An Answer to a Question of Zeilberger and Zeilberger about Fractional Counting of Partitions

Johnson's bijections and their application to counting simultaneous core partitions

Johnson recently proved Armstrong's conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of $(b_1,b_2,\cdots, b_n)$-core partitions where $\{b_1,b_2,\cdots,b_n\}$ contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate $(s,s+1,s+2)$-core partition.

Counting Partitions on the Abacus

In 2003, Maroti showed that one could use the machinery of l-cores and l-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case l=2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss.

关于受限划分的带符号计数及三个恒等式的组合证明

Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities

该研究受相关组合方法启发，对\(l\)-正则划分及其变体的三个恒等式给出组合证明，其中两个恒等式此前由他人通过解析方法建立，此为恒等式证明提供了新的组合证明方式。

The Algorithm Z and Ramanujan's $_1ψ_1$ Summation

We use the Algortihm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan's $_1ψ_1$ summation formula.

计数集合划分中的对称峰与非对称峰

Counting symmetric and non-symmetric peaks in a set partition

该研究针对恰好含k个块的[n]集合划分，分别推导其对称峰与非对称峰总数的显式公式，并通过生成函数理论和组合工具两种方式给出结果。

Counting genus one partitions and permutations

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Ou…

Counting self-conjugate (s,s+1,s+2)-core partitions

We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection about the line $x=n/2$ is itself. In this paper, we show that the number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of symmetric Motzkin paths of length $s$, and give a closed formula for this number.

草图、移动与划分：反射排列形变的区域计数

Sketches, moves and partitions: counting regions of deformations of reflection arrangements

本文利用移动和草图概念，为反射排列的Catalan形变区域与非嵌套划分建立统一显式双射，并通过指数公式刻画特征多项式系数统计量，同时研究了阈值排列及其形变。

Counting Connected Set Partitions of Graphs

Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $π=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $π$ is connected for $1\le j\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is then $Q(G, x)=\sum_{i=0}^n q_{i}(G)x^i$. This paper presents a splitting approach to the partition polynomial on a separating vertex set $X$ in $G$ and summarizes some properties of the bond lattice. Furthermore the bivariate partition p…

Counting birthday collisions using partitions

We use partitions to provide some formulae for counting s-collisions and other events in various forms of the Birthday Problem.

Systematic Counting of Restricted Partitions

Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".

Approximate counting of standard set-valued tableaux

We present a randomized algorithm for generating standard set-valued tableaux by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of asymptotically rank two partitions, we use this algorithm to give a fully polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux. This FPAUS is then used to construct a fully polynomial randomized approximation scheme (FPRAS) for counting the number of standard set-valued tableaux for such shapes. We also construct a FPAUS and FPRAS for standard set-valued tableaux when either the size of the partition or the difference betw…

A bijection between self-conjugate and ordinary partitions and counting simultaneous cores as its application

We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.

在 n 的所有部分重数小于 k 的分拆中统计可被 k 整除的部分数

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。

带符号的划分矩阵计数

Signed counting of partition matrices

研究带符号划分矩阵计数与逆序序列子类的大小关系，引入不规范划分矩阵概念，并证明其与Motzkin路径的等分布性。

Refined counting of core partitions into $d$-distinct parts

Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s, m s \pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these partitions, by considering $d$-distinct partitions, and by allowing more general $(s, m s \pm r)$-core partitions. As an application of our approach, we obtain the average and maximum number of parts of these core partitions.

Counting partitions of $G_{n,1/2}$ with degree congruence conditions

For $G=G_{n, 1/2}$, the Erdős--Renyi random graph, let $X_n$ be the random variable representing the number of distinct partitions of $V(G)$ into sets $A_1, \ldots, A_q$ so that the degree of each vertex in $G[A_i]$ is divisible by $q$ for all $i\in[q]$. We prove that if $q\geq 3$ is odd then $X_n\xrightarrow{d}{\mathrm{Po}(1/q!)}$, and if $q \geq 4$ is even then $X_n\xrightarrow{d}{\mathrm{Po}(2^q/q!)}$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in $G[A_i]$ to be congruent to $x_i$ modulo $q$ for each $i\in[q]$, where the residue…

Let's Expand Rota's Twelvefold Way For Counting Partitions!

Rota's Twelvefold Way gave formulas for the numbers of partitions which could be formed in twelve scenarios. This proposed AMM article expands Rota's 4 x 3 table. The resulting 6 x 5 table considers a broader collection of splitting-distributing-grouping-arranging scenarios, each of which can be visualized with the distribution of m items into certain kinds of bins. The additional counts or scenarios include: the Bell numbers B(m), the partition numbers p(m), arrangements of m books on b shelves, standings of m teams in a league, arrangements of m books into b scattered stacks, and pairings of…

n色分拆函数与若干计数定理

The $n$-Color Partition Function and Some Counting Theorems

本文定义n色分拆上的计数函数，将其与Möbius函数、Liouville函数等算术函数及除数和关联，并借助Erdös计数方法得到与Andrews和Deutsch结果类似的n色分拆计数定理。

The partial-fractions method for counting solutions to integral linear systems

We present a new tool to compute the number $ϕ_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $ϕ_\A (\b)$ is often described as a \emph{vector partition function}. Our methods use partial fraction expansions of Euler's generating function for $ϕ_\A (\b)$. A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes.

Congruences for Generalized Frobenius Partitions of Nonzero Row Difference

We extend George Andrew's general principle for counting generalized Frobenius partitions to include arrays with nonzero row difference and establish some congruences for these arrays.

Counting Labelled Trees with Given Indegree Sequence

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $λ\vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $λ$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.

Two Questions about the Fractional Counting of Partitions

We recall the notion of fractional enumeration and immediately focus on the fractional counting of integer partitions, where each partition gets credit equal to the reciprocal of the product of its parts. We raise two intriguing questions regarding this count, and for each of these questions, we are pledging a 100 US dollars donation to the OEIS in honor of the first solver. In this revised version we announce that both questions have been answered. The first was first answered by Will Sawin and the second was answered by Christopher Ryba. A donation to the OEIS, of 100 US dollars each, in the…

On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts

Tewodros Amdeberhan and Armin Straub initiated the study of enumerating subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core partitions into odd parts remains elusive. Straub computed the first eleven terms of that sequence, and asked for a "formula," or at least a fast way, to compute many terms. While we are unable to find a "fast" algorithm, we did manage to find a "faster" algorithm, which enabled us to compute 23 terms of…

Notes On U(1) Instanton Counting On $A_{l-1}$ ALE Spaces

In this note, we investigate the detailed relationship between the orbifold partition counting and the (l-quotient, l-core) pair counting. We show that the orbifold partition counting is exactly the same as the (l-quotient, l-core) pair counting.

Statistical model and BPS D4-D2-D0 counting

We construct a statistical model that correctly reproduces the BPS partition function of D4-D2-D0 bound states on the resolved conifold. We prove that the known partition function of the BPS indices is reproduced by the counting "triangular partitions" problem. The wall-crossing phenomena in our model are also studied.

Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting behavior of their zeros as sets and densities.

Counting chains in the noncrossing partition lattice via the W-Laplacian

We give an elementary, case-free, Coxeter-theoretic derivation of the formula $h^nn!/|W|$ for the number of maximal chains in the noncrossing partition lattice $NC(W)$ of a real reflection group $W$. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the $W$-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

Some applications of Wagner's weighted subgraph counting polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have length at most $4$. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

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…

按亏格计数分划 I. 亏格0到2

Counting partitions by genus. I. Genus 0 to 2

该研究将亏格0分划计数的泛函方程推广到高亏格，结合（半）本原分划重构方法，得到亏格1、2下所有分划类型的生成函数显式结果，为普通与自由累积量展开提供二阶插值。

图的连通划分计数

Counting Connected Partitions of Graphs

受Győri-Lovász定理启发，本文研究了图的连通划分计数问题，分别针对边划分和点划分情形，证明了关于可实现连通子图边数或顶点数的无序正整数解个数的多个紧致下界。

Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previous…

The sets of flattened partitions with forbidden patterns

The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well as a pair of two patterns. Several counting sequences, namely Catalan numbers, powers of two, Fibonacci numbers and Motzkin numbers arise. We also consider other combinatorial statistics, namely runs and inversions, and establish some bijections in situations where the statistics coincide.

图的支配划分数计数

Counting the Number of Domatic Partition of a Graph

本文引入支配划分多项式 $DP(G,x)$ 来计数图的支配划分，并基于弱 2-着色数为树设计了二次时间计算算法，同时给出了路径和某些图乘积上的具体结果。

Counting hypergraph matchings up to uniqueness threshold

We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter $λ$, each matching $M$ is assigned a weight $λ^{|M|}$. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity $λ_c= \frac{d^d}{k (d-1)^…

Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chroma…

A new sieve for restricted multiset counting

The Li--Wan sieve is extended to multisets when the underlying set is symmetric. The main ingredient of the proof is the Mobius inversion formula on the poset of partitions of $\{1,2,\dots,k\}$ ordered by refinement. As illustrative applications, we investigate the problems of partitions over finite fields and zero-sum multisets over the additive group $\mathbb{Z}/n\mathbb{Z}$. .

按亏格计数划分：结果汇编

Counting partitions by genus: a compendium of results

本文研究集合划分按长度、部分数、循环类型和亏格的枚举，引入亏格依赖的贝尔数、斯特林数和Faà di Bruno系数，总结已知结果并给出新通用结果与数据，引出新猜想。

通过模分区计算线性扩展和有向 clique 计数

Linear extensions and directed clique counts via modular partitions

本文提出通过模分区结构递归计算偏序集线性扩展的新方法，并探讨其在有向图和神经回路中的应用。

集合划分算法的基准测试

Methods for Counting and Enumerating Set Partitions

本文提出了一种用于集合划分问题的算法基准测试方法，推荐Djokic等人提出的算法用于实际应用。

Counting partitions inside a rectangle

We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout the regime $\ell = Θ(m)$ and $n = Θ(m^2)$. Previously, sharp asymptotics were derived by Takács only in the regime where $|n - \ell m /2| = O(\sqrt{\ell m (\ell + m)})$ using a local central limit theorem. Our approach is to solve a related large deviation problem: we describe the tilted measure that produces configurations whose bounding rectangle has the g…

Some observations and speculations on partitions into $d$-th powers

The aim of this note is to provoke discussion concerning arithmetic properties of function $p_{d}(n)$ counting partitions of an positive integer $n$ into $d$-th powers, where $d\geq 2$. Besides results concerning the asymptotic behavior of $p_{d}(n)$ a little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into $d$-th powers. The second part of the paper has experimental nature and contains questions and conjectures concerning arithmetic behavior of the sequence $(p_{d}(n))_{n\in\N}$. They based on our computations…

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Important illustration to the principle ``partition functions in string theory are $τ$-functions of integrable equations'' is the fact that the (dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve $q$-difference equations of non-autonomous dynamics of the ``cluster-algebraic'' integrable systems. We explain in details the ``solutions'' side of the proposal. In t…

关于包含分拆函数的某一数列的值域与遗漏值

The range and omitted values of a certain sequence involving the partition function

研究分拆函数p(n)与n之差构成数列的值域，给出组合解释，证明其稀疏性（计数函数阶为(log x)^2，自然密度为零），并扩展到Durfee正方形约束的分拆。

Families of Quasimodular Forms and Jacobi Forms: The Crank Statistic for Partitions

Families of quasimodular forms arise naturally in many situations such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this note we give examples of such expansions that arise in the study of partition statistics. The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two variable generating …

d重分拆菱形

d-Fold Partition Diamonds

本文定义$d$重分拆菱形及Schmidt型变体，利用分拆分析求得生成函数并联系Eulerian多项式，从而证明无穷多个Ramanujan型同余式。

Cohen Lenstra Partitions and Mutually Annihilating Matrices over a Finite Field

Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure.