Yi Wang
Simion had a unimodality conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. Hildebrand recently showed the stronger result that these numbers are log concave. Here we present a simple proof of Hildebrand's result.
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Simion conjectured the unimodality of a sequence counting lattice paths in a grid with a Ferrers diagram removed from the northwest corner. Recently, Hildebrand and then Wang proved the stronger result that this sequence is actually log concave. Both proofs were mainly algebraic in nature. We give two combinatorial proofs of this theorem.
Bonin, Shapiro and Simion (1993) gave two formulas on the distribution of major index for Schröder paths, and proved their result for the case $E<D<N$. In this short note, we correct an error in their proof, and give a complete proof for all cases.
We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a "one-line" combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.
We provide a proof of the Borwein Conjecture using analytic methods.
We prove the Aharoni Berger Conjecture
We prove a conjecture of Meszaros and Morales on the volume of a flow polytope. Independently from our work, Zeilberger sketched a proof of their conjecture. In fact, our proof is the same as Zeilberger's proof. The purpose of this note is to give a more detailed proof of the conjecture.
We present a new proof of the Joints Theorem without taking derivatives. Then we generalize the proof to prove the Multijoints Conjecture and Carbery's generalization. All results are in any dimension over an arbitrary field.
Error in proof of theorem 10.
Proof of Union-Closed Sets Conjecture
该研究完成了并闭集猜想的证明工作。
A simple proof of Atanassov's Conjecture is presented. Atanassov's Conjecture is a generalization of Sperner's Lemma, a lemma which has been used to prove Brouwer's Fixed Point Theorem, among other fixed point theorems. The proof of Atanassov's Conjecture is based on the Brouwer Degree of maps and is extremely elementary. It is much simpler than the original proofs given for the conjecture and provides some insight into the nature of the conjecture. Furthermore, a generalization of the conjecture is presented and finally a new theorem, similar to the original Sperner Lemma, is proved.
We settle in the affirmative the Graham-Sloane conjecture.
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
The well-known Steinberg's conjecture asserts that any planar graph without 4- and 5-cycles is 3 colorable. In this note we have given a short algorithmic proof of this conjecture based on the spiral chains of planar graphs proposed in the proof of the four color theorem by the author in 2004.
We prove a conjecture by Guo-Niu Han which interpolates between two known hook expansion formulas.
We present a proof of determinant of special nonsymmetric Toeplitz matrices conjectured by Anđelić and Fonseca in \cite{andjelic2020some}. A proof is also demonstrated for a more general theorem. The two conjectures are therefore just two possible results, under two specific settings. Numerical examples validating the theorem are provided.
We prove two conjectures on correlation inequalities for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions
Fred Galvin's amazing proof of the Dinitiz conjecture is used to illustrate the method of undetermined generalization and specialization.
We give a proof of a conjecture of A. Lacasse in his doctoral thesis which has applications in machine learning algorithms. The proof relies on some interesting binomial sums identities introduced by Abel (1839), and on their generalization to the multinomial case by Hurwitz (1902).
Rosenfeld Conjectured in 1972 that there exists an integer K $\geq$ 8 such that any tournament of order n $\geq$ K contains any Hamiltonian oriented path. In 2000, Havet and Thomassé proved this conjecture for any tournament with exactly 3 exceptions. We give a simplified proof of this fact.
We give a new proof of a_4ϕ_3 summation due to G.E. Andrews and confirm another_4ϕ_3 summation conjectured by him recently. Some variations of these two_4ϕ_3 summations are also given.
Proof of the High Girth Existence Conjecture via Refined Absorption
本文利用精细吸收法证明高围长存在性猜想，统一解决了组合设计存在性猜想与Erdős高围长Steiner三元系猜想的共同推广问题。
We give a proof of Willcocks's Conjecture, stating that if $p - q$ and $p + q$ are relatively prime, then there exists a Hamiltonian tour of a $(p, q)$-leaper on a square chessboard of side $2(p + q)$. The conjecture was formulated by T. H. Willcocks in 1976 and has been an open problem since.
We give a new proof of Vassiliev's planarity criterion for framed four-valent graphs (and more generally, *-graphs), which is based on Pontryagin-Kuratowski theorem.
An elementary proof of Sierksma's conjecture for seven points in the plane
本文通过几何方法证明了平面中七点存在四组Tverberg划分，这是Sierksma猜想中唯一已确认的非平凡情况，不同于之前依赖拓扑论的证明方法。
Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new 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)
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck's conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher's bijection, we give the combinatorial proofs…
Halin conjectured 20 years ago that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. We prove Halin's conjecture. This implies a forbidden minor characterisation for the property of having a normal spanning tree.
By considering the fundamental equation $x=y-y^2=z-z^3$, Somos conjectured that the Hankel determinants for the generating series $y(z)$ are the Somos-4 numbers. We prove this conjecture by using the quadratic transformation for Hankel determinants of Sulanke and Xin.
Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces of size at most 6 are Hamiltonian. We prove the conjecture.
A conjecture of Chunwei Song on a limiting case of the q,t-Schröder theorem is proved combinatorially. The proof matches pairs of tableaux to Catalan words in a manner that preserves differences in the maj statistic.
In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they made a similar conjecture for fractional dimension. This paper settles this latter conjecture.
We prove a theorem on how to count induced subgraphs in neighborhoods of graphs. Then we use it to prove a subgraph counting identity conjectured by McKay and Radziszowski in there work on Ramsey theory.
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 this paper the circulant Hadamard conjecture is proved.
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。
We prove Dejean's conjecture. Specifically, we show that Dejean's conjecture holds for the last remaining open values of n, namely 15 <= n <= 26.
We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in a succession of simplifications of the cohomological approach. Our proof is inductive, and requires Tardos' previous result settling the rank two case of the conjecture.
In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.
In this work we tried to prove the lonely runner conjecture also known as the view obstruction problem.
Padberg introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.
Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey-Wilson and Rogers polynomials. We also obtain some related results, including a new proof of a partition identity conjectured by Capparelli and fir…
We prove a conjecture of Ohba which says that every graph $G$ on at most $2χ(G)+1$ vertices satisfies $χ_\ell(G)=χ(G)$.
The VPN Tree Routing Conjecture states that there always exists an optimal solution to the symmetric Virtual Private Network Design (sVPND) problem where the paths between all terminals form a tree. Only recently, Hurkens, Keijsper, and Stougie gave a proof of this conjecture for the special case of ring networks. Their proof is based on a dual pair of linear programs and is somewhat in- volved. We present a short proof of a slightly stronger conjecture which might also turn out to be useful for proving the VPN Tree Routing Conjecture for general networks.
An elementary proof of the bunkbed conjecture for forests
针对一般图上不成立的上下铺猜想，本文给出其对任意无环图（森林）成立的简短自洽初等证明，该论证具备一定教学意义。
The Hom-complexes were introduced by Lovasz to study topological obstructions to graph colorings. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K_n) is (n-d-2)-connected, where d is the maximal degree of a vertex of G. We give a short proof of the conjecture.
In terms of Sear's transformation formula for $_4ϕ_3$-series, we give new proofs of a summation formula for ${_4ϕ_3}$-series due to Andrews [2] and another summation formula for${_4ϕ_3}$-series conjectured in the same paper. Meanwhile, other several related results are also derived.
A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power series and their (matrix) logarithms.
The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product formula, has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.
We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb{R}$ with non-negative values on Macdonald symmetric functions $P_λ$, that was conjectured by S.V. Kerov in 1992.
Proof of Irvine's Conjecture via Mechanized Guessing
本文利用机械化猜测与验证方法，结合定理证明器Walnut，成功证明了Sean A. Irvine关于非线性递归的猜想。
We extend the Lee-Schiffler Dyck path model to give a proof of the Kontsevich non-commutative cluster positivity conjecture with unequal parameters.
Recently Dekking conjectured the form of the subword complexity function for the Fibonacci-Thue-Morse sequence. In this note we prove his conjecture by purely computational means, using the free software Walnut.
We prove a 2018 conjecture of Krawchuk and Rampersad on the extremal behavior of $c(n)$, where $c(n)$ counts the number of length-$n$ factors of the Thue-Morse word $\mathbf{t}$, up to cyclic rotation.
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