We give a simple proof of a generalization of an inequality for homomorphism counts by Sidorenko (1994). A special case of our inequality says that if $d_v$ denotes the degree of a vertex $v$ in a graph $G$ and $textrm{Hom}_Delta(H,G)$ denotes the number of homomorphisms from a connected graph $H$ on $h$ vertices to $G$ which map a particular vertex of $H$ to a vertex $v$ in $G$ with $d_v ge Delta$, then $textrm{Hom}_Delta(H,G) le sum_{vin G} d_v^{h-1}mathbf{1}_{d_vge Delta}$. We use this inequality to study the minimum sample size needed to estimate the number of copies of $H$ in $G$ by sampling vertices of $G$ at random.
我们给出了Sidorenko(1994)关于同态计数不等式推广的一个简单证明。我们不等式的一个特殊情况是，如果$d_v$表示图中顶点$v$的度，$G$和$textrm{Hom}_Delta(H,G)$表示连接图$H$在$h$上的顶点到$G$的同态数，这些同态数将$H$的特定顶点映射到$G$中的顶点$v$与$d_v ge Delta$，则$textrm{Hom}_Delta(H,G) le sum_{vin G} d_v^{h-1}mathbf{1}_{d_vge Delta}$。我们使用这个不等式来研究通过随机采样$G$的顶点来估计$G$中$H$的副本数量所需的最小样本量。
{"title":"Estimating Global Subgraph Counts by Sampling","authors":"S. Janson, Valentas Kurauskas","doi":"10.37236/11618","DOIUrl":"https://doi.org/10.37236/11618","url":null,"abstract":"We give a simple proof of a generalization of an inequality for homomorphism counts by Sidorenko (1994). A special case of our inequality says that if $d_v$ denotes the degree of a vertex $v$ in a graph $G$ and $textrm{Hom}_Delta(H,G)$ denotes the number of homomorphisms from a connected graph $H$ on $h$ vertices to $G$ which map a particular vertex of $H$ to a vertex $v$ in $G$ with $d_v ge Delta$, then $textrm{Hom}_Delta(H,G) le sum_{vin G} d_v^{h-1}mathbf{1}_{d_vge Delta}$. \u0000We use this inequality to study the minimum sample size needed to estimate the number of copies of $H$ in $G$ by sampling vertices of $G$ at random.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86267636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Luiz Emilio Allem, Elismar R. Oliveira, Fernando Tura
A graph is said to be $I$-eigenvalue free if it has no eigenvalues in the interval $I$ with respect to the adjacency matrix $A$. In this paper we present twoalgorithms for generating $I$-eigenvalue free threshold graphs.
{"title":"Generating $I$-Eigenvalue Free Threshold Graphs","authors":"Luiz Emilio Allem, Elismar R. Oliveira, Fernando Tura","doi":"10.37236/11225","DOIUrl":"https://doi.org/10.37236/11225","url":null,"abstract":"A graph is said to be $I$-eigenvalue free if it has no eigenvalues in the interval $I$ with respect to the adjacency matrix $A$. In this paper we present twoalgorithms for generating $I$-eigenvalue free threshold graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135626385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $rm{Hom}_mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.
{"title":"The Complexity of the Matroid Homomorphism Problem","authors":"Cheolwon Heo, Hyobin Kim, Siggers Mark","doi":"10.37236/11119","DOIUrl":"https://doi.org/10.37236/11119","url":null,"abstract":"We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $rm{Hom}_mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"86 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78187789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paweł Gawrychowski, Tomasz Kociumaka, Wojciech Rytter, Tomasz Waleń
For an undirected tree with edges labeled by single letters, we consider its substrings, which are labels of the simple paths between two nodes. A palindrome is a word $w$ equal to its reverse $w^R$. We prove that the maximum number of distinct palindromic substrings in a tree of $n$ edges satisfies $text{pal}(n)=O(n^{1.5})$. This solves an open problem of Brlek, Lafrenière, and Provençal (DLT 2015), who showed that $text{pal}(n)=Omega(n^{1.5})$. Hence, we settle the tight bound of $Theta(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for trees that are simple paths, the maximum palindromic complexity is exactly $n+1$.
We also propose an $O(n^{1.5} log^{0.5}{n})$-time algorithm reporting all distinct palindromes and an $O(n log^2 n)$-time algorithm finding the longest palindrome in a tree.
{"title":"Tight Bound for the Number of Distinct Palindromes in a Tree","authors":"Paweł Gawrychowski, Tomasz Kociumaka, Wojciech Rytter, Tomasz Waleń","doi":"10.37236/10842","DOIUrl":"https://doi.org/10.37236/10842","url":null,"abstract":"For an undirected tree with edges labeled by single letters, we consider its substrings, which are labels of the simple paths between two nodes. A palindrome is a word $w$ equal to its reverse $w^R$. We prove that the maximum number of distinct palindromic substrings in a tree of $n$ edges satisfies $text{pal}(n)=O(n^{1.5})$. This solves an open problem of Brlek, Lafrenière, and Provençal (DLT 2015), who showed that $text{pal}(n)=Omega(n^{1.5})$. Hence, we settle the tight bound of $Theta(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for trees that are simple paths, the maximum palindromic complexity is exactly $n+1$.
 We also propose an $O(n^{1.5} log^{0.5}{n})$-time algorithm reporting all distinct palindromes and an $O(n log^2 n)$-time algorithm finding the longest palindrome in a tree.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135463998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rigidity is the property of a structure that does not flex under an applied force. In the past several decades, the rigidity of graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained a combinatorial characterization of rigid graphs in $mathbb{R}^2$. Lovász and Yemini (1982) proved that every $6$-connected graph is rigid in $mathbb{R}^2$. Jackson and Jordán (2005) strengthened this result, and showed that every $6$-connected graph is globally rigid in $mathbb{R}^2$. Thus every graph with algebraic connectivity greater than $5$ is globally rigid in $mathbb{R}^2$. In 2021, Cioabă, Dewar and Gu improved this bound, and proved that every graph with minimum degree at least $6$ and algebraic connectivity greater than $2+frac{1}{delta-1}$ (resp., $2+frac{2}{delta-1}$) is rigid (resp., globally rigid) in $mathbb{R}^2$. In this paper, we study the rigidity of graphs in $mathbb{R}^2$ from the viewpoint of adjacency eigenvalues. Specifically, we provide a spectral radius condition for the rigidity (resp., globally rigidity) of $2$-connected (resp., $3$-connected) graphs with given minimum degree. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order $n$.
{"title":"Spectral Radius Conditions for the Rigidity of Graphs","authors":"Dandan Fan, Xueyi Huang, Huiqiu Lin","doi":"10.37236/11308","DOIUrl":"https://doi.org/10.37236/11308","url":null,"abstract":"Rigidity is the property of a structure that does not flex under an applied force. In the past several decades, the rigidity of graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained a combinatorial characterization of rigid graphs in $mathbb{R}^2$. Lovász and Yemini (1982) proved that every $6$-connected graph is rigid in $mathbb{R}^2$. Jackson and Jordán (2005) strengthened this result, and showed that every $6$-connected graph is globally rigid in $mathbb{R}^2$. Thus every graph with algebraic connectivity greater than $5$ is globally rigid in $mathbb{R}^2$. In 2021, Cioabă, Dewar and Gu improved this bound, and proved that every graph with minimum degree at least $6$ and algebraic connectivity greater than $2+frac{1}{delta-1}$ (resp., $2+frac{2}{delta-1}$) is rigid (resp., globally rigid) in $mathbb{R}^2$. In this paper, we study the rigidity of graphs in $mathbb{R}^2$ from the viewpoint of adjacency eigenvalues. Specifically, we provide a spectral radius condition for the rigidity (resp., globally rigidity) of $2$-connected (resp., $3$-connected) graphs with given minimum degree. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order $n$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90893445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study two building games and two removing games played on a finite hypergraph. In each game two players take turns selecting vertices of the hypergraph until the set of jointly selected vertices satisfies a condition related to the edges of the hypergraph. The winner is the last player able to move. The building achievement game ends as soon as the set of selected vertices contains an edge. In the building avoidance game the players are not allowed to select a set that contains an edge. The removing achievement game ends as soon as the complement of the set of selected vertices no longer contains an edge. In the removing avoidance game the players are not allowed to select a set whose complement does not contain an edge. We develop some generic tools for finding the nim-value of these games and show that the nim-value can be an arbitrary nonnegative integer. The outcome of many of these games were previously determined for several special cases in algebraic and combinatorial settings. We provide several examples and show how our tools can be used to refine these results by finding nim-values.
{"title":"Impartial Hypergraph Games","authors":"Nándor Sieben","doi":"10.37236/11665","DOIUrl":"https://doi.org/10.37236/11665","url":null,"abstract":"We study two building games and two removing games played on a finite hypergraph. In each game two players take turns selecting vertices of the hypergraph until the set of jointly selected vertices satisfies a condition related to the edges of the hypergraph. The winner is the last player able to move. The building achievement game ends as soon as the set of selected vertices contains an edge. In the building avoidance game the players are not allowed to select a set that contains an edge. The removing achievement game ends as soon as the complement of the set of selected vertices no longer contains an edge. In the removing avoidance game the players are not allowed to select a set whose complement does not contain an edge. We develop some generic tools for finding the nim-value of these games and show that the nim-value can be an arbitrary nonnegative integer. The outcome of many of these games were previously determined for several special cases in algebraic and combinatorial settings. We provide several examples and show how our tools can be used to refine these results by finding nim-values.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81032360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper considers ribbon tilings of large regions and their per-tile entropy (the logarithm of the number of tilings divided by the number of tiles). For tilings of general regions by tiles of length $n$, we give an upper bound on the per-tile entropy as $n - 1$. For growing rectangular regions, we prove the existence of the asymptotic per tile entropy and show that it is bounded from below by $log_2 (n/e)$ and from above by $log_2(en)$. For growing generalized "Aztec Diamond" regions and for growing "stair" regions, the asymptotic per-tile entropy is calculated exactly as $1/2$ and $log_2(n + 1) - 1$, respectively.
{"title":"On Enumeration and Entropy of Ribbon Tilings","authors":"Yinsong Chen, V. Kargin","doi":"10.37236/10991","DOIUrl":"https://doi.org/10.37236/10991","url":null,"abstract":"The paper considers ribbon tilings of large regions and their per-tile entropy (the logarithm of the number of tilings divided by the number of tiles). For tilings of general regions by tiles of length $n$, we give an upper bound on the per-tile entropy as $n - 1$. For growing rectangular regions, we prove the existence of the asymptotic per tile entropy and show that it is bounded from below by $log_2 (n/e)$ and from above by $log_2(en)$. For growing generalized \"Aztec Diamond\" regions and for growing \"stair\" regions, the asymptotic per-tile entropy is calculated exactly as $1/2$ and $log_2(n + 1) - 1$, respectively.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"59 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74589673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $Gtimes P_{2n+1}$ ($ngeqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G times P_{n}$ by the eigenvalues of $G$.
如果$ G-V(H) $有完美匹配，那么图形$G$的子图$ H $就很好。有向图中的偶循环$ C $是奇异有向的，如果对于$ C $周围的任意一个遍历方向的选择，$C$沿遍历方向的边数是奇数。具有偶数个顶点的图形$ G $的方向$ D $是Pfaffian，如果$ G $的每个好循环在$ D $中都有奇怪的方向。设$ P_{n} $表示$ n $顶点上的路径。Pfaffian图$G times P_{2n} $是由Lu和Zhang确定的[图的笛卡尔积的Pfaffian性质，J. Comb.]。优化，27(2014)530—540]。在本文中，我们描述了关于$G$的禁止子图的Pfaffian图$ G times P_{2n+1} $。我们首先给出$Gtimes P_{2n+1}$ ($ngeqslant 2$)是可行的充要条件。然后我们刻画了$G$为二部图时的Pfaffian图$ G times P_{3} $，并将这一结果推广到$G$只包含一个奇循环的情况。根据这些结果，我们根据$G$的方向图的特征值枚举了Pfaffian图$G times P_{n}$的完美匹配次数，并通过$G$的特征值计算了某些Pfaffian图$G times P_{n}$的完美匹配次数。
{"title":"Enumeration of Perfect Matchings of the Cartesian Products of Graphs","authors":"Wei Li, Yao Wang","doi":"10.37236/11141","DOIUrl":"https://doi.org/10.37236/11141","url":null,"abstract":"A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $Gtimes P_{2n+1}$ ($ngeqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G times P_{n}$ by the eigenvalues of $G$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"28 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81591769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Half graphs and their variants, such as semi-ladders and co-matchings, are configurations that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabiański et al. (STACS; 2019) prove that in powers of sparse graphs, one cannot find arbitrarily large configurations of this kind. However, these proofs either are non-constructive, or provide only loose upper bounds on the orders of half graphs and semi-ladders.In this work we provide nearly tight asymptotic lower and upper bounds on the maximum order of half graphs, parameterized by the power, in the following classes of sparse graphs: planar graphs, graphs with bounded maximum degree, graphs with bounded pathwidth or treewidth, and graphs excluding a fixed clique as a minor.
The most significant part of our work is the upper bound for planar graphs. Here, we employ techniques of structural graph theory to analyze semi-ladders in planar graphs via the notion of cages, which expose a topological structure in semi-ladders. As an essential building block of this proof, we also state and prove a new structural result, yielding a fully polynomial bound on the neighborhood complexity in the class of planar graphs.
{"title":"Bounds on Half Graph Orders in Powers of Sparse Graphs","authors":"Marek Sokołowski","doi":"10.37236/11063","DOIUrl":"https://doi.org/10.37236/11063","url":null,"abstract":"Half graphs and their variants, such as semi-ladders and co-matchings, are configurations that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabiański et al. (STACS; 2019) prove that in powers of sparse graphs, one cannot find arbitrarily large configurations of this kind. However, these proofs either are non-constructive, or provide only loose upper bounds on the orders of half graphs and semi-ladders.In this work we provide nearly tight asymptotic lower and upper bounds on the maximum order of half graphs, parameterized by the power, in the following classes of sparse graphs: planar graphs, graphs with bounded maximum degree, graphs with bounded pathwidth or treewidth, and graphs excluding a fixed clique as a minor.
 The most significant part of our work is the upper bound for planar graphs. Here, we employ techniques of structural graph theory to analyze semi-ladders in planar graphs via the notion of cages, which expose a topological structure in semi-ladders. As an essential building block of this proof, we also state and prove a new structural result, yielding a fully polynomial bound on the neighborhood complexity in the class of planar graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135742548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $n>1$ be an odd integer, and let $zeta$ be a primitive $n$th root of unity in the complex field. Via the Eigenvector-eigenvalue Identity, we show that$$sum_{tauin D(n-1)}mathrm{sign}(tau)prod_{j=1}^{n-1}frac{1+zeta^{j-tau(j)}}{1-zeta^{j-tau(j)}}=(-1)^{frac{n-1}{2}}frac{((n-2)!!)^2}{n},$$where $D(n-1)$ is the set of all derangements of $1,ldots,n-1$.This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $delta=0,1$ we determine the value of $det[x+m_{jk}]_{1leqslant j,kleqslant n-1}$ completely, where$$m_{jk}=begin{cases}(1+zeta^{j-k})/(1-zeta^{j-k})&text{if} jnot=k,delta&text{if} j=k.end{cases}$$
{"title":"Proof of a Conjecture Involving Derangements and Roots of Unity","authors":"H. Wang, Zhi-Wei Sun","doi":"10.37236/11377","DOIUrl":"https://doi.org/10.37236/11377","url":null,"abstract":"Let $n>1$ be an odd integer, and let $zeta$ be a primitive $n$th root of unity in the complex field. Via the Eigenvector-eigenvalue Identity, we show that$$sum_{tauin D(n-1)}mathrm{sign}(tau)prod_{j=1}^{n-1}frac{1+zeta^{j-tau(j)}}{1-zeta^{j-tau(j)}}=(-1)^{frac{n-1}{2}}frac{((n-2)!!)^2}{n},$$where $D(n-1)$ is the set of all derangements of $1,ldots,n-1$.This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $delta=0,1$ we determine the value of $det[x+m_{jk}]_{1leqslant j,kleqslant n-1}$ completely, where$$m_{jk}=begin{cases}(1+zeta^{j-k})/(1-zeta^{j-k})&text{if} jnot=k,delta&text{if} j=k.end{cases}$$","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79945746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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