Ralf Stephan
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.
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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.
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.
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。
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…
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.
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。
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。
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…
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。
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。
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。
The sum-product conjecture is false for real numbers
通过构造任意大的实数子集A（其元素是代数整数），证明和积猜想对于实数是错误的，并推广到多个和与积的情形。
A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.
Rudolph conjectures that for permutations $p$ and $q$ of the same length, $A_n(p) \le A_n(q)$ for all $n$ if and only if the spine structure of $T(p)$ is less than or equal to the spine structure of $T(q)$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $T(p)$ is less than or equal to the spine structure of $T(q)$, then $A_n(p) \le A_n(q)$ for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.
In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) about undirected graphs. He proved that there exists an EMSO sentence $ϕ$ such that ${\sf P}(G_n \modelsϕ)$ does not converge as $n\to\infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1, \dots, n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law …
We disprove a conjecture of Frank stating that each weakly 2k-connected has a k-vertex-connected orientation. For k at least 3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.
Disproof of the Odd Hadwiger Conjecture
该研究通过证明存在特定图不包含奇极小且色数较高，从而否定了奇Hadwiger猜想。
We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \, : \, n=a+b+c \text{ and } a,b,c \in A \}|$, counting the number of $3$-compositions using elements of $A$ only, is strictly increasing.
The weighted union-set conjecture is false
本文构造反例否证加权并集猜想，证明在满足特定权重条件下，每个元素可仅占极小权重比例。
In this paper we disprove three conjectures from [M. Dehmer, F. Emmert-Streib, Y. Shi, Interrelations of graph distance measures based on topological indices, PLoS ONE 9 (2014) e94985] on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities. The constructions are based on the families of trees that have the same Wiener index, graph energy or Randic index - but different degree sequences.
In this paper, we disprove a recent monotonicity conjecture of C. Procesi on the generating function for monotone walks on the symmetric group, an object which is equivalent to the Weingarten function of the unitary group.
We disprove a conjecture of Nagy on the maximum number of copies N(G,H) of a fixed graph G in a large graph H with prescribed edge density. Nagy conjectured that for all G, the quantity N(G,H) is asymptotically maximised by either a quasi-star or a quasi-clique. We show this is false for infinitely many graphs, the smallest of which has 6 vertices and 6 edges. We also propose some new conjectures for the behaviour of N(G,H), and present some evidence for them.
Turán Densities for Daisies and Hypercubes
本文推翻了 $r$-daisy 的 Turán 密度随 $r$ 趋于无穷而趋于零的猜想，并调整构造进一步推翻了关于超立方体 Turán 密度的民间猜想，证明了其渐近密度的更紧上界。
In this paper we disprove a conjecture of Lidický and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most $2$ in a triangle-free graph.
We disprove a conjecture proposed in [Gaspers et al., Discrete Applied Mathematics, 2010] and provide a new upper bound for the minimum number of brushes required to continually parallel clean a clique.
Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the conjecture still holds. The proof is based on properties of extension graphs.
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a combinatorial type of 5-dimensional polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.
A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding
该研究推翻了树的等距ℓ_∞嵌入维数相关猜想，构造出32叶树其最小等距ℓ_∞维数为6，证明≤31叶树符合原猜想并回答了相关学者的问题。
Conjecture on Supersequence Lower Bound related to Connell Sequence
本文证明八元素集合的超序列最小大小为52，否证了Connell序列下界猜想，并提供了内部结构分析以助理解一般下界问题。
Cubulating the sphere with many facets
本研究构造了具有$\Omega(n^{5/4})$个面的$d$维球面立方体复形，推翻了Kalai关于邻接立方多面体边界最大化立方球面面数的猜想，证明其增长阶也有误。
On Ziegler's conjectures for logarithmic derivations of arrangements
本文研究了Ziegler关于超平面排列的两个猜想，证明了自由排列的通用切割猜想，并否定了最小次数生成元的猜想。
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