitions. For small values of a, we give formulas for the number of (a,b)-core partitions corresponding to numerical semigroups. We also study the number of partitions with a given hook set.

整数分拆可检测素数

Integer partitions detect the primes

本研究解答Schneider的问题，证明整数分拆可通过MacMahon分拆函数的特殊常系数方程检测素数，且此类素数检测方程有无穷多个。

Partition-theoretic formulas for arithmetic densities

If $\gcd(r,t)=1$, then a theorem of Alladi offers the Möbius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} μ(n)n^{-1}= \frac{1}{φ(t)}. $$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of $n$. The right-hand side represents the proportion of primes in a fixed arithmetic progression modulo $t$. Locus generalized this to Chebotarev densities for Galois extensions. Answering a question of Alladi, we obtain analogs of these results to arithmetic densities of subsets of positive integers using $q$-series and integer partitions. For suitable subsets $§$ of th…

一些可分离整数分拆类

Some Separable integer partition classes

本文基于 Andrews 的可分离整数分拆类，研究按奇偶性分隔部分的分拆，并将其推广到过划分，探讨与 Rogers-Ramanujan 恒等式相关的可分离过划分类。

Partition Identities and the Coin Exchange Problem

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.

Sequences, q-Multinomial Identities, Integer Partitions with Kinds, and Generalized Galois Numbers

Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.

On the dimension of downsets of integer partitions and compositions

We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension.

虚二次整数环中的分拆正则性

Partition regularity in imaginary quadratic rings of integers

本文研究了具有良好线性因子分解的二次 homogeneous 方程在虚二次整数环中的分拆正则性，提出了数论结果，如无周期完全乘法函数的特征化、Turán-Kubilius 不等式和乘法函数的新集中估计。

关于双分拆的 Cayley 算法

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。

同余类中最小划分排除数矩的渐近性质

Asymptotics for moments of the minimal partition excludant in congruence classes

本文研究整数划分中频率小于给定数的最小正整数的矩，针对特定同余类建立渐近公式，并证明固定模数下各矩渐近相等。

An efficient Algorithm to partition a Sequence of Integers into Subsets with equal Sums

To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add up to the same sum.

可分离整数分拆类与带同余条件的分拆

Separable integer partition classes and partitions with congruence conditions

本文利用可分离整数分拆类研究模k同余分拆，引入(k,r)-过划分并建立其与部分第k次出现可加划线分拆的等数关系，进而推广可分离分拆类并给出满足同余条件的(k,r)-模过划分的生成函数。

Maximum entropy and integer partitions

We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions.

M-partitions: Optimal partitions of weight for one scale pan

An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the p…

A new order on integer partitions

Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions equivalent to the classical Foulkes conjecture. We establish some of the properties of this relation via the construction of explicit inclusion of modules whose characters correspond to the plethysms considered. We also prove some stability properties for the number of irreducible occurring in these modules as $n$ grows.

开花的树：整数分拆树与整数复合树目录及其渐近分析

Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis

本文研究节点携带整数分拆或整数复合“花朵”的组合树类，给出花瓣权重与花朵权重的生成函数，并推导系数渐近增长及集中分布期望的通用公式。

Partitions of an Integer into Powers

In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

Combinatorics of Integer Partitions With Prescribed Perimeter

We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.

A theorem about partitioning consecutive numbers

In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an interesting statement about triangular numbers, those positive integers which can be partitioned into consecutive numbers beginning at 1. For every partition of a triangular number n into consecutive numbers we can partition the sequence of numbers beginning at 1, adding up to n again, such that every part of this partition adds up to exactly one number of the chose…

Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.

The Lattice of integer partitions and its infinite extension

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of an integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an e…

整数划分中统计配置和的多项式玻色形式及奇偶最小排除量

A polynomial bosonic form of statistical configuration sums and the odd/even minimal excludant in integer partitions

本文提出两种新的整数划分统计量sqrank和rerank，并证明其与奇偶最小排除量在划分数目上具有等价性。