Pub Date : 2019-08-29DOI: 10.4310/joc.2023.v14.n2.a1
Gidon Orelowitz
The $textit{Edelman-Greene statistic}$ of S. Billey-B. Pawlowski measures the "shortness" of the Schur expansion of a Stanley symmetric function. We show that the maximum value of this statistic on permutations of Coxeter length $n$ is the number of involutions in the symmetric group $S_n$, and explicitly describe the permutations that attain this maximum. Our proof confirms a recent conjecture of C. Monical, B. Pankow, and A. Yong: we give an explicit combinatorial injection between a certain collections of Edelman-Greene tableaux and standard Young tableaux.
S. billey的$textit{Edelman-Greene statistic}$。Pawlowski测量了Stanley对称函数的Schur展开的“短度”。我们证明了这个统计量在cox长度$n$的排列上的最大值是对称群$S_n$中的对合数，并明确地描述了达到这个最大值的排列。我们的证明证实了C. Monical, B. Pankow和a . Yong最近的一个猜想:我们在Edelman-Greene的某些集合和标准Young的集合之间给出了一个明确的组合注入。
{"title":"Maximizing the Edelman–Greene statistic","authors":"Gidon Orelowitz","doi":"10.4310/joc.2023.v14.n2.a1","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n2.a1","url":null,"abstract":"The $textit{Edelman-Greene statistic}$ of S. Billey-B. Pawlowski measures the \"shortness\" of the Schur expansion of a Stanley symmetric function. We show that the maximum value of this statistic on permutations of Coxeter length $n$ is the number of involutions in the symmetric group $S_n$, and explicitly describe the permutations that attain this maximum. Our proof confirms a recent conjecture of C. Monical, B. Pankow, and A. Yong: we give an explicit combinatorial injection between a certain collections of Edelman-Greene tableaux and standard Young tableaux.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"71 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76546108","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 : 2019-07-23DOI: 10.4310/joc.2021.v12.n4.a3
Anum Khalid, T. Prellberg
In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Melou, who expressed generating functions of discrete excursions in terms of rectangular Schur functions.
{"title":"Skew Schur function representation of directed paths in a slit","authors":"Anum Khalid, T. Prellberg","doi":"10.4310/joc.2021.v12.n4.a3","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n4.a3","url":null,"abstract":"In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Melou, who expressed generating functions of discrete excursions in terms of rectangular Schur functions.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88563224","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 : 2019-06-27DOI: 10.4310/joc.2022.v13.n4.a4
M. Furuya, Tamae Kawasaki
For $cin mathbb{R}^{+}cup {infty }$ and a graph $G$, a function $f:V(G)rightarrow {0,1,c}$ is called a $c$-self dominating function of $G$ if for every vertex $uin V(G)$, $f(u)geq c$ or $max{f(v):vin N_{G}(u)}geq 1$ where $N_{G}(u)$ is the neighborhood of $u$ in $G$. The minimum weight $w(f)=sum _{uin V(G)}f(u)$ of a $c$-self dominating function $f$ of $G$ is called the $c$-self domination number of $G$. The $c$-self domination concept is a common generalization of three domination-type invariants; (original) domination, total domination and Roman domination. In this paper, we study a behavior of the $c$-self domination number in random graphs for small $c$.
{"title":"Small domination-type invariants in random graphs","authors":"M. Furuya, Tamae Kawasaki","doi":"10.4310/joc.2022.v13.n4.a4","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n4.a4","url":null,"abstract":"For $cin mathbb{R}^{+}cup {infty }$ and a graph $G$, a function $f:V(G)rightarrow {0,1,c}$ is called a $c$-self dominating function of $G$ if for every vertex $uin V(G)$, $f(u)geq c$ or $max{f(v):vin N_{G}(u)}geq 1$ where $N_{G}(u)$ is the neighborhood of $u$ in $G$. The minimum weight $w(f)=sum _{uin V(G)}f(u)$ of a $c$-self dominating function $f$ of $G$ is called the $c$-self domination number of $G$. The $c$-self domination concept is a common generalization of three domination-type invariants; (original) domination, total domination and Roman domination. In this paper, we study a behavior of the $c$-self domination number in random graphs for small $c$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"35 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85096723","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 : 2019-05-20DOI: 10.4310/joc.2020.v11.n3.a7
Paul Jung, Greg Markowsky
A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in question is not regular.
{"title":"Remarks on the recurrence and transience of non-backtracking random walks","authors":"Paul Jung, Greg Markowsky","doi":"10.4310/joc.2020.v11.n3.a7","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n3.a7","url":null,"abstract":"A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in question is not regular.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"13 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74024714","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 : 2019-05-09DOI: 10.4310/joc.2021.v12.n3.a6
J. Balogh, William Linz, Leticia Mattos
Define $T_k$ as the minimal $tin mathbb{N}$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for all $nin mathbb{N}$. Jungic, Licht (Fox), Mahdian, Nesetril and Radoicic proved that $lfloor{frac{k^2}{4}rfloor}le T_k$. We almost close the gap between the upper and lower bounds by proving that $T_k le k^2e^{(lnln k)^2(1+o(1))}$. Conlon, Fox and Sudakov have independently shown a stronger statement that $T_k=O(k^2log k)$.
{"title":"Long rainbow arithmetic progressions","authors":"J. Balogh, William Linz, Leticia Mattos","doi":"10.4310/joc.2021.v12.n3.a6","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n3.a6","url":null,"abstract":"Define $T_k$ as the minimal $tin mathbb{N}$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for all $nin mathbb{N}$. Jungic, Licht (Fox), Mahdian, Nesetril and Radoicic proved that $lfloor{frac{k^2}{4}rfloor}le T_k$. We almost close the gap between the upper and lower bounds by proving that $T_k le k^2e^{(lnln k)^2(1+o(1))}$. Conlon, Fox and Sudakov have independently shown a stronger statement that $T_k=O(k^2log k)$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87190225","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 : 2019-05-05DOI: 10.4310/JOC.2021.v12.n2.a1
D. Mubayi, S. Mukherjee
Given a family of hypergraphs $mathcal H$, let $f(m,mathcal H)$ denote the largest size of an $mathcal H$-free subgraph that one is guaranteed to find in every hypergraph with $m$ edges. This function was first introduced by Erdős and Komlos in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families ${mathcal H_m}$ have bounded $f(m,mathcal H_m)$ as $mtoinfty$? A variety of bounds for $f(m,mathcal H_m)$ are obtained which answer this question in some cases. Obtaining a complete description of sequences ${mathcal H_m}$ for which $f(m,mathcal H_m)$ is bounded seems hopeless.
{"title":"Maximum $mathcal{H}$-free subgraphs","authors":"D. Mubayi, S. Mukherjee","doi":"10.4310/JOC.2021.v12.n2.a1","DOIUrl":"https://doi.org/10.4310/JOC.2021.v12.n2.a1","url":null,"abstract":"Given a family of hypergraphs $mathcal H$, let $f(m,mathcal H)$ denote the largest size of an $mathcal H$-free subgraph that one is guaranteed to find in every hypergraph with $m$ edges. This function was first introduced by Erdős and Komlos in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families ${mathcal H_m}$ have bounded $f(m,mathcal H_m)$ as $mtoinfty$? A variety of bounds for $f(m,mathcal H_m)$ are obtained which answer this question in some cases. Obtaining a complete description of sequences ${mathcal H_m}$ for which $f(m,mathcal H_m)$ is bounded seems hopeless.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"102 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87859560","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 : 2019-04-04DOI: 10.4310/joc.2020.v11.n3.a5
Colin Defant
Let $s$ denote West's stack-sorting map. A permutation is called $t-textit{sorted}$ if it is of the form $s^t(mu)$ for some permutation $mu$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can have is $leftlfloorfrac{n-t}{2}rightrfloor$. When $n$ and $t$ have the same parity and $tgeq 2$, we give a simple characterization of those $t$-sorted permutations in $S_n$ that attain this maximum. In particular, the number of such permutations is $(n-t-1)!!$.
{"title":"Descents in $t$-sorted permutations","authors":"Colin Defant","doi":"10.4310/joc.2020.v11.n3.a5","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n3.a5","url":null,"abstract":"Let $s$ denote West's stack-sorting map. A permutation is called $t-textit{sorted}$ if it is of the form $s^t(mu)$ for some permutation $mu$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can have is $leftlfloorfrac{n-t}{2}rightrfloor$. When $n$ and $t$ have the same parity and $tgeq 2$, we give a simple characterization of those $t$-sorted permutations in $S_n$ that attain this maximum. In particular, the number of such permutations is $(n-t-1)!!$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"4 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73310718","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 : 2019-03-25DOI: 10.4310/joc.2022.v13.n1.a4
A. Bonato, Jane Breen, Boris Brimkov, Joshua Carlson, Sean English, Jesse T. Geneson, L. Hogben, K. Perry, Carolyn Reinhart
The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= Omega(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(sqrt{n})$. We also introduce the product cop throttling number $th_c^{times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{times}=1+rad(G)$.
{"title":"Optimizing the trade-off between number of cops and capture time in Cops and Robbers","authors":"A. Bonato, Jane Breen, Boris Brimkov, Joshua Carlson, Sean English, Jesse T. Geneson, L. Hogben, K. Perry, Carolyn Reinhart","doi":"10.4310/joc.2022.v13.n1.a4","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n1.a4","url":null,"abstract":"The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= Omega(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(sqrt{n})$. We also introduce the product cop throttling number $th_c^{times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{times}=1+rad(G)$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"139 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77989553","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 : 2019-03-11DOI: 10.4310/joc.2021.v12.n2.a3
K. Markstrom, Carsten Thomassen
In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A striking difference between the graph, $r=2$, and the hypergraph, $ r geq 3 $, cases is that in the first case both the existence threshold and the counting function are non-linear in the involved densities, whereas for hypergraphs they are given by a linear function. Also, the smallest density of the $r$-partite parts needed to ensure the existence of a complete $r$-graph with $(r+1)$ vertices is equal to the golden ratio $tau=0.618ldots$ for $r=2$, while it is $frac{r}{r+1}$for $rgeq3$.
{"title":"Partite Turán-densities for complete $r$-uniform hypergraphs on $r+1$ vertices","authors":"K. Markstrom, Carsten Thomassen","doi":"10.4310/joc.2021.v12.n2.a3","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n2.a3","url":null,"abstract":"In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A striking difference between the graph, $r=2$, and the hypergraph, $ r geq 3 $, cases is that in the first case both the existence threshold and the counting function are non-linear in the involved densities, whereas for hypergraphs they are given by a linear function. Also, the smallest density of the $r$-partite parts needed to ensure the existence of a complete $r$-graph with $(r+1)$ vertices is equal to the golden ratio $tau=0.618ldots$ for $r=2$, while it is $frac{r}{r+1}$for $rgeq3$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90187268","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 : 2019-02-24DOI: 10.4310/joc.2021.v12.n4.a5
D. Kroes, Sam Spiro
We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce various properties of these random graphs. In particular, we asymptotically determine the expected number of edges and isolated vertices, and more generally we determine the expected number of (induced) subgraphs isomorphic to a given connected graph.
{"title":"Random graphs induced by Catalan pairs","authors":"D. Kroes, Sam Spiro","doi":"10.4310/joc.2021.v12.n4.a5","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n4.a5","url":null,"abstract":"We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce various properties of these random graphs. In particular, we asymptotically determine the expected number of edges and isolated vertices, and more generally we determine the expected number of (induced) subgraphs isomorphic to a given connected graph.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"62 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85621015","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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