Pub Date : 2021-01-01DOI: 10.4310/joc.2021.v12.n3.a1
J. Schroeder
{"title":"A $2$-regular graph has a prime labeling if and only if it has at most one odd component","authors":"J. Schroeder","doi":"10.4310/joc.2021.v12.n3.a1","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n3.a1","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"160 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76627344","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 : 2020-12-31DOI: 10.4310/joc.2022.v13.n3.a4
J. Koenig, H. Nguyen
A central question in random matrix theory is universality. When an emergent phenomena is observed from a large collection of chosen random variables it is natural to ask if this behavior is specific to the chosen random variable or if the behavior occurs for a larger class of random variables. The rank statistics of random matrices chosen uniformly from Mat(Fq) over a finite field are well understood. The universality properties of these statistics are not yet fully understood however. Recently Wood [39] and Maples [26] considered a natural requirement where the random variables are not allowed to be too close to constant and they showed that the rank statistics match with the uniform model up to an error of type e−cn. In this paper we explore a condition called near uniform, under which we are able to prove tighter bounds q−cn on the asymptotic convergence of the rank statistics. Our method is completely elementary, and allows for a small number of the entries to be deterministic, and for the entries to not be identically distributed so long as they are independent. More importantly, the method also extends to near uniform symmetric, alternating matrices. Our method also applies to two models of perturbations of random matrices sampled uniformly over GLn(Fq): subtracting the identity or taking a minor of a uniformly sampled invertible matrix.
{"title":"Rank of near uniform matrices","authors":"J. Koenig, H. Nguyen","doi":"10.4310/joc.2022.v13.n3.a4","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n3.a4","url":null,"abstract":"A central question in random matrix theory is universality. When an emergent phenomena is observed from a large collection of chosen random variables it is natural to ask if this behavior is specific to the chosen random variable or if the behavior occurs for a larger class of random variables. The rank statistics of random matrices chosen uniformly from Mat(Fq) over a finite field are well understood. The universality properties of these statistics are not yet fully understood however. Recently Wood [39] and Maples [26] considered a natural requirement where the random variables are not allowed to be too close to constant and they showed that the rank statistics match with the uniform model up to an error of type e−cn. In this paper we explore a condition called near uniform, under which we are able to prove tighter bounds q−cn on the asymptotic convergence of the rank statistics. Our method is completely elementary, and allows for a small number of the entries to be deterministic, and for the entries to not be identically distributed so long as they are independent. More importantly, the method also extends to near uniform symmetric, alternating matrices. Our method also applies to two models of perturbations of random matrices sampled uniformly over GLn(Fq): subtracting the identity or taking a minor of a uniformly sampled invertible matrix.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"72 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85587608","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 : 2020-12-17DOI: 10.4310/joc.2022.v13.n4.a6
D. Brewster, Reuven Hodges, A. Yong
We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation is proper goes to zero in the limit.
{"title":"Proper permutations, Schubert geometry, and randomness","authors":"D. Brewster, Reuven Hodges, A. Yong","doi":"10.4310/joc.2022.v13.n4.a6","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n4.a6","url":null,"abstract":"We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation is proper goes to zero in the limit.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90151605","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 : 2020-12-03DOI: 10.4310/joc.2022.v13.n4.a3
A. R. Mayorova, E. Vassilieva
Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $pi$ in $S_n$ such that for any $1leq j leq n$, ${pi(1),pi(2),dots,pi(j)}$ is an interval in $mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and generating functions for domino tableaux. As an important characteristic, our development is compatible with the works of Solomon regarding the descent algebra of Coxeter groups. In particular, we design descent preserving bijections between signed arc permutations and sets of domino tableaux to show that they are indeed type B Schur-positive.
{"title":"A domino tableau-based view on type B Schur-positivity","authors":"A. R. Mayorova, E. Vassilieva","doi":"10.4310/joc.2022.v13.n4.a3","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n4.a3","url":null,"abstract":"Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $pi$ in $S_n$ such that for any $1leq j leq n$, ${pi(1),pi(2),dots,pi(j)}$ is an interval in $mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and generating functions for domino tableaux. As an important characteristic, our development is compatible with the works of Solomon regarding the descent algebra of Coxeter groups. In particular, we design descent preserving bijections between signed arc permutations and sets of domino tableaux to show that they are indeed type B Schur-positive.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"132 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75929557","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 : 2020-11-19DOI: 10.4310/joc.2023.v14.n2.a4
Sunita Chepuri, Feiyang Lin
The birational $R$-matrix is a transformation that appears in the theory of geometric crystals, the study of total positivity in loop groups, and discrete dynamical systems. This $R$-matrix gives rise to an action of the symmetric group $S_m$ on an $m$-tuple of vectors. While the birational $R$-matrix is precisely the formula corresponding to the action of the simple transposition $s_i$, explicit formulas for the action of other permutations are generally not known. One particular case was studied by Lam and Pylyavskyy as it relates to energy functions of crystals. In this paper, we will discuss formulas for several additional cases, including transpositions, and provide combinatorial interpretations for the functions that appear in our work.
birational R -矩阵是出现在几何晶体理论、环群总正性研究和离散动力系统中的一种变换。这个$R$-矩阵产生了对称群$S_m$对向量元组$m$的作用。虽然双象R -矩阵正是简单转置s_i作用的对应公式，但其他置换作用的显式公式通常是未知的。Lam和pylyavsky研究了一个特殊的例子，因为它与晶体的能量函数有关。在本文中，我们将讨论几种其他情况的公式，包括换位，并为我们工作中出现的函数提供组合解释。
{"title":"Symmetric group action of the birational $R$-matrix","authors":"Sunita Chepuri, Feiyang Lin","doi":"10.4310/joc.2023.v14.n2.a4","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n2.a4","url":null,"abstract":"The birational $R$-matrix is a transformation that appears in the theory of geometric crystals, the study of total positivity in loop groups, and discrete dynamical systems. This $R$-matrix gives rise to an action of the symmetric group $S_m$ on an $m$-tuple of vectors. While the birational $R$-matrix is precisely the formula corresponding to the action of the simple transposition $s_i$, explicit formulas for the action of other permutations are generally not known. One particular case was studied by Lam and Pylyavskyy as it relates to energy functions of crystals. In this paper, we will discuss formulas for several additional cases, including transpositions, and provide combinatorial interpretations for the functions that appear in our work.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"13 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87539282","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 : 2020-11-19DOI: 10.4310/JOC.2022.v13.n2.a3
C. Elbracht, Jakob Kneip, Maximilian Teegen
We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems by using it to prove tree-of-tangles theorems. This approach allows us to strengthen some of the existing tree-of-tangles theorems by bounding the node degrees in them. We also present a slight strengthening and simplified proof of the duality theorem, which allows us to derive a tree-of-tangles theorem also for tangles of different orders.
{"title":"Obtaining trees of tangles from tangle-tree duality","authors":"C. Elbracht, Jakob Kneip, Maximilian Teegen","doi":"10.4310/JOC.2022.v13.n2.a3","DOIUrl":"https://doi.org/10.4310/JOC.2022.v13.n2.a3","url":null,"abstract":"We demonstrate the versatility of the tangle-tree duality theorem for abstract separation systems by using it to prove tree-of-tangles theorems. This approach allows us to strengthen some of the existing tree-of-tangles theorems by bounding the node degrees in them. We also present a slight strengthening and simplified proof of the duality theorem, which allows us to derive a tree-of-tangles theorem also for tangles of different orders.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"8 3","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72582594","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 : 2020-11-16DOI: 10.4310/joc.2023.v14.n3.a4
Satyan L. Devadoss, Mia Smith
Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.
{"title":"Colorful graph associahedra","authors":"Satyan L. Devadoss, Mia Smith","doi":"10.4310/joc.2023.v14.n3.a4","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n3.a4","url":null,"abstract":"Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"44 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86664345","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 : 2020-10-29DOI: 10.4310/joc.2023.v14.n1.a4
Samantha Dahlberg, Young-Hie Kim
It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $nldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs on commutation and long braid classes. Also, these results give a recursive formula for the diameter of the longest permutation, which matches that of Reiner, Roichman and Assaf. Lastly, We make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement, and find families of permutations that achieve the upper bound and lower bound of the conjecture. In particular permutations that avoid 312 or 231 have graphs that achieve the upper bound.
{"title":"Diameters of graphs on reduced words of $12$ and $21$-inflations","authors":"Samantha Dahlberg, Young-Hie Kim","doi":"10.4310/joc.2023.v14.n1.a4","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n1.a4","url":null,"abstract":"It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $nldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs on commutation and long braid classes. Also, these results give a recursive formula for the diameter of the longest permutation, which matches that of Reiner, Roichman and Assaf. Lastly, We make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement, and find families of permutations that achieve the upper bound and lower bound of the conjecture. In particular permutations that avoid 312 or 231 have graphs that achieve the upper bound.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"98 1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89548147","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 : 2020-09-22DOI: 10.4310/joc.2022.v13.n3.a5
A. Jafari
Let $nge 1$ and $sge 1$ be integers. An almost $s$-stable subset $A$ of $[n]={1,dots,n}$ is a subset such that for any two distinct elements $i, jin A$, one has $|i-j|ge s$. For a family $cal F$ of subsets in $[n]$ and $rge 2$, the chromatic number of the $r$-uniform Kneser hypergraph $mbox{KG}^r({cal F})$, whose vertex set is $cal F$ and whose edges set is the set of ${A_1,dots, A_r}$ of pairwise disjoint elements of $cal F$, has been studied extensively in the literature and Abyazi Sani and Alishahi were able to give a lower bound for it in terms of the equatable $r$-colorability defect, $mbox{ecd}^r({cal F})$. In this article, the methods of Chen for the special family of all $k$-subsets of $[n]$, are modified to give lower bounds for the chromatic number of almost stable general Kneser hypergraph $mbox{KG}^r({cal F}_s)$ in terms of $mbox{ecd}^s({cal F})$. Here ${cal F}_s$ is he collection of almost $s$-stable elements of $cal F$. We also, propose a generalization of conjecture of Meunier.
{"title":"On the chromatic number of almost stable general Kneser hypergraphs","authors":"A. Jafari","doi":"10.4310/joc.2022.v13.n3.a5","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n3.a5","url":null,"abstract":"Let $nge 1$ and $sge 1$ be integers. An almost $s$-stable subset $A$ of $[n]={1,dots,n}$ is a subset such that for any two distinct elements $i, jin A$, one has $|i-j|ge s$. For a family $cal F$ of subsets in $[n]$ and $rge 2$, the chromatic number of the $r$-uniform Kneser hypergraph $mbox{KG}^r({cal F})$, whose vertex set is $cal F$ and whose edges set is the set of ${A_1,dots, A_r}$ of pairwise disjoint elements of $cal F$, has been studied extensively in the literature and Abyazi Sani and Alishahi were able to give a lower bound for it in terms of the equatable $r$-colorability defect, $mbox{ecd}^r({cal F})$. In this article, the methods of Chen for the special family of all $k$-subsets of $[n]$, are modified to give lower bounds for the chromatic number of almost stable general Kneser hypergraph $mbox{KG}^r({cal F}_s)$ in terms of $mbox{ecd}^s({cal F})$. Here ${cal F}_s$ is he collection of almost $s$-stable elements of $cal F$. We also, propose a generalization of conjecture of Meunier.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"23 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75699976","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 : 2020-09-17DOI: 10.4310/joc.2023.v14.n2.a5
Attila Jo'o
. We give a common matroidal generalisation of ‘A Cantor-Bernstein theorem for paths in graphs’ by Diestel and Thomassen and ‘A Cantor-Bernstein-type theorem for spanning trees in infinite graphs’ by ourselves.
{"title":"A Cantor–Bernstein theorem for infinite matroids","authors":"Attila Jo'o","doi":"10.4310/joc.2023.v14.n2.a5","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n2.a5","url":null,"abstract":". We give a common matroidal generalisation of ‘A Cantor-Bernstein theorem for paths in graphs’ by Diestel and Thomassen and ‘A Cantor-Bernstein-type theorem for spanning trees in infinite graphs’ by ourselves.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"53 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84586332","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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