Aqsa Bashir, Alfred Geroldinger, Qinghai Zhong
We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull monoids over finite abelian groups.
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A note on $m$-factorizations of complete multigraphs arising from designs
给出了完全多重图$λK_v$的简单不可分解的$m$ -因子分解的新的无限族.大多数构造来自于有限几何。
In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized dihedral groups, and we use them to construct uniform $2$-factorizations that solve infinitely many open cases of the Hamilton-Waterloo problem, thus filling up large parts of the gaps in th…
We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{μ\leλ} g^{(k)}_μ$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by $\widetilde{g}^{(k)}_λ$. As an application of this, we also give a $k$-rectangle factorization formula $\widetilde{g}^{(k)}_{R_t\cupλ}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_λ$ where $R_t=(t^{k+1-t})$, analogous to that of $k$-Schur functions $s^{(k)}_{R_t\cupλ}=s^{(k)}_{R_t}s^{(k)}_λ$.
Factorization of alternating sums of Virasoro characters
G. Andrews证明了如果$n$是素数，则乘积$(q,q)_\infty/(q^n,q^n)_\infty=\sum_k a_kq^k$的系数$a_k$和$a_{k+n}$具有相同的符号，参见[A1]。我们把这个结果推广到几个方向。我们的结果是基于这样的观察，即许多产品可以写为交替和Virasoro模块的字符。
Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities
本文研究了由求和-乘积恒等式产生的两个划分类中t-hooks数量的不等式，通过生成函数和渐进行式证明了t=1,2时的不等式。
Ordered Factorizations with $k$ Factors
我们给出了一个概述的顺序$k$ -整数的因子分解$f_k(n,l)$，其中每个因素是大于或等于$l$的数量的组合性质。我们表明，对于大量的$k$的因素，值的累积和$F_k(x,l)=\sum\nolimits_{n\leq x} f_k(n,l)$是一个多项式在$\lfloor \log_l x \rfloor$，并给出明确的表达式，这个多项式的次数和系数。从除数问题的已知结果导出对于大于或等于2的固定数目$k$的因子的有序因式分解的数目的平均阶。
In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the negative by giving counterexamples. W…
Paley-like quasi-random graphs arising from polynomials
本文提出了一种基于多项式构造的伪随机图家族，该家族既非Cayley图也非Cayley求和图，为研究有限域上多项式相关的图提供了统一视角，并给出了clique和独立集数的下界。
Factorization of Basic Hypergeometric Series
本文研究基本超几何级数的因式分解问题，重点分析一般 $_2ψ_2$ 级数，并建立与根系上基本超几何级数理论的联系，同时给出若干经典求和与变换公式的新证明。
In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form \[ \sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \] where $f_1, f_2$ are functions of $n$. Such sums appear in the analyses of the Mabinogion urn and the Ehrenfest urn in probability. Using hypergeometric functions, we are able to simplify these sums, and in some cases express them in terms of the harmonic numbers.
Algebraic Birkhoff Factorization and the Euler-Maclaurin Formula on Cones
我们给格锥空间配上一个余积，使它成为一个连通余阶化余代数。格锥上的指数和与指数积分可以看作是这个空间上的线性映射，其值在亚纯芽空间中，且线性极点为零。我们调查的细分属性-让人想起的包含-排除原则的基数有限集-这样的线性映射，并建立一个兼容性这些属性的卷积商的余代数。在所考虑的线性映射上实现代数Birkhoff因子分解过程，我们将指数和因子分解为两个映射的卷积商，因子分解中的每个映射都满足细分性质。因此，代数Birkhoff分解专门针对格锥上的Euler-Maclaurin公式，并通过投影映射提供了一个简单的插值因子公式。
We present a constructive proof of Jacobi's identity for the sum of two squares. We present a combinatorial proof of the Jacobi Triple Product and combine with a proof of Hirschhorn to define an algorithm. The input is a factorization $n=dN$ with $d \equiv1\mod 4$ plus two bits of data, and whose output is either another factorization $n=d'N'$ and $d' \equiv3\mod 4$ with two more bits of data, or a pair of integers whose squares sum to $n$. We phrase this algorithm in terms of integer partitions and matchings on an infinite graph.
Near-central Permutation Factorization and Strahov's Generalized Murnaghan-Nakayama Rule
$(p,q,n)$-偶极问题是一个映射枚举问题，出现在微扰杨-米尔斯理论中，其中参数$p$和$q$，在每个顶点，指定的边缘分离的两个区别的边缘的数量。组合，这是值得注意的是一个置换因式分解问题，不在于中心的$\mathbb{C}[\mathfrak{S}_n]$，使问题无法通过字符理论的方法，往往采用研究这类问题。本文在$q=n-1$的所有可定向曲面上给出了该问题的一个解，这是一个组合上有意义的特例：它是一个\emph{near-central}问题.我们给出了一个编码的$(p,n-1,n)$-偶极问题作为产品的标准基元素的中心化$Z_1(n)$的群代数$\mathbb{C}[\mathfrak{S}_n]$关于子群$\mathfrak{S}_{n-1}$。在$(p,n-1,n)$-偶极问题的解中出现的广义特征是Gel 'fand对$(\mathfrak{S}_n\times \mathfrak{S}_{n-1}, \mathrm{diag}(\mathfrak{S}_{n-1}))$的带状球函数，并被明确地计算。这个解决方案是用来证明，对于一个给定的表面，$(p,n-1,n)$-偶极子和$(n+1-p,n-1,n)$-偶极子的数量是相等的，事实上，我们没有组合的解释。这些技术也给出了一个解决方案，近中央模拟的问题，分解成两个因素的指定周期类型的完整的周期。
Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the Minkowski factorization of polytopes. Using our construction we show that for any given polytopal fan there is a polytope factorization basis, i.e. a finite set of polytopes with respect to which any polytope whose normal fan is refined by the original fan can be uniquely written as a signed Minkowski sum. We explicitly study the factorization of polymatroids a…
Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
利用高斯随机矩阵积分的图解方法，我们找到了多项式李群积分在曲面上的无限和表示。所涉及的映射满足一个特定的条件：它们有一些标记的顶点，并且没有避开这些顶点的闭合行走。我们还制定我们的结果的排列，到达新的类型的因式分解问题。
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