Benjamin Nasmith
Three-graded root systems can be arranged into nested sequences. One exceptional sequence provides a natural means to recover some structures and symmetries familiar in the context of particle physics.
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Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya's work) to classify exceptional sequences of representations of Q, the linearly-ordered quiver with n vertices. We also show how to use variations of this model to classify c-matrices of Q, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of c-matrices, we also give an interpretation …
Combinatorial Approaches to Exceptional Sequences for Weighted Projective Lines of Type $(p,q)$
该研究通过带标记环面对(p,q)型加权射影线凝聚层范畴中的态射给出组合描述，实现例外序列的几何表示，进而分类完全例外序列、提出扩充方法、证明辫群作用传递性，并描述倾斜丛及计算倾斜层数量。
The notion of a $τ$-exceptional sequence was introduced by Buan and Marsh in 2018 as a generalisation of an exceptional sequence for finite dimensional algebras. We calculate the number of complete $τ$-exceptional sequences over certain classes of Nakayama algebras. In some cases, we obtain closed formulas which also count other well known combinatorial objects and exceptional sequences of path algebras of Dynkin quivers.
Exceptional sequences are important sequences of quiver representations in the study of representation theory of algebras. They are also closely related to the theory of cluster algebras and the combinatorics of Coxeter groups. We combinatorially classify exceptional sequences of a family of type D Dynkin quivers, and we show how our model for exceptional sequences connects to the combinatorics of type D noncrossing partitions.
Exceptional sequences are certain ordered sequences of quiver representations. We introduce a class of objects called strand diagrams and use this model to classify exceptional sequences of representations of a quiver whose underlying graph is a type $A_n$ Dynkin diagram. We also use variations of this model to classify c-matrices of such quivers, to interpret exceptional sequences as linear extensions of posets, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This work extends a classification of exceptional sequences f…
Maximal green sequences are particular sequences of mutations of quivers which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Córdova-Vafa in the context of supersymmetric gauge theory. The existence of maximal green sequences for exceptional finite mutation type quivers has been shown by Alim-Cecotti-Córdova-Espahbodi-Rastogi-Vafa except for the quiver $X_7$. In this paper we show that the quiver $X_7$ does not have any maximal green sequences. We also generalize the idea of the proof to give sufficient conditions for the non-existence …
We prove that there is a one to one correspondence between the following three sets: idempotent functions on a set of size $n$, complete exceptional sequences of linear radical square zero Nakayama algebras of rank $n$ and rooted labeled forests with $n$ nodes and height of at most one. Therefore, the number of exceptional sequences is given by the sum $\sum\limits^n_{j=1}\binom{n}{j}j^{n-j}$.
Probability distribution for exceptional sequences of type $A_n$
本文确定了类型 $A_n$ 例外序列中相对投射对象的概率分布，证明相关事件彼此独立且与序列长度无关，从而为例外序列及簇的计数乘积公式提供了概率解释。
Short history of signed exceptional sequences
本文回顾了有向例外序列的概念起源及其在代数K理论、链接不变量和簇组合学中的发展，探讨其在有限维代数中的推广。
The structure of exceptional sequences on toric varieties of Picard rank two
本文分类了皮卡秩为二的光滑射影环面簇上极大长度的可逆层例外序列，并证明其在字典序重排下保持例外性、满足强高度约束且生成导出范畴。
In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category $\mathsf{K^b(A)}$ of a gentle algebra $\mathsf{A}$. We show that full exceptional sequences in $\mathsf{K^b(A)}$ exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterise when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we sh…
Bijection between positive clusters and projectively signed exceptional sequences
本文证明例外序列中每一项要么相对投射、要么相对内射或两者兼有，并将Igusa–Todorov双射细化为投影符号化例外序列与有序部分正团簇之间的双射，同时用支撑给出相对投射/内射对象的刻画。
Para-exceptional sequences for tame hereditary algebras and McCammond-Sulway lattices
本文利用仿型遗传代数表示论中的准例外序列与准例外子范畴，构造并证明了与 McCammond–Sulway 格同构的组合 Garside 结构。
We investigate maximal exceptional sequences of line bundles on (P^1)^3, i.e. those consisting of 2^r elements. For r=3 we show that they are always full, meaning that they generate the derived category. Everything is done in the discrete setup: Exceptional sequences of line bundles appear as special finite subsets s of the Picard group Z^r of (P^1)^r, and the question of generation is understood like a process of contamination of the whole Z^r out of an infectious seed s.
The concept of scattered polynomials is generalized to those of exceptional scattered sequences which are shown to be the natural algebraic counterpart of $\mathbb{F}_{q^n}$-linear MRD codes. The first infinite family in the first nontrivial case is also provided and equivalence issues are considered. As a byproduct, a new infinite family of MRD codes is obtained.
An upper bound on the exceptional characteristics for Lusztig's character formula
我们发展和研究了与根系相关的组合范畴中的Lefschetz理论，并导出了约化代数群的单有理特征标的Lusztig公式的例外特征的上界.我们的界与考克斯特数相比是巨大的。然而，它是由一个明确的公式给出的。
The purpose of this paper is to show that some combinatorial sequences, such as second-order Eulerian numbers and Eulerian numbers of type $B$, can be generated by context-free grammars.
Enumerating m-clusters using exceptional sequences
本文引入了m-例外序列的概念，证明了所有遗传Artin代数中有序m-簇与完全m-例外序列之间存在双射，推广了m=1情形的已有结果。
In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonngeative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve bounds on Perron values of rooted trees in combinatorial settings, in order to find characteristic sets of trees. The other is to generate log-concave and log-convex sequences through the monotone sequences.
$τ$-exceptional sequences and the shard intersection order in type A
本文通过Bancroft组合模型将碎片交序的向下覆盖关联到唯一碎片，证明其与Jasso的$\ au$-倾斜约化一致，从而为$\ au$-例外序列提供弧图组合模型，并给出碎片交序在A型中EL-可壳的新表示论证明。
In 1962 O. A. Gross proved that the last digits of the Fubini numbers (or surjective numbers) have a simple periodicity property. We extend this result to a wider class of combinatorial numbers coming from restricted set partitions.
We consider Dynkin algebras, these are the hereditary artin algebras of finite representation type. The indecomposable modules for a Dynkin algebra correspond bijectively to the positive roots of a Dynkin diagram. Given a Dynkin algebra with n simple modules, a complete exceptional sequence is a sequence M_1,..., M_n of indecomposable modules such that Hom(M_i,M_j) = 0 = Ext(M_i,M_j) for i > j. The aim of this paper is to determine the number of complete exceptional sequences for any Dynkin algebra. There are direct connections between the representation theory of a Dynkin algebra A and the la…
Here we prove some conjectures on the monotony of combinatorial sequences from the recent preprint of Zhi--Wei Sun.
Logarithmic behavior of some combinatorial sequences
给出了建立递归定义的实数序列的对数性态的两种一般方法。一种是交错方法，另一种是基于微积分的方法。这两种方法都被用来证明一些组合相关序列的对数行为，如Motzkin和Schröder数，一些经典正交多项式的值序列，以及许多其他的。微积分方法也扩展到两个（或更多）索引序列。
Multiple analogues of certain families of combinatorial numbers are recently constructed by the author in terms of well poised Macdonald functions, and some of their fundamental properties are developed. In this paper, we present combinatorial formulas for the well poised Macdonald functions, the multiple binomial coefficients, the multiple bracket function, and the multiple Catalan and Lah numbers.
On the Number of Exceptional Pairs Over a Class of Nakayama Algebras
本文利用组合与同调方法，为若干Nakayama代数及根平方零代数的Auslander代数推导出例外对数量的显式公式，并揭示其与OEIS整数序列的联系。
We give a combinatorial interpretation in terms of bicolored ordered trees for the sequence (a_n)_{n>=1}=(1, 1, 1, 2, 3, 6, 10, 20, 36, 73,... ), A345973 in OEIS, whose generating function satisfies the defining identity Sum_{n>=1}a_n x^n = x + x^2/Product_{n>=1}(1 - a_n x^n).
Exceptional sequences and rooted labeled forests
本文通过表示论建立了有根标记森林与直线定向 $A_n$ 型箭图完全例外序列之间的双射，并研究了扩展辫群及 Garside 元素在这些对象上的自然作用及其与丛理论的联系。
Log-balanced combinatorial sequences
我们考虑对数凸序列，满足一个额外的约束施加在他们的增长率。我们称这样的序列为对数平衡的。证明了所有这类序列都满足一对双重不等式。对于序列满足两项（或多项）线性递归的情形，给出了对数平衡性的充分条件。它表明，许多组合有趣的序列属于这一类，因此，上述双重不等式是有效的。
Exceptional sequences of type $B_n/C_n$ and those in the abelian tube
本文通过阿贝尔管中的例外序列研究簇管中的簇，建立了类型 $B_n/C_n$ 模范畴与秩为 $n+1$ 的阿贝尔管之间带符号例外序列的双射，并借此重新解释了 Buan-Marsh-Vatne 公式。
Let $Λ$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $τ$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated $Λ$-modules. When $Λ$ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with…
We give a combinatorial interpretation of a classical meta-Fibonacci sequence defined by G(n) = n - G(G(n-1)) with the initial condition G(1) = 1, which appears in Hofstadter's 'Godel, Escher, Bach: An Eternal Golden Braid'. The interpretation is in terms of an infinite labelled tree. We then show a few corollaries about the behaviour of the sequence G(n) directly from the interpretation.
Convergent sequences of combinatorial submodular setfunctions
本文通过构造多类拟阵和次模集合函数并证明其序列收敛，论证次模函数收敛概念可合理纳入图的极限理论。
We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.
In this short note we present a set of interesting and useful properties of a one-parameter family of sequences including factorial and subfactorial, and their relations to the Gamma function and the incomplete Gamma function.
A Combinatorial Proof of Cayley's Formula via Degree Sequences
通过度数序列提供卡伊利公式的组合证明，强调标号树的结构特性，并探讨其与其他计数问题的关联。
This article is a sequel to a recent article by Eric Rowland and Reem Yassawi, presenting yet another approach to the fast determination of congruence properties of `famous' combinatorial sequences. The present approach can be taught to a computer, and our beloved servant, Shalosh B. Ekhad, was able to generate many new theorems, for famous sequences, of course, but also for many obscure ones!
On combinatorial compexity of convex sequences
得到了由实数的严格凸序列所产生的丢番图方程的解的个数的一个好的上界。
Combinatorics of Exceptional Sequences of Type $\tilde{\mathbb{A}}_n$
本文通过链图、弦图与环面弧图等组合工具，将类型 $\tilde{\mathbb{A}}_n$ 的无穷多例外序列分类为有限多个参数化族，并给出其与小型弧图等价类的双射及导出范畴代数解释。
On the log-convexity of combinatorial sequences
本文主要研究组合序列的对数凸性。证明了对数凸性在分量和、二项卷积以及由二项式系数矩阵和两类Stirling数矩阵给出的线性变换下保持不变。我们开发的技术来处理满足三项递归的序列的对数凸性。我们还引入了$q$ -对数凸性的概念，并建立了与保持对数凸性的线性变换的联系。作为应用，我们证明了许多著名的组合数列和多项式的对数凸性和$q$ -对数凸性。
We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.
A Combinatorial Interpretation for Certain Relatives of the Conolly Sequence
对于任意整数s >= 0，我们得到了由递归（s参数化）h_s（n）= h_s（n-s- h_s（n - 1））+ h_s（n - 2 -s-h_s（n - 3）），n > s + 3生成的序列族的组合解释，初始条件h_s（1）= h_s（2）=... = h_s（s+2）= 1和h_s（s+3）= 2。我们展示了这些序列如何计算某个无限树结构的叶子数。利用这种解释，我们证明了h_s序列是“慢增长的”，即h_s序列是单调非减的，连续项增加0或1，因此每个序列命中每一个正整数。此外，对于固定的s，序列h_s（n）命中每个正整数两次，除了2的幂，所有这些都命中s+2次。我们的组合解释提供了一个简单的方法来推导这些序列的普通生成函数。
In this paper we introduce {\em weak ascent sequences}, a class of number sequences that properly contains ascent sequences. We show how these sequences uniquely encode each of the following objects: permutations avoiding a particular length-4 bivincular pattern; upper-triangular binary matrices that satisfy a column-adjacency rule; factorial posets that are weakly (3+1)-free. We also show how weak ascent sequences are related to a class of pattern avoiding inversion sequences that has been a topic of recent research by Auli and Elizalde. Finally, we consider the problem of enumerating these n…
We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of integer sequences related to the discussed problem. The obtained sequences are new and they are not described in the On-Line Encyclopedia of Integer Sequences (OEIS). We show a relationship between some particular values of the parameters and the Fibonacci sequence.
Coherent States from Combinatorial Sequences
我们构造相干态使用序列的组合数，如各种二项式和三项数，贝尔和卡塔兰数。我们表明，这些国家满足一个自然的方式统一的决议的条件。在每种情况下，正权重函数作为相关Stieltjes或Hausdorff矩问题的解给出，其中矩是组合数。
Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by C{n,k}={n}!/({k}!{n-k}!) where {n}!={1}{2}...{n}. It is easy to see that C{n,k} is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k by n-k rectangle. When s=t=1 we obtain combinatorial interpretations of the fibonomial coeffic…
In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.
A combinatorial interpretation of the eigensequence for composition
在与自身的卷积下左移的monic序列是具有130+组合解释的Catalan数。在这里，我们建立了一个组合解释的一元序列，左移下组成：它计数排列，包含3241模式的一部分，只有35241模式。我们给两个递归，第一个允许相对快速的计算，第二个类似于一个加泰罗尼亚数字。其中4倍4！= 96个包含4个字母的类似限制模式（例如4 \underline{2} 31：431模式仅作为4231的一部分出现），产生四个不同的计数序列：64给出加泰罗尼亚数，16给出贝尔数，12给出OEIS中的序列A051295，4给出具有显式公式的新序列。
The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix $B \in \mathbb{F}^{n \times n}$ is defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_j \in \{\tt{A}, \tt{S}, \tt{N}\}$ according to whether all, some but not all, or none of the principal minors of order $j$ of $B$ are nonzero. Building upon the second author's recent classification of the epr-sequences of symmetric matrices over the field $\mathbb{F}=\mathbb{F}_2$, we initiate a study of the case $\mathbb{F}=\mathbb{F}_3$. Moreover, epr-sequences over finite fields are shown to have connections to Ram…
Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which is an implementation of a decision procedure for proving various properties of automatic sequences. In this paper we explore some results (old and new) that can be proved using this method.
A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio log-convexity of ratio sequences. As applications, we prove that the ratio sequences of numbers, including the derangement numbers, the Motzkin numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio log-convex, respectively. Finally, we not only prove that the sequence of derangement numbers is asymptotically infinitely log-monotonic, but …
Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such a sequence by $a_k$, we obtain a general formula that determines the congruence class, modulo $p$, of the indefinite sum $\sum_{k=0}^{rp -1} a_k$, for {\it any} prime $p$, and any positive integer $r$, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our c…
Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power series in such a triple are closely linked by identities coming from the theory of symmetric functions. We extend the work of Z-H. Sun, who developed similar ideas, and Macdonald, revealing more of the structure of these triples. De Moivre polynomials play a key role in this study.
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus of the present article is the analysis of combinatorial and arithmetical properties of them. The results include several combinatorial identities and recurrences as well as some properties of their $p$-adic valuations.
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial objects, e.g., undirected graphs, permutations, matchings etc. In the second step, we use algorithms for lifted first-order model counting to generate integer sequences that count the objects encoded by the first-order logic formulas generated in the first step. For instance, if the first-order sentence defines permutations then the generated integer sequence is…
A q-Analog of Dual Sequences with Applications
本文给出了q-对偶序列或系数为q-对偶序列的多项式的组合恒等式。进一步推导了q-二项式系数（高斯系数）、q-Stirling数和q-Bernoulli数与多项式的组合恒等式.
We present combinatorial and analytical results concerning a Sheffer sequence with an exponential generating function of the form $G(s,z)=e^{czs+αz^{2}+βz^{4}}$, where $α, β, c \in \mathbb{R}$ with $β<0$ and $c\neq 0$. We demonstrate that the zeros of all polynomials in such a Sheffer sequence are either real, or purely imaginary. Additionally, using the properties of Riordan matrices we show that our Sheffer sequence satisfies a three-term recurrence relation of order 4, and we also exhibit a connection between the coefficients of these Sheffer polynomials and the number of nodes with a a giv…
We confirm Sun's conjecture that $(\root{n+1}\of{F_{n+1}}/\root{n}\of{F_n})_{n\ge 4}$ is strictly decreasing to the limit 1, where $(F_n)_{n\ge0}$ is the Fibonacci sequence. We also prove that the sequence $(\root{n+1}\of{D_{n+1}}/\root{n}\of{D_n})_{n\ge3}$ is strictly decreasing with limit $1$, where $D_n$ is the $n$-th derangement number. For $m$-th order harmonic numbers $H_n^{(m)}=\sum_{k=1}^n 1/k^m\ (n=1,2,3,\ldots)$, we show that $(\root{n+1}\of{H^{(m)}_{n+1}}/\root{n}\of{H^{(m)}_n})_{n\ge3}$ is strictly increasing.
Asymptotic probability of irreducibles II: sequence
本文研究了大组合对象渐近概率的结构，揭示了不可约性概率的渐近展开系数与相关组合类计数序列的关系，并应用于多种组合结构。
In this article, we present a short, non-exhaustive study of an important and well-known property of combinatorial sequences - unimodality. We shall have a look at a sample of classical results on unimodality and related properties, and then proceed to understand the unimodality of the Gaussian polynomial in more detail. We will look at an outline of O'Hara's proof of the unimodality of the Gaussian polynomial. In order to grasp the challenge of the problem of obtaining an injective proof of the unimodality of the Gaussian polynomial (which is still an open question), we make several attempts …
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