Pub Date : 2020-09-11DOI: 10.4310/joc.2023.v14.n4.a1
Joseph S. Alameda, J. Kritschgau, Michael Young
Vertex leaky forcing was recently introduced as a new variation of zero forcing in order to show how vertex leaks can disrupt the zero forcing process in a graph. An edge leak is an edge that is not allowed to be forced across during the zero forcing process. The $ell$-edge-leaky forcing number of a graph is the size of a smallest zero forcing set that can force the graph blue despite $ell$ edge leaks. This paper contains an analysis of the effect of edge leaks on the zero forcing process instead of vertex leaks. Furthermore, specified $ell$-leaky forcing is introduced. The main result is that $ell$-leaky forcing, $ell$-edge-leaky forcing, and specified $ell$-leaky forcing are equivalent. Furthermore, all of these different kinds of leaks can be mixed so that vertex leaks, edge leaks, and specified leaks are used. This mixed $ell$-leaky forcing number is also the same as the (vertex) $ell$-leaky forcing number.
最近引入了顶点泄漏强迫作为零强迫的新变体，以显示顶点泄漏如何在图中破坏零强迫过程。边缘泄漏是在零强迫过程中不允许被强迫穿过的边缘。图的$ well $-edge-leaky强迫数是一个最小的零强迫集的大小，它可以在$ well $ edge泄漏的情况下迫使图变成蓝色。本文分析了边缘泄漏对零强迫过程的影响，而不是顶点泄漏。此外，还引入了指定的$ well $泄漏强迫。主要结果是$ well $-leaky强迫、$ well $-edge-leaky强迫和指定的$ well $-leaky强迫是等效的。此外，所有这些不同类型的泄漏都可以混合使用，这样就可以使用顶点泄漏、边缘泄漏和指定泄漏。这个混合$ well $-leaky强迫数也与(顶点)$ well $-leaky强迫数相同。
{"title":"Generalizations of leaky forcing","authors":"Joseph S. Alameda, J. Kritschgau, Michael Young","doi":"10.4310/joc.2023.v14.n4.a1","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n4.a1","url":null,"abstract":"Vertex leaky forcing was recently introduced as a new variation of zero forcing in order to show how vertex leaks can disrupt the zero forcing process in a graph. An edge leak is an edge that is not allowed to be forced across during the zero forcing process. The $ell$-edge-leaky forcing number of a graph is the size of a smallest zero forcing set that can force the graph blue despite $ell$ edge leaks. This paper contains an analysis of the effect of edge leaks on the zero forcing process instead of vertex leaks. Furthermore, specified $ell$-leaky forcing is introduced. The main result is that $ell$-leaky forcing, $ell$-edge-leaky forcing, and specified $ell$-leaky forcing are equivalent. Furthermore, all of these different kinds of leaks can be mixed so that vertex leaks, edge leaks, and specified leaks are used. This mixed $ell$-leaky forcing number is also the same as the (vertex) $ell$-leaky forcing number.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"54 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90109430","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 : 2020-09-06DOI: 10.4310/joc.2022.v13.n1.a5
Connor Ahlbach, Jacob David, Suho Oh, Christopher Wu
If one attaches shifted copies of a skew tableau to the right of itself and rectifies, at a certain point the copies no longer experience vertical slides, a phenomenon called tableau stabilization. While tableau stabilization was originally developed to construct the sufficiently large rectangular tableaux fixed by given powers of promotion, the purpose of this paper is to improve the original bound on tableau stabilization to the number of rows of the skew tableau. In order to prove this bound, we encode increasing subsequences as lattice paths and show that various operations on these lattice paths weakly increase the maximum combined length of the increasing subsequences.
{"title":"Tableau stabilization and lattice paths","authors":"Connor Ahlbach, Jacob David, Suho Oh, Christopher Wu","doi":"10.4310/joc.2022.v13.n1.a5","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n1.a5","url":null,"abstract":"If one attaches shifted copies of a skew tableau to the right of itself and rectifies, at a certain point the copies no longer experience vertical slides, a phenomenon called tableau stabilization. While tableau stabilization was originally developed to construct the sufficiently large rectangular tableaux fixed by given powers of promotion, the purpose of this paper is to improve the original bound on tableau stabilization to the number of rows of the skew tableau. In order to prove this bound, we encode increasing subsequences as lattice paths and show that various operations on these lattice paths weakly increase the maximum combined length of the increasing subsequences.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"220 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76602508","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 : 2020-09-04DOI: 10.4310/joc.2022.v13.n3.a3
Stephen Lacina
Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called $s$-weak order. They also generalized the Tamari lattice by defining a particular sublattice of $s$-weak order called the $s$-Tamari lattice. We prove that the homotopy type of each open interval in $s$-weak order and in the $s$-Tamari lattice is either a ball or sphere. We do this by giving $s$-weak order and the $s$-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to spheres and which are homotopy equivalent to balls; we also determine the dimension of the spheres for the intervals yielding spheres.
{"title":"Poset topology of $s$ weak order via SB-labelings","authors":"Stephen Lacina","doi":"10.4310/joc.2022.v13.n3.a3","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n3.a3","url":null,"abstract":"Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called $s$-weak order. They also generalized the Tamari lattice by defining a particular sublattice of $s$-weak order called the $s$-Tamari lattice. We prove that the homotopy type of each open interval in $s$-weak order and in the $s$-Tamari lattice is either a ball or sphere. We do this by giving $s$-weak order and the $s$-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to spheres and which are homotopy equivalent to balls; we also determine the dimension of the spheres for the intervals yielding spheres.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90382787","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 : 2020-05-26DOI: 10.4310/joc.2022.v13.n3.a2
Hiranya Kishore Dey, S. Sivasubramanian
Central Limit Theorems are known for the Eulerian statistic "descent" (or "excedance") in the symmetric group $SSS_n$. Recently, Fulman, Kim, Lee and Petersen gave a Central Limit Theorem for "descent" over the alternating group $AAA_n$ and also gave a Carlitz identity in $AAA_n$ using descents. In this paper, we give a Central Limit Theorem in $AAA_n$ involving excedances. We extend these to the positive elements in type B and type D Coxeter groups. Boroweic and Mlotkowski enumerated type B descents over $DD_n$, the type D Coxeter group and gave similar results. We refine their results for both the positive and negative part of $DD_n$. Our results are a consequence of signed enumeration over these subsets.
中心极限定理以对称群$SSS_n$中的欧拉统计量“下降”(或“超越”)而闻名。最近，Fulman, Kim, Lee和Petersen给出了交替群$AAA_n$上“下降”的中心极限定理，并利用下降给出了$AAA_n$上的Carlitz恒等式。本文给出了$AAA_n$中一个涉及超越的中心极限定理。我们将这些扩展到B型和D型考克斯特组中的正元素。Boroweic和Mlotkowski列举了D型Coxeter组$DD_n$上的B型下降，并给出了类似的结果。我们针对$DD_n$的正负部分改进了他们的结果。我们的结果是对这些子集进行有符号枚举的结果。
{"title":"Eulerian central limit theorems and Carlitz identities in positive elements of classical Weyl groups","authors":"Hiranya Kishore Dey, S. Sivasubramanian","doi":"10.4310/joc.2022.v13.n3.a2","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n3.a2","url":null,"abstract":"Central Limit Theorems are known for the Eulerian statistic \"descent\" (or \"excedance\") in the symmetric group $SSS_n$. Recently, Fulman, Kim, Lee and Petersen gave a Central Limit Theorem for \"descent\" over the alternating group $AAA_n$ and also gave a Carlitz identity in $AAA_n$ using descents. \u0000In this paper, we give a Central Limit Theorem in $AAA_n$ involving excedances. We extend these to the positive elements in type B and type D Coxeter groups. Boroweic and Mlotkowski enumerated type B descents over $DD_n$, the type D Coxeter group and gave similar results. We refine their results for both the positive and negative part of $DD_n$. Our results are a consequence of signed enumeration over these subsets.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"2006 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82436112","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 : 2020-05-17DOI: 10.4310/joc.2023.v14.n1.a6
C. J. Casselgren, Herman Goransson
Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, is completable.
{"title":"Completing partial Latin squares with two filled rows and three filled columns","authors":"C. J. Casselgren, Herman Goransson","doi":"10.4310/joc.2023.v14.n1.a6","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n1.a6","url":null,"abstract":"Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, is completable.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75068560","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 : 2020-05-06DOI: 10.4310/joc.2021.v12.n4.a2
D. Mubayi, Andrew Suk
Erdős and Hajnal constructed a 4-coloring of the triples of an $N$-element set such that every $n$-element subset contains 2 triples with distinct colors, and $N$ is double exponential in $n$. Conlon, Fox and Rodl asked whether there is some integer $qge 3$ and a $q$-coloring of the triples of an $N$-element set such that every $n$-element subset has 3 triples with distinct colors, and $N$ is double exponential in $n$. We make the first nontrivial progress on this problem by providing a $q$-coloring with this property for all $qgeq 9$, where $N$ is exponential in $n^{2+cq}$ and $c>0$ is an absolute constant.
{"title":"Cliques with many colors in triple systems","authors":"D. Mubayi, Andrew Suk","doi":"10.4310/joc.2021.v12.n4.a2","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n4.a2","url":null,"abstract":"Erdős and Hajnal constructed a 4-coloring of the triples of an $N$-element set such that every $n$-element subset contains 2 triples with distinct colors, and $N$ is double exponential in $n$. Conlon, Fox and Rodl asked whether there is some integer $qge 3$ and a $q$-coloring of the triples of an $N$-element set such that every $n$-element subset has 3 triples with distinct colors, and $N$ is double exponential in $n$. We make the first nontrivial progress on this problem by providing a $q$-coloring with this property for all $qgeq 9$, where $N$ is exponential in $n^{2+cq}$ and $c>0$ is an absolute constant.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"5 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78863954","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":"Distributions and Combinatorial\u0000 Proofs","authors":"","doi":"10.1090/text/055/02","DOIUrl":"https://doi.org/10.1090/text/055/02","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":" 6","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72381920","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":"Counting Under Equivalence","authors":"","doi":"10.1090/text/055/05","DOIUrl":"https://doi.org/10.1090/text/055/05","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86009474","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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