Jeremy M. Dover
Stephan (Prove or Disprove 100 Conjectures from the OES, arXiv:math/0409509v4 [math.CO])enumerates a number of conjectures regarding integer sequences contained in Sloane's On-line Encyclopedia of Integer Sequences (N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org). In this paper, we prove two of these conjectures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
I conjecture a certain explicit determinant evaluation, whose proof would imply the solution of certain enumeration problem that I have been working on, and that I find interesting. I am pledging \$500 to the OEIS Foundation (in honor of the prover!) for a proof, and \$50 (in honor of the disprover)
Proofs of several OEIS conjectures on determinants and permanents
利用行和列操作、块行列式公式、柯西行列式、Sylvester行列式定理和LU分解，证明了在线整数序列百科中关于特殊矩阵（如Toeplitz矩阵、交叉矩阵、Kronecker幂矩阵及差幂矩阵）的行列式和积和式的几个猜想，并得到了若干相关整数序列的闭式公式。
We verify the conjecture that the sixth binary partition function is equal (aside from the initial zero term) to the partial sums of the Stern-Brocot sequence.
Presented here are over one hundred conjectures ranging from easy to difficult, from many mathematical fields. I also summarize briefly methods and tools that have led to this collection.
Three short proofs of Mathar's 2014 conjecture for OEIS A002627
本文通过三种简短方法证明了OEIS A002627序列的二阶齐次递推关系，展示了通过恒等式消去常数、指数生成函数和二项式求和形式的推导过程。
Hardinian Arrays
本文证明了Kauers和Koutschan关于Hardin序列的递推猜想及OEIS中的旧猜想，并提出了关于序列渐近性的新猜想。
We prove an existing conjecture that the sequence defined recursively by $a_1=1, a_2=2, a_n=4a_{n-1}-2a_{n-2}$ counts the number of length-$n$ permutations avoiding the four generalized permutation patterns 1-32-4, 1-42-3, 2-31-4, and 2-41-3.
The On-Line Encyclopedia Of Integer Sequences , that wonderful resource that most combinatorialists, and many other mathematicians and scientists, use at least once a day, is a treasure trove of mathematical information, and, one of its charms is that it contains many intriguing conjectures. But one should be on one's guard, because some of the conjectures are either already theorems, or can be routinely proved. In this case study we demonstrate, and actually fully implement (in an accompanying Maple package), how to turn many conjectures made in the OEIS by R.J. Mathar, regarding linear recur…
Some D-finite and Some Possibly D-finite Sequences in the OEIS
本文通过自动搜索在 OEIS 中发现若干此前未被识别的 D-finite 或可能 D-finite 序列，并按可证明猜想递推、仅能证明递推存在、来源不明三类加以区分。
We show that sequences A026737 and A111279 in The On-Line Encyclopedia of Integer Sequences are the same by giving a bijection between two classes of Grand Schroder paths.
In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to b…
Sequence A000975 in the Online Encyclopedia of Integer Sequences (OEIS) starts out 1, 2, 5, 10, 21, 42, 85, ... . As of July 1, 2016, the description in the OEIS lists several characterizations of this sequence and numerous examples of instances where this sequence occurs. It also presents a "not yet proved" result, a conjecture, and an unanswered question concerning this sequence. In this paper we show that all of these proposed results are in fact true.
An application of Jacobi's residue formula and proofs of some other conjectures on binomial coefficients
该研究证明了七个涉及二项式系数的OEIS猜想，应用雅可比留数公式，并通过麦克马斯特主定理独立给出了一个新近结果的新证明。
Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS
本文综述了有限偏序集序多面体的 Ehrhart 多项式，探讨了偏序集运算下的组合解释，并利用特定划分的 Ferrers 图证明 OEIS 序列猜想，重新发现 Kreweras 行列式公式。
Real-rooted Eulerian polynomials from permutations, words, and paths
本文研究六个Eulerian型多项式族，证明错排下降多项式实根性等猜想，并加强或解决多个相关实根性与交错性问题。
In this paper, we consider nine OEIS sequences, the analysis of which allows us to find a connection between Motzkin numbers and Fibonacci numbers. In each Motzkin number, we distinguish an even component and an odd component, the difference of these two components is called the shadow of the Motzkin number. Reverse of the Motzkin shadows give us a sequence from the family of Fibonacci numbers.
Proof of Conjecture 19 of Ballantine, Beck, Merca, and Sagan on Elementary Symmetric Partitions
本文证明了Ballantine等人关于整数划分上pre_k映射像与OEIS序列关联的猜想19，无条件证明了(i)(iii)部分，在单射假设下证明(iv)并证明等价性，部分证明(ii)并归约为Hickerson猜想，同时修正了(iii)中的符号错误。
Proofs of some OEIS conjectures on Wythoff sums, Fibonacci and Lucas words
本文用Walnut证明器证明了多个关于Wythoff序列、Fibonacci词和Lucas词的OEIS猜想，包括Kimberling的表示计数猜想与间隙猜想，并提供了完整命令文件。
Riordan Arrays and Shifted Hankel Determinants of OEIS A005773
本文通过边界分解统一两个 Riordan 阵列，得到 Motzkin 端点分布的周期三变换及按高度模 3 的路径公式，并导出第三、第四 Hankel 行列式的闭式解。
The $O(1/n^{85})$ Asymptotic expansion of OEIS sequence A85
本文为致敬 Neil Sloane 和 Don Knuth 的 85 岁生日，将对合数序列 OEIS A85 的渐近公式从 Knuth 的 O(1/n) 精度推进到 O(1/n^{85})。
Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted per…
A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123
本文通过Burnside引理证明了A032123序列的递推猜想，推导出闭式表达式并验证了Mathar的五阶递推关系。
A short proof of Mathar's 2013 recurrence conjecture for the Laguerre sequence~A025166
本文给出 OEIS 序列 A025166 的递推关系的简短证明，通过生成函数和微分方程推导出 Mathar 的猜想。
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…
We study two families of sequences, listed in the On-Line Encyclopedia of Integer Sequences (OEIS), which are associated to invariant theory of Lie algebras. For the first family, we prove combinatorially that the sequences A059710 and A108307 are related by a binomial transform. Based on this, we present two independent proofs of a recurrence equation for A059710, which was conjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs' conjecture by the method of algebraic residues. As a consequence, closed formulae for the generating function of sequence A059710 are obtained in…
On three conjectures of Kimberling concerning the array $\lfloor kφ^n\rfloor$
本文证明了 Kimberling 关于数组 $\lfloor k\varphi^n\rfloor$ 的三个猜想：前两个由 Skolem-Bang 定理推出，第三个通过确定包含奇下标 Lucas 数的行及 Lucas 移位引理证明，并给出计算证据。
Universal Shift Congruences for Weighted Permutation Sequences
本文提出加权置换序列的通用移位同余规则，确定其适用最大模数，证明素数2的特殊性，应用该规则证明Bala猜想及多个OEIS序列的同余与整除性，涉及101个OEIS序列。
A short proof of Mathar's 2020 recurrence conjecture for the generalized-Stirling sequence A001711
本文提供了一个简短的证明，验证了Mathar对广义斯特林序列A001711的递推猜想，通过谐波数闭式表达式简化了递推关系，证明了其成立。
In his July 1974 Scientific American column, Martin Gardner mentioned the Handbook of Integer Sequences, which then contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (the OEIS) contains 140000 sequences. This paper discusses eight of them, suggested by the theme of the Eighth Gathering For Gardner: they are all infinite, and all 'ateful in one way or another. Each one is connected with an unsolved problem. The sequences are related to: hateful numbers, Angelini's 1995 puzzle, the persistence of a number, Alekseyev's 123 sequence, the curling number conjecture, Quet'…
Full Grid Lattice Polygons with Maximal Sum of Squares of Edge-Lengths
本文研究 n×n 格点上不自交 n² 边形的边长平方和最大值，给出 OEIS 序列 A358212 的下界及最优性猜想，并对禁止平角的真正多边形得到类似结果。
A Proof of Bala's Congruence Conjecture for A028342
研究序列A028342的巴拉同余猜想，通过建立乘积同余式\(a(n + k)\equiv a(n)a(k)\pmod{k}\)，计算素数幂的\(a(p^r)\bmod p^r\)，证明了关于\(a(n)\)的一族同余式，包括奇数\(k\)及特定模8同余的\(k\)的情况。
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).
Non-orientable Nurikabe
本文研究了非定向表面上的奈奇拜谜题，提出了两种非定向奈奇拜谜题变体，并在莫比乌斯带、克莱因瓶和大小为1×n的射影平面中研究其组合学。研究结果建立了OEIS序列A101946、A213387、A123203和A001045（雅各布斯序列）之间的新联系。
A short proof of Mathar's 2021 recurrence conjecture for the Lehmer-Comtet diagonal A045406
本文证明了Mathar 2021年提出的关于Lehmer-Comtet三角形A008296列2对角线A045406的二阶P-递推关系，利用Harmonic数闭形式简化了递推式，验证了其正确性。
A Proof of Bala's Congruence Conjectures for A158690
本文证明了Bala关于OEIS序列A158690的两个同余猜想，通过一般定理和Stirling数公式，得到模周期性与移位Gauss同余。
The Hofstadter consecutive-sum sequence omits infinitely many positive integers
研究Hofstadter连续求和序列的渐进行为，证明序列下界与上界，证明序列排除无限多个正整数，解决OEIS A005243猜想。
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…
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。
Counterexamples to two conjectures about matroids
本文通过构造反例，否证了关于拟阵的两个著名猜想：White关于对称交换二项式生成环面理想的猜想，以及Mason关于给定秩的平面计数对数凹性的猜想。
We prove two conjectures on correlation inequalities for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions
By extending Lv-Xin-Zhou's first layer formulas of the $q$-Dyson product, we prove Kadell's conjecture for the Dyson product and show the error of his $q$-analogous conjecture. With the extended formulas we establish a $q$-analog of Kadell's conjecture for the Dyson product.
We prove the following conjecture of Narayana: there are no dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime.
In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with certain permutation polynomials of finite fields. Two natural strengthenings of Conjecture~2, referred to as Conjectures~A and B in the present paper, were also insinuated. In a recent development, Conjecture~2 and hence Conjecture~1 have been confirmed. The present paper gives a proof of Conjecture~A.
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. In this paper, we introduce two stronger versions of Frankl's conjecture and give a partial proof. Three related questions are introduced.
The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other.
Reed Conjecture is open for more than 20 years now. Here we prove that Reed Conjecture is valid for (1) {P4UnionK1, Kite}-free graphs (2) {Chair, Kite}-free graphs (3) {K2UnionK2complement , H}-free graphs and (4) {2K2, M}-free graphs where H and M are graphs on six vertices each. Reed conjecture is still open in general.
In 1974 Chvátal conjectured that no intersecting family $\mathcal{F}$ in a downset can be larger than the largest star. In the same year Kleitman and Magnanti proved the conjecture when $\mathcal{F}$ is contained in the union of two stars, and Sterboul when $\operatorname{rank}(\mathcal{F})\le 3$. We give short self-contained proofs of these two statements.
Regular triangle unions with maximal number of sides
研究n个三角形的并集的最大边数，提出序列A375986并证明其线性上下界，引入三角形转换方法。
Convolutive sequences, I: Through the lens of integer partition functions
本文研究了满足特定卷积性质的序列，通过组合学方法和生成函数操作，探讨了整数划分函数与eta积的关系，并给出了相关序列的实例。
We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its $q$-analogue.
The classical Berge-Fulkerson conjecture states that any bridgeless cubic graph $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to two of the perfect matchings from the list. In this short note, we discuss two statements that are consequences of this conjecture. We show that the first statement is equivalent to Fan-Raspaud conjecture. We also show that the smallest counter-example to the second one is a cyclically $4$-edge-connected cubic graph.
Two new proofs of partial Godbersen's Conjecture
本文为Godbersen猜想的部分结论提供了两个新证明，分别利用Helly定理简洁证明其不等式，以及利用Brunn-Minkowski不等式全新证明质心在原点的凸体包含关系，从而证明该猜想。
Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the free cumulants of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov-Biane, recently proved by Feray.
The trapezoidal Fox conjecture states that the coefficient sequence of the Alexander polynomial of an alternating knot is unimodal. We are motivated by a harder question, the strong Fox conjecture, which asks whether the coefficient sequence of the Alexander polynomial of alternating knots is actually log-concave. Our approach is to introduce a polynomial $Δ(t)$ associated to a Christoffel word and to prove that its coefficient sequence is log-concave. This implies the strong Fox conjecture for two-bridge knots.
A logical implication between two conjectures on matrix permanents
本研究证明了Bapat与Sunder提出的两个半正定矩阵积和式猜想间存在逻辑蕴含，既推导了满足首个猜想的矩阵类也满足第二个，还通过构造第二个猜想的新反例得到首个猜想的新反例，建立了两条件的非平凡关联。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。