Shane Chern
We study the generating function for overpartitions with bounded differences between largest and smallest parts, which is analogous to a result of Breuer and Kronholm on integer partitions. We also connect this problem with over $q$-binomial coefficients introduced by Dousse and Kim.
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Overpartitions with repeated smallest non-overlined part
本文扩展了 Andrews 和 Bachraoui 关于重复最小部分的划分研究，研究具有最小非重叠部分恰好出现k次且每个重叠部分都比该部分大的划分，证明了这些划分的生成函数可表示为q-Pochhammer 符号的线性组合。
Extending Congruences for the number of smallest parts in overpartitions with smallest part even
本文扩展了超划分中最小部分为偶数的最小部分数量的同余式，证明了新的个体同余式，并给出了无限族拉马努金型同余式。
In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_ω(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of $5$ for $\overline{\textrm{spt}}_ω(n)$. Moreover, they conjectured a family of internal congruences modulo any powers of $5$ and two families of congruences modulo any even powers of $5$. In this paper, we confirm three …
We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over Gaussian polynomials or over $q$-binomial coefficients. We investigate basic properties and applications of over $q$-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan type partition theorem.
We generalize recent results of Breuer and Kronholm, and Chern on partitions and overpartitions with bounded differences between largest and smallest parts. We prove our generalization both analytically and combinatorially.
On an Overpartition Analogue of $SOME(n)$
本文研究了划分类比的过划分函数，推导了生成函数并获得模3、5及2的幂的同余式。
In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…
In 1967, Andrews found a combinatorial generalization of the Göllnitz-Gordon theorem, which can be called the Andrews-Göllnitz-Gordon theorem. In 1980, Bressoud derived a multisum Rogers-Ramanujan-type identity, which can be considered as the generating function counterpart of the Andrews-Göllnitz-Gordon theorem. Lovejoy gave an overpartition analogue of the Andrews-Göllnitz-Gordon theorem for $i=k$. In this paper, we give an overpartition analogue of this theorem in the general case. By using Bailey's lemma and a change of base formula due to Bressoud, Ismail and Stanton, we obtain an overpar…
The Rogers-Ramanujan-Gordon Theorem for Overpartitions
设$B_{k,i}(n)$为$n$在某种差分条件下的划分数，设$A_{k,i}(n)$为$n$在某种同余条件下的划分数。Rogers-Ramanujan-Gordon定理指出，$B_{k,i}(n)=A_{k,i}(n)$。Lovejoy对$i=1$和$i=k$的情形得到了Rogers-Ramanujan-Gordon定理的一个过分割类比。在一般情况下，我们发现了Rogers-Ramanujan-Gordon定理的一个过度分割类似物。设$D_{k,i}(n)$为满足一定差分条件的$n$的超划分数，$C_{k,i}(n)$为满足一定同余条件的$n$的非上划线部分的超划分数.我们表明，$C_{k,i}(n)=D_{k,i}(n)$。利用Andrews引入的一个函数，我们得到了一个递推关系，它意味着$D_{k,i}(n)$的母函数等于$C_{k,i}(n)$的母函数.我们还发现了一个生成函数公式的$D_{k,i}(n)$通过使用戈登标记表示的超划分，它可以被认为是一个单位的安德鲁斯为普通分区的超划分模拟。
A Combinatorial Proof of an Identity of Malik and Sarma on Overpartitions with Repeated Smallest Non-Overlined Part
本文为Malik和Sarma提出的关于具有重复最小非上划线部分的超分拆的四个恒等式中剩余的一个提供了组合证明，解决了该开放问题。
An Overpartition Analogue of Bressoud's Theorem of Rogers-Ramanujan Type
对于$k\geq i\geq 1$，令$B_{k,i}(n)$表示$n$的分区数，使得部分1最多出现$i-1$次，两个连续整数l和$l+1$最多出现$k-1$次，并且如果l和$l+1$恰好出现$k-1$次，则部分l和$l+1$的总和与$i-1$模2全等。令$A_{k,i}(n)$表示具有与$i$、$2k-i$和$2k$模不全等的部分的分区的数目。Bressoud定理指出，$A_{k,i}(n)=B_{k,i}(n)$。Corteel、Lovejoy和Mallet发现了一个类似于Bressoud定理的$i=1$的过分区，也就是说，对于不包含非上划线部分1的分区。在一般情况下，我们得到了Bressoud定理的一个过分割类似物。对于$k\geq i\geq 1$，令$D_{k,i}(n)$表示$n$的上分区数，使得对于任何整数$l$，非上划线部分1最多出现$i-1$次，$l$和非上划线的$l+1$出现的次数最多为$k-1$次数，并且如果部分$l$和非上划线的部分$l+1$恰好出现$k-1$次数，则部分$l$和非上划线的部分$l+1$的总和等于小于$l+1$加上$i-1$模2的上划线部分。令$C_{k,i}(n)$表示具有不全等于$\pm i$和$2k-1$模$2k-1$的非上划线部分的过划分的数目。我们表明，$C_{k,i}(n)=D_{k,i}(n)$。这个关系也可以被认为是Roger…
An overpartition analogue of Bressoud conjecture for even moduli
本文通过引入新的分拆函数 $\overline{B}_0$ 并构造其与 $B_1$ 之间的双射，证明了 $j=0$ 情形下 Bressoud 猜想的 overpartition 类比。
Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part
本文通过组合方法证明了关于带有重复最小非重线部分的划分恒等式。
In 2015, Bringmann, Lovejoy and Mahlburg considered certain kinds overpartitions, which can been seen as the overpartition analogue of Schur's partition. The motivation of their work is that the difference between the generating function of Schur's classical partitions and the generating functions of the partitions in which the smallest part is excluded. The difference between the two generating functions of partitions is a specialization of the "universal" mock theta function g_3 which introduced by Hickerson. To give an analogue of this, by using another universal mock theta function g_2 ins…
Three types of the minimal excludant size of an overpartition
本文研究超划分的最小排除数大小，定义了三种相关类型，作为划分最小排除数的超划分类比。
In this paper, we find an identity which connects the overpartition function and the function of Rogers--Ramanujan--Gordon type overpartitions by considering the weights and gaps. This identity can be seen as an analogue of the weighted identity of Alladi.
We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in te…
We use the $q$-binomial theorem, the $q$-Gauss sum, and the ${}_2ϕ_1 \rightarrow {}_2ϕ_2$ transformation of Jackson to discover and prove many new weighted partition identities. These identities involve unrestricted partitions, overpartitions, and partitions with distinct even parts. Smallest part of the partitions plays an important role in our analysis. This work was motivated in part by the research of Krishna Alladi.
We give an overpartition analogue of Bressoud's combinatorial generalization of the Göllnitz-Gordon theorem for even moduli in general case. Let $\widetilde{O}_{k,i}(n)$ be the number of overpartitions of $n$ whose parts satisfy certain difference condition and $\widetilde{P}_{k,i}(n)$ be the number of overpartitions of $n$ whose non-overlined parts satisfy certain congruence condition. We show that $\widetilde{O}_{k,i}(n)=\widetilde{P}_{k,i}(n)$ for $1\leq i<k$.
Arithmetic properties of $t$-Schur overpartitions
本文延续对Nadji和Ahmia提出的$t$-Schur超分拆（经典Schur分拆的推广）的研究，证明了$t=3、9$及$t$为2或3的幂时的若干算术性质。
Overpartitions and Bressoud's conjecture, II
本文通过推广 Kim 的方法建立 $j=1$ 情形的过划分类比，并利用普通划分与过划分函数之间的联系，最终完整解决了 Bressoud 猜想在 $j=0$ 情形下的证明。
Some Comments on Regular Overpartitions modulo $2^k$
研究\(\overline{R}_{\ell,\mu}(n)\)及单模类似物\(\overline{R}_{\ell}(n)\)，记录根据基础普通分割不同部分大小数量确定这两个函数模\(2\)各幂次的组合机制，还根据完全平方数推导其模\(4\)的简洁特征。
In 1980, Bressoud conjectured a combinatorial identity $A_j=B_j$ for $j=0$ or $1$, where the function $A_j$ counts the number of partitions with certain congruence conditions and the function $B_j$ counts the number of partitions with certain difference conditions. Bressoud's conjecture specializes to a wide variety of well-known theorems in the theory of partitions. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim resolved Bressoud's conjecture for the case $j=1$. In this paper, we introduce a new partition function $\bar{B}_j$ whi…
Elementary proofs of congruences modulo 5 for overpartitions with restricted odd differences
本文针对Hanson和Smith在2023年利用模形式证明的关于重分拆函数$\overline{t}(n)$的两个模5同余式，给出了一个完全初等的证明方法。
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。
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…
Extensions of some results of Jovovic and Dhar
本文将Dhar关于最小部分出现次数和最大最小部分差的整数分拆公式从特定情形推广至一般情形，并探讨了overpartitions和$\ell$-regular partitions的类似结果。
In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The proof is based on what we call Hoffman's limit theorem which we formulate for Hermitian matrices, and also the extension of the concept of Hoffman graph and line graph for the setting of signed graphs.
On Partition Classes Arising from Parity, Differences, and Repeated Smallest Parts
本文研究了与奇偶性、差值和重复最小部分相关的划分类，推导了相关身份并扩展了欧拉划分定理，同时给出了组合和q-级数的证明。
Generalization of the Extended Minimal Excludant of Andrews and Newman
本文推广Andrews与Newman的扩展mex分拆函数，建立其与任意rank、crank界分拆数的联系，给出最小部分函数的有限和表达式及受限分拆恒等式，并引入overpartition的新mex。
Diagrammatics for the smallest quantum coideal and Jones--Wenzl projectors
本文研究量子 $\mathfrak{sl}_2$ 张量幂对最小余理想子代数的限制，以弦图和生成元关系刻画其模范畴，并构造 $B/D$ 型 Jones–Wenzl 投影子及 $\Theta$-网络递归公式。
$k$-Pairing: A Generalization of the Partition Pairing Theorems
本文推广Andrews-Dastidar划分配对理论至$k$-配对，通过两个保权双射给出配对指标-宽度分布与负秩枚举的组合解释，并扩展至Young图元组等价类与平面划分。
Rank, two-color partitions and Mock theta function
该研究建立了正奇秩自然数分拆与特定双色分拆的数目等价关系，由此得到三阶mock theta函数新表示及Spt函数恒等式类似形式，还探讨了对应奇最小部分版本的生成函数关联的mock theta函数。
A note on dilation coefficient, plane-width, and resolution coefficient of graphs
本文研究并比较了与平面图的“紧性”有关的三个图不变量：膨胀系数，定义为最长边与最短边的最小可能商，平面宽度，定义为任意两点之间的最大距离与最短边的最小可能商;和分辨率系数，最长边长度和任意两点之间的最小距离之间的最小可能商。这三个不变量对于完全图是一致的。我们证明了具有大膨胀系数或大平面宽度的图有一个大效价的顶点，但存在具有任意大分辨系数的三次图。令人惊讶的是，这三个不变量的一维类似物允许我们重新审视三个众所周知的图形参数：圆色数，色数和带宽。我们还研究了有界分辨率系数和小闭图类之间的联系。
Let $P$ be a graded poset of rank $r$ and let $\mathbf{c}$ be a $c$-element chain. For an order ideal $I$ of $P \times \mathbf{c}$, its rowmotion $ψ(I)$ is the smallest ideal containing the minimal elements of the complementary filter of $I$. The map $ψ$ defines invertible dynamics on the set of ideals. We say that $P$ has NRP ("not relatively prime") rowmotion if no $ψ$-orbit has cardinality relatively prime to $r+c+1$. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product $P = \mathbf{a} \times \mathbf{b}…
Untrodden pathways in the study of partitions with parts separated by parity
本文研究Andrews引入的奇偶部分分隔分拆函数，证明其生成函数可表示为特定模形式，推导最小部分分拆函数的生成函数形式，给出新组合解释与量子q级数恒等式及渐近项。
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