{"title":"Combinatorics on Graphs","authors":"","doi":"10.1090/text/055/06","DOIUrl":"https://doi.org/10.1090/text/055/06","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"38 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80644383","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}
{"title":"Famous Number Families","authors":"","doi":"10.1090/text/055/04","DOIUrl":"https://doi.org/10.1090/text/055/04","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79977032","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 : 2020-01-14DOI: 10.4310/joc.2022.v13.n1.a6
B. E. Tenner
We study the appearance of notable interval structures---lattices, modular lattices, distributive lattices, and boolean lattices---in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetric group, for each of the four structure types, in each of the two posets. Each category can be enumerated, with intriguing connections to Fibonacci and Catalan numbers. We conclude with suggestions for further directions and questions, including an interesting analysis of the intervals formed between a permutation and each generator in its support.
{"title":"Interval structures in the Bruhat and weak orders","authors":"B. E. Tenner","doi":"10.4310/joc.2022.v13.n1.a6","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n1.a6","url":null,"abstract":"We study the appearance of notable interval structures---lattices, modular lattices, distributive lattices, and boolean lattices---in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetric group, for each of the four structure types, in each of the two posets. Each category can be enumerated, with intriguing connections to Fibonacci and Catalan numbers. We conclude with suggestions for further directions and questions, including an interesting analysis of the intervals formed between a permutation and each generator in its support.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"9 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76284009","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 : 2020-01-01DOI: 10.4310/joc.2020.v11.n2.a5
P. Horn, Adam Purcilly
Consider a connected graph G , with a pebble placed on each vertex of G . The routing number, rt ( G ), of G is the minimum number of steps needed to route any permutation on the vertices of G , where a step consists of selecting a matching in the graph and swapping the pebbles on the endpoints of each edge. Alon, Chung, and Graham [ SIAM J. Discrete Math. , 7 (1994), pp. 516–530.] introduced this parameter, and (among other results) gave a bound based on the spectral gap for general graphs. The bound they obtain is poly-logarithmic for graphs with a sufficiently strong spectral gap. In this paper, we use spectral properties and probablistic methods to investigate when this upper bound can be improved to be constant depending on the gap and the vertex degrees.
考虑一个连通图G，在G的每个顶点上都有一个小石子。G的路由数rt (G)是在G的顶点上路由任何排列所需的最小步数，其中一个步骤包括在图中选择一个匹配并交换每个边端点上的鹅卵石。Alon, Chung, and Graham [SIAM J.离散数学]。， 7(1994)，第516-530页。]引入了这个参数，并且(在其他结果中)给出了基于一般图的谱间隙的界。对于具有足够强谱隙的图，他们得到的界是多对数的。在本文中，我们利用谱性质和概率方法来研究该上界何时可以根据间隙和顶点度改进为常数。
{"title":"Routing number of dense and expanding graphs","authors":"P. Horn, Adam Purcilly","doi":"10.4310/joc.2020.v11.n2.a5","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n2.a5","url":null,"abstract":"Consider a connected graph G , with a pebble placed on each vertex of G . The routing number, rt ( G ), of G is the minimum number of steps needed to route any permutation on the vertices of G , where a step consists of selecting a matching in the graph and swapping the pebbles on the endpoints of each edge. Alon, Chung, and Graham [ SIAM J. Discrete Math. , 7 (1994), pp. 516–530.] introduced this parameter, and (among other results) gave a bound based on the spectral gap for general graphs. The bound they obtain is poly-logarithmic for graphs with a sufficiently strong spectral gap. In this paper, we use spectral properties and probablistic methods to investigate when this upper bound can be improved to be constant depending on the gap and the vertex degrees.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"430 ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72423462","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 : 2020-01-01DOI: 10.4310/JOC.2020.V11.N1.A8
E. Oguz
We introduce a shifted analogue of the ribbon tableaux defined by James and Kerber. For any positive integer $k$, we give a bijection between the $k$-ribbon fillings of a shifted shape and regular fillings of a $lfloor k/2rfloor$-tuple of shapes called its $k$-quotient. We also define the corresponding generating functions, and prove that they are symmetric, Schur positive and Schur $Q$-positive.
{"title":"A shifted analogue to ribbon tableaux","authors":"E. Oguz","doi":"10.4310/JOC.2020.V11.N1.A8","DOIUrl":"https://doi.org/10.4310/JOC.2020.V11.N1.A8","url":null,"abstract":"We introduce a shifted analogue of the ribbon tableaux defined by James and Kerber. For any positive integer $k$, we give a bijection between the $k$-ribbon fillings of a shifted shape and regular fillings of a $lfloor k/2rfloor$-tuple of shapes called its $k$-quotient. We also define the corresponding generating functions, and prove that they are symmetric, Schur positive and Schur $Q$-positive.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"24 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80210231","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 : 2019-12-18DOI: 10.4310/joc.2021.v12.n1.a1
J. Fox, Sammy Luo, Yuval Wigderson
Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $Goverset{r}{longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] overset{r}{longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponential constant is indeed unnecessary, but conjecture that some dependence on $G$ is unavoidable. An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.
{"title":"Extremal and Ramsey results on graph blowups","authors":"J. Fox, Sammy Luo, Yuval Wigderson","doi":"10.4310/joc.2021.v12.n1.a1","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n1.a1","url":null,"abstract":"Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $Goverset{r}{longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] overset{r}{longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponential constant is indeed unnecessary, but conjecture that some dependence on $G$ is unavoidable. \u0000An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"62 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87934093","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 : 2019-12-10DOI: 10.4310/JOC.2021.v12.n2.a7
Shoni Gilboa, E. Lapid
We show that any smooth permutation $sigmain S_n$ is characterized by the set ${mathbf{C}}(sigma)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{leqsigma}$, and that $sigma$ is the product (in a certain order) of the transpositions in ${mathbf{C}}(sigma)$. We also characterize the image of the map $sigmamapsto{mathbf{C}}(sigma)$. As an application, we show that $sigma$ is smooth if and only if the intersection of $(S_n)_{leqsigma}$ with every conjugate of a parabolic subgroup of $S_n$ admits a maximum. This also gives another approach for enumerating smooth permutations and subclasses thereof. Finally, we relate covexillary permutations to smooth ones and rephrase the results in terms of the (co)essential set in the sense of Fulton.
{"title":"Some combinatorial results on smooth permutations","authors":"Shoni Gilboa, E. Lapid","doi":"10.4310/JOC.2021.v12.n2.a7","DOIUrl":"https://doi.org/10.4310/JOC.2021.v12.n2.a7","url":null,"abstract":"We show that any smooth permutation $sigmain S_n$ is characterized by the set ${mathbf{C}}(sigma)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{leqsigma}$, and that $sigma$ is the product (in a certain order) of the transpositions in ${mathbf{C}}(sigma)$. We also characterize the image of the map $sigmamapsto{mathbf{C}}(sigma)$. As an application, we show that $sigma$ is smooth if and only if the intersection of $(S_n)_{leqsigma}$ with every conjugate of a parabolic subgroup of $S_n$ admits a maximum. This also gives another approach for enumerating smooth permutations and subclasses thereof. Finally, we relate covexillary permutations to smooth ones and rephrase the results in terms of the (co)essential set in the sense of Fulton.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79234274","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 : 2019-12-04DOI: 10.4310/joc.2022.v13.n2.a1
P. Harris, Margaret Rahmoeller, Lisa Schneider
Kostant's partition function counts the number of distinct ways to express a weight of a classical Lie algebra $mathfrak{g}$ as a sum of positive roots of $mathfrak{g}$. We refer to each of these expressions as decompositions of a weight and our main result establishes that the (normalized) distribution of the number of positive roots in the decomposition of the highest root of a classical Lie algebra of rank $r$ converges to a Gaussian distribution as $rtoinfty$. We extend these results to an infinite family of weights, irrespective of Lie type, for which we establish a closed formula for the $q$-analog of Kostant's partition function and then prove that the analogous distribution also converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We end our analysis with some directions for future research.
{"title":"On the asymptotic behavior of the $q$-analog of Kostant's partition function","authors":"P. Harris, Margaret Rahmoeller, Lisa Schneider","doi":"10.4310/joc.2022.v13.n2.a1","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n2.a1","url":null,"abstract":"Kostant's partition function counts the number of distinct ways to express a weight of a classical Lie algebra $mathfrak{g}$ as a sum of positive roots of $mathfrak{g}$. We refer to each of these expressions as decompositions of a weight and our main result establishes that the (normalized) distribution of the number of positive roots in the decomposition of the highest root of a classical Lie algebra of rank $r$ converges to a Gaussian distribution as $rtoinfty$. We extend these results to an infinite family of weights, irrespective of Lie type, for which we establish a closed formula for the $q$-analog of Kostant's partition function and then prove that the analogous distribution also converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We end our analysis with some directions for future research.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87139782","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 : 2019-12-04DOI: 10.4310/joc.2021.v12.n4.a4
A. Czygrinow, T. Molla, B. Nagle, Roy Oursler
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $ell$-cycles $C_{ell}$: if every vertex $v in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n geq n_0(ell)$ is sufficiently large, then $G$ admits an even rainbow $ell$-cycle $C_{ell}$. This result is best possible whenever $ell notequiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer $ell geq 4$, every large $n$-vertex oriented graph $vec{G} = (V, vec{E})$ with minimum outdegree at least $(n+1)/3$ admits a (consistently) directed $ell$-cycle $vec{C}_{ell}$. Our latter result relates to one of Kelly, Kuhn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree. Our proofs are based on the stability method.
{"title":"On even rainbow or nontriangular directed cycles","authors":"A. Czygrinow, T. Molla, B. Nagle, Roy Oursler","doi":"10.4310/joc.2021.v12.n4.a4","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n4.a4","url":null,"abstract":"Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $ell$-cycles $C_{ell}$: if every vertex $v in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n geq n_0(ell)$ is sufficiently large, then $G$ admits an even rainbow $ell$-cycle $C_{ell}$. This result is best possible whenever $ell notequiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer $ell geq 4$, every large $n$-vertex oriented graph $vec{G} = (V, vec{E})$ with minimum outdegree at least $(n+1)/3$ admits a (consistently) directed $ell$-cycle $vec{C}_{ell}$. Our latter result relates to one of Kelly, Kuhn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree. Our proofs are based on the stability method.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"56 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72526004","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 : 2019-11-11DOI: 10.4310/joc.2021.v12.n3.a2
T. Mansour, R. Rastegar
Let $f_n$ be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides. Suppose ${mathcal T}_n$ is a random triangulation, sampled uniformly out of all possible triangulations of $P_n$. We study the sum of weights of triangles in ${mathcal T}_n$ and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of $f_n$ in which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new proofs for already known results such as the degree of a fixed vertex and the number of ears in ${mathcal T}_n,$ as well as, provide new results on the number of "blue" angles and refined information on the distribution of angles at a fixed vertex. We note that our approach is systematic and can be applied to many other new examples while generalizing the existing results.
{"title":"On typical triangulations of a convex $n$-gon","authors":"T. Mansour, R. Rastegar","doi":"10.4310/joc.2021.v12.n3.a2","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n3.a2","url":null,"abstract":"Let $f_n$ be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides. Suppose ${mathcal T}_n$ is a random triangulation, sampled uniformly out of all possible triangulations of $P_n$. We study the sum of weights of triangles in ${mathcal T}_n$ and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of $f_n$ in which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new proofs for already known results such as the degree of a fixed vertex and the number of ears in ${mathcal T}_n,$ as well as, provide new results on the number of \"blue\" angles and refined information on the distribution of angles at a fixed vertex. We note that our approach is systematic and can be applied to many other new examples while generalizing the existing results.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"62 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84662730","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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