In this paper, we give a complete, explicit and constructive solution to the double generalized majorization problem. Apart from purely combinatorial interest, double generalized majorization problem has strong impact in Matrix and Matrix Pencils Completion Problems, Bounded Rank Perturbation Problems, and it has additional nice interpretation in Representation Theory of Kronecker Quivers.
{"title":"Double Generalized Majorization","authors":"M. Dodig, M. Stosic","doi":"10.37236/11127","DOIUrl":"https://doi.org/10.37236/11127","url":null,"abstract":"\u0000 \u0000 \u0000 \u0000 \u0000 \u0000In this paper, we give a complete, explicit and constructive solution to the double generalized majorization problem. Apart from purely combinatorial interest, double generalized majorization problem has strong impact in Matrix and Matrix Pencils Completion Problems, Bounded Rank Perturbation Problems, and it has additional nice interpretation in Representation Theory of Kronecker Quivers. \u0000 \u0000 \u0000 \u0000 \u0000 \u0000","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"354 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78115759","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}
Alvaro Carbonero, Willem Fletcher, Jing Guo, A. Gyárfás, Rona Wang, Shiyu Yan
A 3-graph is a pair H = (V, E) of sets, where elements of V are called points or vertices and E contains some 3-element subsets of V , called edges. A 3-graph is called linear if any two distinct edges intersect in at most one vertex.There is a recent interest in extremal properties of 3-graphs containing no crown, three pairwise disjoint edges and a fourth edge which intersects all of them. We show that every linear 3-graph with minimum degree 4 contains a crown. This is not true if 4 is replaced by 3.
{"title":"Crowns in Linear $3$-Graphs of Minimum Degree $4$","authors":"Alvaro Carbonero, Willem Fletcher, Jing Guo, A. Gyárfás, Rona Wang, Shiyu Yan","doi":"10.37236/11037","DOIUrl":"https://doi.org/10.37236/11037","url":null,"abstract":"A 3-graph is a pair H = (V, E) of sets, where elements of V are called points or vertices and E contains some 3-element subsets of V , called edges. A 3-graph is called linear if any two distinct edges intersect in at most one vertex.There is a recent interest in extremal properties of 3-graphs containing no crown, three pairwise disjoint edges and a fourth edge which intersects all of them. We show that every linear 3-graph with minimum degree 4 contains a crown. This is not true if 4 is replaced by 3.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"75 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90332238","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}
Liam Armstrong, Bryan Ducasse, Thomas Meyer, H. Swisher
In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $n$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities.
{"title":"Generalized Alder-Type Partition Inequalities","authors":"Liam Armstrong, Bryan Ducasse, Thomas Meyer, H. Swisher","doi":"10.37236/11606","DOIUrl":"https://doi.org/10.37236/11606","url":null,"abstract":"In 2020, Kang and Park conjectured a \"level $2$\" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a \"shift\" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $n$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84110856","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 classify cocovers of a given element of the double affine Weyl semigroup $W_{mathcal{T}}$ with respect to the Bruhat order, specifically when $W_{mathcal{T}}$ is associated to a finite root system that is irreducible and simply laced. We do so by introducing a graphical representation of the length difference set defined by Muthiah and Orr and identifying the cocovering relations with the corners of those graphs. This new method allows us to prove that there are finitely many cocovers of each $x in W_{mathcal{T}}$. Further, we show that the Bruhat intervals in the double affine Bruhat order are finite.
{"title":"Classification of Cocovers in the Double Affine Bruhat Order","authors":"Amanda Welch","doi":"10.37236/10745","DOIUrl":"https://doi.org/10.37236/10745","url":null,"abstract":"We classify cocovers of a given element of the double affine Weyl semigroup $W_{mathcal{T}}$ with respect to the Bruhat order, specifically when $W_{mathcal{T}}$ is associated to a finite root system that is irreducible and simply laced. We do so by introducing a graphical representation of the length difference set defined by Muthiah and Orr and identifying the cocovering relations with the corners of those graphs. This new method allows us to prove that there are finitely many cocovers of each $x in W_{mathcal{T}}$. Further, we show that the Bruhat intervals in the double affine Bruhat order are finite.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74721508","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}
Pub Date : 2022-10-02DOI: 10.48550/arXiv.2210.00508
Siddharth Berera, Andr'es G'omez-Colunga, Joey Lakerdas-Gayle, John L'opez, Mauditra Matin, Daniel Roebuck, E. Rowland, Noam Scully, Juliet Whidden
The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence. For other prefixes, the structure is significantly more complicated. In this paper, we show that $L(p)$ reflects the structure of the ruler sequence for several words $p$. We provide morphisms that generate $L(n)$ for letters $n=1$ and $ngeq3$, and $L(p)$ for most families of two-letter words $p$.
{"title":"The lexicographically least square-free word with a given prefix","authors":"Siddharth Berera, Andr'es G'omez-Colunga, Joey Lakerdas-Gayle, John L'opez, Mauditra Matin, Daniel Roebuck, E. Rowland, Noam Scully, Juliet Whidden","doi":"10.48550/arXiv.2210.00508","DOIUrl":"https://doi.org/10.48550/arXiv.2210.00508","url":null,"abstract":"The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence. For other prefixes, the structure is significantly more complicated. In this paper, we show that $L(p)$ reflects the structure of the ruler sequence for several words $p$. We provide morphisms that generate $L(n)$ for letters $n=1$ and $ngeq3$, and $L(p)$ for most families of two-letter words $p$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"87 8 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83448958","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 $Isubset S=K[x_1,dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}ne(0)$ is called the monomial grade of $I$ and it is denoted by $nu(I)$. Let $d_k$ be the minimum degree of the monomials belonging to $I^{[k]}$. Then, $text{depth}(S/I^{[k]})ge d_k-1$ for all $1le klenu(I)$. The normalized depth function of $I$ is defined as $g_{I}(k)=text{depth}(S/I^{[k]})-(d_k-1)$, $1le klenu(I)$. It is expected that $g_I(k)$ is a non-increasing function for any $I$. In this article we study the behaviour of $g_{I}(k)$ under various operations on monomial ideals. Our main result characterizes all cochordal graphs $G$ such that for the edge ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex. As a far-reaching application, for given integers $1le s
设$Isubset S=K[x_1,dots,x_n]$为无平方项理想，$K$为场。$I$的$k$无平方幂$I^{[k]}$是由所有属于$I^k$的无平方单项式生成的$S$的单项式理想。最大的整数$k$使得$I^{[k]}ne(0)$称为$I$的单项等级，用$nu(I)$表示。设$d_k$为属于$I^{[k]}$的单项式的最小次。然后，$text{depth}(S/I^{[k]})ge d_k-1$为所有$1le klenu(I)$。归一化深度函数$I$定义为$g_{I}(k)=text{depth}(S/I^{[k]})-(d_k-1)$, $1le klenu(I)$。可以预期$g_I(k)$对于任何$I$都是一个不增加的函数。本文研究了$g_{I}(k)$在单项式理想的各种运算下的行为。我们的主要结果表征了所有的弦图$G$，对于$G$的边理想$I(G)$，我们有$g_{I(G)}(1)=0$。它们都是弦图$G$，它们的互补图$G^c$是连通的，并且有一个切顶点。作为一个深远的应用，对于给定的整数$1le s
{"title":"Behaviour of the Normalized Depth Function","authors":"A. Ficarra, J. Herzog, T. Hibi","doi":"10.37236/11611","DOIUrl":"https://doi.org/10.37236/11611","url":null,"abstract":"Let $Isubset S=K[x_1,dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}ne(0)$ is called the monomial grade of $I$ and it is denoted by $nu(I)$. Let $d_k$ be the minimum degree of the monomials belonging to $I^{[k]}$. Then, $text{depth}(S/I^{[k]})ge d_k-1$ for all $1le klenu(I)$. The normalized depth function of $I$ is defined as $g_{I}(k)=text{depth}(S/I^{[k]})-(d_k-1)$, $1le klenu(I)$. It is expected that $g_I(k)$ is a non-increasing function for any $I$. In this article we study the behaviour of $g_{I}(k)$ under various operations on monomial ideals. Our main result characterizes all cochordal graphs $G$ such that for the edge ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex. As a far-reaching application, for given integers $1le s<m$ we construct a graph $G$ such that $nu(I(G))=m$ and $g_{I(G)}(k)=0$ if and only if $k=s+1,dots,m$. Finally, we show that any non-increasing function of non-negative integers is the normalized depth function of some squarefree monomial ideal.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79092203","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 analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest.From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic planar maps, which allow us to recover known results in a more general and transparent way.This approach allows us to obtain new enumerative results. Concerning random maps, we first obtain the distribution of the degree of the root face, which has an exponential tail as for other classes of random maps. Our main result is a limiting map-Airy distribution law for the size of the largest block $L$, whose expectation is asymptotically $n/sqrt{3}$ in a random cubic map with $n+2$ faces.We prove analogous results for the size of the largest cubic block, obtained from $L$ by erasing all vertices of degree two, and for the size of the largest 3-connected component, whose expected values are respectively $n/2$ and $n/4$.To obtain these results we need to analyse a new type of composition scheme which has not been treated by Banderier et al. [Random Structures Algorithms 2001].
{"title":"Random Cubic Planar Maps","authors":"M. Drmota, M. Noy, Cl'ement Requil'e, Juanjo Ru'e","doi":"10.37236/11619","DOIUrl":"https://doi.org/10.37236/11619","url":null,"abstract":"We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest.From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic planar maps, which allow us to recover known results in a more general and transparent way.This approach allows us to obtain new enumerative results. \u0000Concerning random maps, we first obtain the distribution of the degree of the root face, which has an exponential tail as for other classes of random maps. Our main result is a limiting map-Airy distribution law for the size of the largest block $L$, whose expectation is asymptotically $n/sqrt{3}$ in a random cubic map with $n+2$ faces.We prove analogous results for the size of the largest cubic block, obtained from $L$ by erasing all vertices of degree two, and for the size of the largest 3-connected component, whose expected values are respectively $n/2$ and $n/4$.To obtain these results we need to analyse a new type of composition scheme which has not been treated by Banderier et al. [Random Structures Algorithms 2001].","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85867165","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 introduce a new family of simple graphs, so called, growing graphs. We investigate ways to modify a given simple graph G combinatorially to obtain a growing graph. One may obtain infinitely many growing graphs from a single simple graph. We show that a growing graph obtained from any given simple graph is Cohen–Macaulay and every Cohen–Macaulay chordal graph is a growing graph. We also prove that under certain conditions, a graph is growing if and only if its clique complex is grafted and give several equivalent conditions in this case. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers and the work of Faridi on grafting of simplicial complexes.
{"title":"Cohen-Macaulay Growing Graphs","authors":"Safyan Ahmad, Fazal Abbas, Shamsa Kanwal","doi":"10.37236/10908","DOIUrl":"https://doi.org/10.37236/10908","url":null,"abstract":"We introduce a new family of simple graphs, so called, growing graphs. We investigate ways to modify a given simple graph G combinatorially to obtain a growing graph. One may obtain infinitely many growing graphs from a single simple graph. We show that a growing graph obtained from any given simple graph is Cohen–Macaulay and every Cohen–Macaulay chordal graph is a growing graph. We also prove that under certain conditions, a graph is growing if and only if its clique complex is grafted and give several equivalent conditions in this case. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers and the work of Faridi on grafting of simplicial complexes.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"50 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74544287","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}
Gian-Carlo Rota conjectured that for any $n$ bases $B_1,B_2,ldots,B_n$ in a matroid of rank $n$, there exist $n$ disjoint transversal bases of $B_1,B_2,ldots,B_n$. The conjecture for graphic matroids corresponds to the problem of an edge-decomposition as follows; If an edge-colored connected multigraph $G$ has $n-1$ colors and the graph induced by the edges colored with $c$ is a spanning tree for each color $c$, then $G$ has $n-1$ mutually edge-disjoint rainbow spanning trees. In this paper, we prove that edge-colored graphs where the edges colored with $c$ induce a spanning star for each color $c$ can be decomposed into rainbow spanning trees.
{"title":"Special Case of Rota's Basis Conjecture on Graphic Matroids","authors":"Shun-ichi Maezawa, Akiko Yazawa","doi":"10.37236/10835","DOIUrl":"https://doi.org/10.37236/10835","url":null,"abstract":"Gian-Carlo Rota conjectured that for any $n$ bases $B_1,B_2,ldots,B_n$ in a matroid of rank $n$, there exist $n$ disjoint transversal bases of $B_1,B_2,ldots,B_n$. The conjecture for graphic matroids corresponds to the problem of an edge-decomposition as follows; If an edge-colored connected multigraph $G$ has $n-1$ colors and the graph induced by the edges colored with $c$ is a spanning tree for each color $c$, then $G$ has $n-1$ mutually edge-disjoint rainbow spanning trees. In this paper, we prove that edge-colored graphs where the edges colored with $c$ induce a spanning star for each color $c$ can be decomposed into rainbow spanning trees.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81159510","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}
Baber and Talbot asked whether there is an irrational Turán density of a single hypergraph. In this paper, we show that the Lagrangian density of a 4-uniform matching of size 3 is an irrational number. Sidorenko showed that the Lagrangian density of an r-uniform hypergraph F is the same as the Turán density of the extension of F. Therefore, our result gives a positive answer to the question of Baber and Talbot. We also determine the Lagrangian densities of a class of r-uniform hypergraphs on n vertices with θ(n2) edges. As far as we know, for every hypergraph F with known hypergraph Lagrangian density, the number of edges in F is less than the number of its vertices.
{"title":"An Irrational Turán Density via Hypergraph Lagrangian Densities","authors":"Biao Wu","doi":"10.37236/10645","DOIUrl":"https://doi.org/10.37236/10645","url":null,"abstract":"Baber and Talbot asked whether there is an irrational Turán density of a single hypergraph. In this paper, we show that the Lagrangian density of a 4-uniform matching of size 3 is an irrational number. Sidorenko showed that the Lagrangian density of an r-uniform hypergraph F is the same as the Turán density of the extension of F. Therefore, our result gives a positive answer to the question of Baber and Talbot. We also determine the Lagrangian densities of a class of r-uniform hypergraphs on n vertices with θ(n2) edges. As far as we know, for every hypergraph F with known hypergraph Lagrangian density, the number of edges in F is less than the number of its vertices.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80273384","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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