Pub Date : 2021-10-25DOI: 10.4310/joc.2023.v14.n2.a3
Christian Gaetz, Ram K. Goel
A permutation is called smooth if the corresponding Schubert variety is smooth. Gilboa and Lapid prove that in the symmetric group, multiplying the reflections below a smooth element $w$ in Bruhat order in a compatible order yields back the element $w$. We strengthen this result by showing that such a product in fact determines a saturated chain $e to w$ in Bruhat order, and that this property characterizes smooth elements.
{"title":"Products of reflections in smooth Bruhat intervals","authors":"Christian Gaetz, Ram K. Goel","doi":"10.4310/joc.2023.v14.n2.a3","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n2.a3","url":null,"abstract":"A permutation is called smooth if the corresponding Schubert variety is smooth. Gilboa and Lapid prove that in the symmetric group, multiplying the reflections below a smooth element $w$ in Bruhat order in a compatible order yields back the element $w$. We strengthen this result by showing that such a product in fact determines a saturated chain $e to w$ in Bruhat order, and that this property characterizes smooth elements.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75726320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-09DOI: 10.4310/joc.2023.v14.n1.a3
Leo Versteegen
A linear configuration is said to be common in a finite Abelian group G if for every 2-coloring of G the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation in an even number of variables over G , then it is common in F n p if and only if the equation’s coefficients can be partitioned into pairs that sum to zero mod p . This was proven by Fox, Pham and Zhao for sufficiently large n . We generalize their result to all sufficiently large Abelian groups G for which the equation’s coefficients are coprime to | G | .
在有限阿贝尔群G中，如果对G的每一个2-着色，该构型的单色实例的数目至少与随机选择的着色相等，则称线性构型是公共的。Saad和Wolf推测，如果一个位形被定义为G上偶数个变量的单个齐次方程的解集，那么当且仅当该方程的系数可以分割成对，对p求和为零时，它在F n p中是公的。Fox, Pham和Zhao在n足够大时证明了这一点。我们将他们的结果推广到所有足够大的阿贝尔群G，对于这些群G，方程的系数是素。
{"title":"Common and Sidorenko equations in Abelian groups","authors":"Leo Versteegen","doi":"10.4310/joc.2023.v14.n1.a3","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n1.a3","url":null,"abstract":"A linear configuration is said to be common in a finite Abelian group G if for every 2-coloring of G the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation in an even number of variables over G , then it is common in F n p if and only if the equation’s coefficients can be partitioned into pairs that sum to zero mod p . This was proven by Fox, Pham and Zhao for sufficiently large n . We generalize their result to all sufficiently large Abelian groups G for which the equation’s coefficients are coprime to | G | .","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"75 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76338896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-12DOI: 10.4310/joc.2023.v14.n1.a5
Kittitat Iamthong, S. Kitaev
A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1rightarrow u_2rightarrow cdots rightarrow u_t$, $t geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_irightarrow u_j$ exist for $1 leq i
一个有向图是半传递的当且仅当它是非循环的，并且对于任何有向路径$u_1rightarrow u_2rightarrow cdots rightarrow u_t$, $t geq 2$，要么没有从$u_1$到$u_t$的边，要么$1 leq i
{"title":"Semi-transitivity of directed split graphs generated by morphisms","authors":"Kittitat Iamthong, S. Kitaev","doi":"10.4310/joc.2023.v14.n1.a5","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n1.a5","url":null,"abstract":"A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1rightarrow u_2rightarrow cdots rightarrow u_t$, $t geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_irightarrow u_j$ exist for $1 leq i<j leq t$. In this paper, we study semi-transitivity of families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving in the limit infinite directed split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. We fully classify semi-transitive infinite directed split graphs when a morphism in question can involve any $ntimes m$ matrices over ${-1,0,1}$ with a single natural condition.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"15 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87746108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-13DOI: 10.4310/joc.2023.v14.n1.a1
Nathan Chapelier-Laget
Let W be an irreducible Weyl group and W a its affine Weyl group. In [4] the author defined an affine variety (cid:2) X W a , called the Shi variety of W a , whose integral points are in bijection with W a . The set of irreducible components of (cid:2) X W a , denoted H 0 ( (cid:2) X W a ), is of some interest and we show in this article that H 0 ( (cid:2) X W a ) has a structure of a semidistributive lattice.
设W是一个不可约Weyl群，W是它的仿射Weyl群。在[4]中定义了一个仿射变量(cid:2) X W a，称为W a的Shi变量，其积分点与W a双射。(cid:2) X wa的不可约分量集h0 ((cid:2) X wa)具有一些有趣的性质，本文证明了h0 ((cid:2) X wa)具有半分配格的结构。
{"title":"Lattice associated to a Shi variety","authors":"Nathan Chapelier-Laget","doi":"10.4310/joc.2023.v14.n1.a1","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n1.a1","url":null,"abstract":"Let W be an irreducible Weyl group and W a its affine Weyl group. In [4] the author defined an affine variety (cid:2) X W a , called the Shi variety of W a , whose integral points are in bijection with W a . The set of irreducible components of (cid:2) X W a , denoted H 0 ( (cid:2) X W a ), is of some interest and we show in this article that H 0 ( (cid:2) X W a ) has a structure of a semidistributive lattice.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"44 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80666802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-23DOI: 10.4310/joc.2023.v14.n2.a6
Christian Winter
The size-Ramsey number ˆ R ( k ) ( H ) of a k -uniform hypergraph H is the minimum number of edges in a k -uniform hypergraph G with the property that every ‘2-edge coloring’ of G contains a monochromatic copy of H . For k ≥ 2 and n ∈ N , a k -uniform tight path on n vertices P ( k ) n is defined as a k -uniform hypergraph on n vertices for which there is an ordering of its vertices such that the edges are all sets of k consecutive vertices with respect to this order. We prove a lower bound on the size-Ramsey number of k -uniform tight paths, which is, considered assymptotically in both the uniformity k and the number of vertices n , ˆ R ( k ) ( P ( k ) n ) = Ω (cid:0) log( k ) n (cid:1) .
《size-Ramseyˆ当家R (k) (H) of a k -uniform hypergraph H最低当家》是edges in a k -uniform hypergraph G和物业的每对2-edge coloring’of G contains a monochromatic复制of s . H。k≥2的a和n∈n, k -uniform紧路径上n vertices P (k)是奈德fi美国k -uniform hypergraph on n有vertices人人平等,这是一个ordering of its vertices edges都让》这样的那个k consecutive vertices和尊重这种秩序。size-Ramsey号码》下束缚在我们证明a k -uniform紧道路,认为这是assymptotically在两者当家》《uniformity k与vertices n,ˆR (k) (P (k) n) =Ω(cid日志:0)(k) n (cid): 1)。
{"title":"Lower bound on the size-Ramsey number of tight paths","authors":"Christian Winter","doi":"10.4310/joc.2023.v14.n2.a6","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n2.a6","url":null,"abstract":"The size-Ramsey number ˆ R ( k ) ( H ) of a k -uniform hypergraph H is the minimum number of edges in a k -uniform hypergraph G with the property that every ‘2-edge coloring’ of G contains a monochromatic copy of H . For k ≥ 2 and n ∈ N , a k -uniform tight path on n vertices P ( k ) n is defined as a k -uniform hypergraph on n vertices for which there is an ordering of its vertices such that the edges are all sets of k consecutive vertices with respect to this order. We prove a lower bound on the size-Ramsey number of k -uniform tight paths, which is, considered assymptotically in both the uniformity k and the number of vertices n , ˆ R ( k ) ( P ( k ) n ) = Ω (cid:0) log( k ) n (cid:1) .","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"110 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77533734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-09DOI: 10.4310/joc.2023.v14.n3.a2
T. Mansour, M. Shattuck
Abstract. In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schröder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.
{"title":"On a conjecture of Lin and Kim concerning a refinement of Schröder numbers","authors":"T. Mansour, M. Shattuck","doi":"10.4310/joc.2023.v14.n3.a2","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n3.a2","url":null,"abstract":"Abstract. In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schröder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"15 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84733309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-17DOI: 10.4310/joc.2023.v14.n4.a7
U. Naumann, Shubhaditya Burela
Second derivatives of mathematical models for real-world phenomena are fundamental ingredients of a wide range of numerical simulation methods including parameter sensitivity analysis, uncertainty quantification, nonlinear optimization and model calibration. The evaluation of such Hessians often dominates the overall computational effort. The combinatorial {sc Hessian Accumulation} problem aiming to minimize the number of floating-point operations required for the computation of a Hessian turns out to be NP-complete. We propose a dynamic programming formulation for the solution of {sc Hessian Accumulation} over a sub-search space. This approach yields improvements by factors of ten and higher over the state of the art based on second-order tangent and adjoint algorithmic differentiation.
{"title":"Hessian chain bracketing","authors":"U. Naumann, Shubhaditya Burela","doi":"10.4310/joc.2023.v14.n4.a7","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n4.a7","url":null,"abstract":"Second derivatives of mathematical models for real-world phenomena are fundamental ingredients of a wide range of numerical simulation methods including parameter sensitivity analysis, uncertainty quantification, nonlinear optimization and model calibration. The evaluation of such Hessians often dominates the overall computational effort. The combinatorial {sc Hessian Accumulation} problem aiming to minimize the number of floating-point operations required for the computation of a Hessian turns out to be NP-complete. We propose a dynamic programming formulation for the solution of {sc Hessian Accumulation} over a sub-search space. This approach yields improvements by factors of ten and higher over the state of the art based on second-order tangent and adjoint algorithmic differentiation.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"53 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88688708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-09DOI: 10.4310/joc.2023.v14.n1.a2
Sami H. Assaf
We introduce a new paradigm for proving the Schur P -positivity of a given quasi-symmetric function. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur P -positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalizations with a new appli-cation to the product of Schur P -functions.
{"title":"Queer dual equivalence graphs","authors":"Sami H. Assaf","doi":"10.4310/joc.2023.v14.n1.a2","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n1.a2","url":null,"abstract":"We introduce a new paradigm for proving the Schur P -positivity of a given quasi-symmetric function. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur P -positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalizations with a new appli-cation to the product of Schur P -functions.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84277041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-04DOI: 10.4310/joc.2023.v14.n3.a1
Kevin P. Costello, Gabriel Elvin
Given an equation, the integers [ n ] = { 1 , 2 , . . . , n } as inputs, and the colors red and blue, how can we color [ n ] in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is only known for a handful of equations, but much progress has been made on improving upper and lower bounds on minima for various equations. A well-studied characteristic an equation, which has its roots in graph Ramsey theory, is to determine if the minimum number of monochromatic solutions can be achieved (asymptotically) by uniformly random colorings. Such equations are called common . We prove that no 3-term equations are common and provide a lower bound for a specific class of 3-term equations.
{"title":"Avoiding monochromatic solutions to 3-term equations","authors":"Kevin P. Costello, Gabriel Elvin","doi":"10.4310/joc.2023.v14.n3.a1","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n3.a1","url":null,"abstract":"Given an equation, the integers [ n ] = { 1 , 2 , . . . , n } as inputs, and the colors red and blue, how can we color [ n ] in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is only known for a handful of equations, but much progress has been made on improving upper and lower bounds on minima for various equations. A well-studied characteristic an equation, which has its roots in graph Ramsey theory, is to determine if the minimum number of monochromatic solutions can be achieved (asymptotically) by uniformly random colorings. Such equations are called common . We prove that no 3-term equations are common and provide a lower bound for a specific class of 3-term equations.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"13 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80466297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4310/joc.2021.v12.n1.a4
Deepak Bal, Patrick Bennett, Sean English, Calum MacRury, P. Prałat
The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of coloured vertices that can eventually force the entire graph to be coloured. The zero forcing number is the size of the smallest zero forcing set. We explore the zero forcing number for random regular graphs, improving on bounds given by Kalinowski, Kam˘cev and Sudakov [15]. We also propose and analyze a degree greedy algorithm for finding small zero forcing sets using the differential equations method.
{"title":"Zero-forcing in random regular graphs","authors":"Deepak Bal, Patrick Bennett, Sean English, Calum MacRury, P. Prałat","doi":"10.4310/joc.2021.v12.n1.a4","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n1.a4","url":null,"abstract":"The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of coloured vertices that can eventually force the entire graph to be coloured. The zero forcing number is the size of the smallest zero forcing set. We explore the zero forcing number for random regular graphs, improving on bounds given by Kalinowski, Kam˘cev and Sudakov [15]. We also propose and analyze a degree greedy algorithm for finding small zero forcing sets using the differential equations method.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89641758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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