BG-ranks and 2-cores

曲柄与秩奇偶函数及相关仿theta函数模$5$和$7$的幂的同余

Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions

本文拓展了秩奇偶函数模5的幂的同余式至模7的幂的情形，推导了相关三阶仿theta函数ω(q)系数的同余式，并证明了曲柄奇偶函数的一族模7的幂的同余式。

Arithmetic Identities and Congruences for Partition Triples with 3-cores

Let ${{B}_{3}}(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for ${{B}_{3}}(n)$. Moreover, let $ω(n)$ denote the number of representations of a nonnegative integer $n$ in the form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\in \mathbb{Z}.$ We find three arithmetic relations between ${{B}_{3}}(n)$ and $ω(n)$, such as $ω(6n+5)=4{{B}_{3}}(6n+4).$

Subsemigroup, ideal and congruence growth of free semigroups

Using Rees index, the subsemigroup growth of free semigroups is investigated. Lower and upper bounds for the sequence are given and it is shown to have superexponential growth of strict type $n^n$ for finite free rank greater than 1. It is also shown that free semigroups have the fastest subsemigroup growth of all finitely generated semigroups. Ideal growth is shown to be exponential with strict type $2^n$ and congruence growth is shown to be at least exponential. In addition we consider the case when the index is fixed and rank increasing, proving that for subsemigroups and ideals this sequen…

Congruences modulo $4$ for Rogers--Ramanujan--Gordon type overpartitions

In a recent work, Andrews defined the singular overpartitions with the goal of presenting an overpartition analogue to the theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo $3$ for the number of singular overpartitions prescribed by parameters $k=3$ and $i=1$. It should be noticed that this number equals the number of the Rogers--Ramanujan--Gordon type overpartitions with $k=i=3$ which come from the overpartition analogue of Gordon's Rogers--Ramanujan partition theorem introduced by Che…

Andrews-Beck Type Congruences Related to the Crank of a Partition

In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a suitable generating function, while the conjectures related to cranks were studied by Shane Chern using weighted partition moments. We revisit the conjectures on the crank of a partition by decomposing the relevant generating function and further explore connections with Apple-Lerch series and tenth order mock theta functions.

对$t$-核心分区的 crank 生成函数的探索

An Exploration of Crank Generating functions for $t$-core partitions

本文提出一族生成函数，解释$t$-核心分区的模数关系，拓展了crank生成函数的应用范围。

Arithmetic Properties of Overpartition Pairs

Bringmann and Lovejoy introduced a rank for overpartition pairs and investigated its role in congruence properties of $\bar{pp}(n)$, the number of overpartition pairs of n. In particular, they applied the theory of Klein forms to show that there exist many Ramanujan-type congruences for the number $\bar{pp}(n)$. In this paper, we shall derive two Ramanujan-type identities and some explicit congruences for $\bar{pp}(n)$. Moreover, we find three ranks as combinatorial interpretations of the fact that $\bar{pp}(n)$ is divisible by three for any n. We also construct infinite families of congruence…

关于恰好两种部分大小的分拆同余式的组合证明

Combinatorial proof of a congruence for partitions into two sizes of part

本文为 $ν_2(16n+14)\equiv 0\pmod{4}$ 这一分拆同余式提供了组合证明，并将整除性细化到与除数函数相关的子类，同时提出关于秩统计量的猜想。

Rank and crank moments for overpartitions

We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n).

Relations between the ranks and cranks of partitions

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

Moments of ranks and cranks, and Quotients of Eisenstein Series and the Dedekind Eta Function

Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions. Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(τ)/η(τ)$, where $E_{2r}(τ)$ is the classical Eisenstein series of weight $2r$ and $η(τ)$ is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for $k \leq 5$, $N_k(n)$ and $M_k(n)$ can be expressed using $e_{2r}(n)$ ($2\leq r \leq k$), $p(n)$ and $N_2(n)$. For $k>5$, additional functions are required for such expressions. For $r\in \{2, 3, 4, 5, 7\}$, by studying…

Zero-sum problems with congruence conditions

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum subsequence $T$ of length $|T| \equiv 0 \mod d$. We determine $\mathsf s_{d \mathbb N} (G)$ for all $d\geq 1$ when $G$ has rank at most two and, under mild conditions on $d$, also obtain precise values in the case of $p$-groups. In the same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv constant provided that, for the $p$-subgroups $G_p$ of $G$,…

On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations

In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0; 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enume…

On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5

In a recent study of sign-balanced, labelled posets Stanley [13], introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0, 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partiti…

On $t$-core towers and $t$-defects of partitions

We study generating functions which count the sizes of $t$-cores of partitions, and, more generally, the sizes of higher rows in $t$-core towers. We then use these results to derive an asymptotic for the average size of the $t$-defect of partitions, as well as some curious congruences.

Congruences for the Coefficients of the Powers of the Euler Product

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2α} n +\frac{k(2^{2α}-1)}{3})$ $(1\leq k\leq 3)$ and $p_{3k} (3^{2β}n+\frac{k(3^{2β}-1)}{8})$ $(1\leq k\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite families of congruences…

Partition statistics and quasiweak Maass forms

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between. In this paper we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of weak Maass forms and their derivatives. In particular, these generating functions may be view…

Rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions

Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal; $(m,n)$-cores are partitions with no hook length equal to $m$ or $n$.Anderson established a bijection between $(m,n)$-Dyck paths and $(m,n)$-cores. We define a new transformation, called rank complement, on rational Dyck paths. We show that rank complement corresponds to conjugation of $(m,n)$-cores under Anderson's bijection. This leads to: i) a …

对称群和交错群的双重覆盖的旋特征的同余

Congruences for spin characters of the double covers of the symmetric and alternating groups

本文从算术角度研究对称群和交错群双重覆盖的旋特征生成函数，利用经典解析工具和初等生成函数操作，推导出许多 Ramanujan-like 同余。

M_2-rank differences for overpartitions

This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their proof of Dyson's conjectures concerning Ramanujan's congruences for the partition function. Since then, other types of rank differences for statistics associated to partitions have been investigated. In this paper, we prove explicit formulas for M_2-rank differences for overpartitions. Additionally, we express a third order mock theta function in terms of rank d…

5核分区对的线性恒等式

Linear identities for partition pairs with $5$-cores

本文通过theta函数恒等式推导出5核分区对数A₅(n)的线性恒等式及同余式。

k-Marked Dyson Symbols and Congruences for Moments of Cranks

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $η_{2k}(n)$ of ranks of partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment $μ_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation of $μ_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that $μ_{2k}(n)$ equals the number of $(k+1)$-marked Dyson symbols of $n$. We then introduce th…

Overpartitions and class numbers of binary quadratic forms

We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as $q$-series identities of the mock theta type.

Multi-cores, posets, and lattice paths

Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer $n$ has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number $t$ is absent from the diagram then the partition is called a $t$-core. A partition is an $(s,t)$-core if it is both an $s$- and a $t$-core. Since the work of Anderson on $(s,t)$-cores, the topic has received a growing attention. This paper expands the discussion to multi…

Monotonicity properties for ranks of overpartitions

The rank of partitions play an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the $D$-rank and $M_2$-rank of an overpartition were introduced by Lovejoy, respectively. Let $\overline{N}(m,n)$ and $\overline{N2}(m,n)$ denote the number of overpartitions of $n$ with $D$-rank $m$ and $M_2$-rank $m$, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of $\overline{N}(m,n)$ and $\overline{N2}(m,n)$. In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for…

Asymptotics, Turán inequalities, and the distribution of the BG-rank and 2-quotient rank of partitions

Let $j,n$ be even positive integers, and let $\overline{p}_j(n)$ denote the number of partitions with BG-rank $j$, and $\overline{p}_j(a,b;n)$ to be the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. We give asymptotics for both statistics, and show that $\overline{p}_j(a,b;n)$ is asymptotically equidistributed over the congruence classes modulo $b$. We also show that each of $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy all higher-order Turán inequalities.

按奇偶性分隔部件的分拆：共轭、同余式与 mock theta 函数

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

本文利用共轭映射的组合方法研究奇偶部件分隔的整数分拆，证明推广 Andrews 二元生成函数的参数化恒等式与两族 Andrews–Beck 型同余式，并揭示共轭稳定分拆子集同三个三阶 mock theta 函数的联系。

非Rascoe分拆与一个与Rogers-Ramanujan分拆相关的秩奇偶函数

Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions

研究Rogers-Ramanujan分拆中奇数秩与偶数秩超出数的生成函数，证明其与非Rascoe分拆（不同部分且部分数非部分）紧密相关，并推导算术性质及模4同余猜想，推广至参数ℓ。

A rank of partitions with overline designated summands

Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the number of partitions of $n$ with designated summands. In a recent work, Lin studied a partition function $PD_{t}(n)$ which counts the number of tagged parts over all the partitions of $n$ with designated summands. He proved that $PD_{t}(3n+2)$ is divisible by $3$. In this paper, we first introduce a structure named partitions with overline designated summands, which is counted by $PD_t(n)$. We then define a generalized rank of partitions with overline designated summands and give a combinatorial interpretation of the c…

Biranks for Partitions into 2 Colors

In 2003, Hammond and Lewis defined a statistic on partitions into 2 colors which combinatorially explains certain well known partition congruences mod 5. We give two analogs of Hammond and Lewis's birank statistic. One analog is in terms of Dyson's rank and the second uses the 5-core crank due to Garvan, Kim and Stanton. We discuss Andrews's bicrank statistic and how it may be extended. We also generalize the Hammond-Lewis birank to a multirank for multipartitions and the Andrews bicrank to a multicrank for extended multipartitions. These both give combinatorial interpretations for multipartit…

具有4-核的分拆对的线性恒等式

Linear identities for partition pairs with 4-cores

利用基本q-级数技巧和3-分拆公式，确定了具有4-核的n的分拆对个数A_4(n)的无穷线性恒等式族，并推导出相应的同余式。

The GBG-Rank and t-Cores I. Counting and 4-Cores

Let r_j(π,s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of πmod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*Π*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all …

New Identities for 7-cores with prescribed BG-rank

A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coe…

关于Lin-Peng-Toh的k-着色分拆统计量的一些恒等式

Some identities on Lin-Peng-Toh's partition statistic of $k$-colored partitions

本文针对Lin-Peng-Toh引入的k-着色分拆统计量$NB_k(r,m,n)$，证明了若干类似于Ramanujan“最美恒等式”的恒等式，并由此重新推出Lin、Peng和Toh先前证明的若干同余式。

Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge- transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively sm…