Evil-avoiding permutations, introduced by Kim and Williams in 2022, arise in the study of the inhomogeneous totally asymmetric simple exclusion process. Rectangular permutations, introduced by Chirivì, Fang, and Fourier in 2021, arise in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the $1$-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n-2$ in a path of seven vertices starting and ending at the middle vertex.
{"title":"A Bijection Between Evil-Avoiding and Rectangular Permutations","authors":"Katherine Tung","doi":"10.37236/11841","DOIUrl":"https://doi.org/10.37236/11841","url":null,"abstract":"Evil-avoiding permutations, introduced by Kim and Williams in 2022, arise in the study of the inhomogeneous totally asymmetric simple exclusion process. Rectangular permutations, introduced by Chirivì, Fang, and Fourier in 2021, arise in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the $1$-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n-2$ in a path of seven vertices starting and ending at the middle vertex.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"27 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567289","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}
In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_3xy+k_1yz+k_2zx=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+2xy^2+ky^2z^2+2xz^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.
{"title":"Generalization of Markov Diophantine Equation via Generalized Cluster Algebra","authors":"Yasuaki Gyoda, Kodai Matsushita","doi":"10.37236/11420","DOIUrl":"https://doi.org/10.37236/11420","url":null,"abstract":"In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_3xy+k_1yz+k_2zx=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+2xy^2+ky^2z^2+2xz^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"40 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135568150","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 exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.
{"title":"Face Posets of Tropical Polyhedra and Monomial Ideals","authors":"Georg Loho, Ben Smith","doi":"10.37236/9999","DOIUrl":"https://doi.org/10.37236/9999","url":null,"abstract":"We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"76 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567086","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 prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $Delta$ has a set $F$ of $O(t^2(log t)^{1/4}sqrt{Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sýkora, and Vrťo (1993) for planar graphs, and of Sýkora and Vrťo (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H boxtimes K_{lfloor p rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = sqrt{(t-3)Delta |E(G)|} + Delta$.
{"title":"Edge Separators for Graphs Excluding a Minor","authors":"Gwenaël Joret, William Lochet, Michał T. Seweryn","doi":"10.37236/11744","DOIUrl":"https://doi.org/10.37236/11744","url":null,"abstract":"We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $Delta$ has a set $F$ of $O(t^2(log t)^{1/4}sqrt{Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sýkora, and Vrťo (1993) for planar graphs, and of Sýkora and Vrťo (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H boxtimes K_{lfloor p rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = sqrt{(t-3)Delta |E(G)|} + Delta$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"27 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567085","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}
This paper introduces the concept of weak $(d,h)$-decomposition of a graph $G$, which is defined as a partition of $E(G)$ into two subsets $E_1,E_2$, such that $E_1$ induces a $d$-degenerate graph $H_1$ and $E_2$ induces a subgraph $H_2$ with $alpha(H_1[N_{H_2}(v)]) le h$ for any vertex $v$. We prove that each planar graph admits a weak $(2,3)$-decomposition. As a consequence, every planar graph $G$ has a subgraph $H$ such that $G-E(H)$ is $3$-paintable and any proper coloring of $G-E(H)$ is a $3$-defective coloring of $G$. This improves the result in [G. Gutowski, M. Han, T. Krawczyk, and X. Zhu, Defective $3$-paintability of planar graphs, Electron. J. Combin., 25(2):#P2.34, 2018] that every planar graph is 3-defective $3$-paintable.
{"title":"Weak (2,3)-Decomposition of Planar Graphs","authors":"Ming Han, Xuding Zhu","doi":"10.37236/11774","DOIUrl":"https://doi.org/10.37236/11774","url":null,"abstract":"This paper introduces the concept of weak $(d,h)$-decomposition of a graph $G$, which is defined as a partition of $E(G)$ into two subsets $E_1,E_2$, such that $E_1$ induces a $d$-degenerate graph $H_1$ and $E_2$ induces a subgraph $H_2$ with $alpha(H_1[N_{H_2}(v)]) le h$ for any vertex $v$. We prove that each planar graph admits a weak $(2,3)$-decomposition. As a consequence, every planar graph $G$ has a subgraph $H$ such that $G-E(H)$ is $3$-paintable and any proper coloring of $G-E(H)$ is a $3$-defective coloring of $G$. This improves the result in [G. Gutowski, M. Han, T. Krawczyk, and X. Zhu, Defective $3$-paintability of planar graphs, Electron. J. Combin., 25(2):#P2.34, 2018] that every planar graph is 3-defective $3$-paintable.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567083","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 the maximum number of edges in a $3$-uniform hypergraph without a Berge cycle of length four is at most $(1+o(1))frac{n^{3/2}}{sqrt{10}}$. This improves earlier estimates by Győri and Lemons and by Füredi and Özkahya.
{"title":"On $3$-uniform hypergraphs avoiding a cycle of length four","authors":"Beka Ergemlidze, Ervin Győri, Abhishek Methuku, Nika Salia, Casey Tompkins","doi":"10.37236/11443","DOIUrl":"https://doi.org/10.37236/11443","url":null,"abstract":"We show that the maximum number of edges in a $3$-uniform hypergraph without a Berge cycle of length four is at most $(1+o(1))frac{n^{3/2}}{sqrt{10}}$. This improves earlier estimates by Győri and Lemons and by Füredi and Özkahya.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302737","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}
Consider a system of $m$ balanced linear equations in $k$ variables with coefficients in $mathbb{F}_q$. If $k geq 2m + 1$, then a routine application of the slice rank method shows that there are constants $beta,gamma geq 1$ with $gamma < q$ such that, for every subset $S subseteq mathbb{F}_q^n$ of size at least $beta cdot gamma^n$, the system has a solution $(x_1,ldots,x_k) in S^k$ with $x_1,ldots,x_k$ not all equal. Building on a series of papers by Mimura and Tokushige and on a paper by Sauermann, this paper investigates the problem of finding a solution of higher non-degeneracy; that is, a solution where $x_1,ldots,x_k$ are pairwise distinct, or even a solution where $x_1,ldots,x_k$ do not satisfy any balanced linear equation that is not a linear combination of the equations in the system.
In this paper, we focus on linear systems with repeated columns. For a large class of systems of this type, we prove that there are constants $beta,gamma geq 1$ with $gamma < q$ such that every subset $S subseteq mathbb{F}_q^n$ of size at least $beta cdot gamma^n$ contains a solution that is non-degenerate (in one of the two senses described above). This class is disjoint from the class covered by Sauermann's result, and captures the systems studied by Mimura and Tokushige into a single proof. Moreover, a special case of our results shows that, if $S subseteq mathbb{F}_p^n$ is a subset such that $S - S$ does not contain a non-trivial $k$-term arithmetic progression (with $p$ prime and $3 leq k leq p$), then $S$ must have exponentially small density.
{"title":"On the Size of Subsets of $mathbb{F}_q^n$ Avoiding Solutions to Linear Systems with Repeated Columns","authors":"Josse van Dobben de Bruyn, Dion Gijswijt","doi":"10.37236/10883","DOIUrl":"https://doi.org/10.37236/10883","url":null,"abstract":"Consider a system of $m$ balanced linear equations in $k$ variables with coefficients in $mathbb{F}_q$. If $k geq 2m + 1$, then a routine application of the slice rank method shows that there are constants $beta,gamma geq 1$ with $gamma < q$ such that, for every subset $S subseteq mathbb{F}_q^n$ of size at least $beta cdot gamma^n$, the system has a solution $(x_1,ldots,x_k) in S^k$ with $x_1,ldots,x_k$ not all equal. Building on a series of papers by Mimura and Tokushige and on a paper by Sauermann, this paper investigates the problem of finding a solution of higher non-degeneracy; that is, a solution where $x_1,ldots,x_k$ are pairwise distinct, or even a solution where $x_1,ldots,x_k$ do not satisfy any balanced linear equation that is not a linear combination of the equations in the system.
 In this paper, we focus on linear systems with repeated columns. For a large class of systems of this type, we prove that there are constants $beta,gamma geq 1$ with $gamma < q$ such that every subset $S subseteq mathbb{F}_q^n$ of size at least $beta cdot gamma^n$ contains a solution that is non-degenerate (in one of the two senses described above). This class is disjoint from the class covered by Sauermann's result, and captures the systems studied by Mimura and Tokushige into a single proof. Moreover, a special case of our results shows that, if $S subseteq mathbb{F}_p^n$ is a subset such that $S - S$ does not contain a non-trivial $k$-term arithmetic progression (with $p$ prime and $3 leq k leq p$), then $S$ must have exponentially small density.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"298 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302171","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}
Anders Claesson, Atli Fannar Franklín, Einar Steingrímsson
A curious generating function $S_0(x)$ for permutations of $[n]$ with exactly $n$ inversions is presented. Moreover, $(xC(x))^iS_0(x)$ is shown to be the generating function for permutations of $[n]$ with exactly $n-i$ inversions, where $C(x)$ is the generating function for the Catalan numbers.
{"title":"Permutations with Few Inversions","authors":"Anders Claesson, Atli Fannar Franklín, Einar Steingrímsson","doi":"10.37236/12075","DOIUrl":"https://doi.org/10.37236/12075","url":null,"abstract":"A curious generating function $S_0(x)$ for permutations of $[n]$ with exactly $n$ inversions is presented. Moreover, $(xC(x))^iS_0(x)$ is shown to be the generating function for permutations of $[n]$ with exactly $n-i$ inversions, where $C(x)$ is the generating function for the Catalan numbers.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"24 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350956","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}
For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the graph obtained by blowing up three pairwise non-adjacent vertices in a 6-cycle to sets of as even size as possible, and that every extremal example closely resembles this graph. This extends previous work by the author which solves the problem for 4-cycles and 5-cycles. The 5-cycle problem was also solved independently by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora.
{"title":"Planar Graphs with the Maximum Number of Induced 6-Cycles","authors":"Michael Savery","doi":"10.37236/11944","DOIUrl":"https://doi.org/10.37236/11944","url":null,"abstract":"For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the graph obtained by blowing up three pairwise non-adjacent vertices in a 6-cycle to sets of as even size as possible, and that every extremal example closely resembles this graph. This extends previous work by the author which solves the problem for 4-cycles and 5-cycles. The 5-cycle problem was also solved independently by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora.
","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302169","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 prove the curious identity in the sense of formal power series:[int_{-infty}^{infty}[y^m]expleft(-frac{t^2}2+sum_{jge3}frac{(it)^j}{j!}, y^{j-2}right)mathrm{d} t= int_{-infty}^{infty}[y^m]expleft(-frac{t^2}2+sum_{jge3}frac{(it)^j}{j}, y^{j-2}right)mathrm{d} t,]for $m=0,1,dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$, which arises from applying the saddle-point method to derive Stirling's formula. The generality of the same approach (saddle-point method over two different contours) is also examined, together with some applications to asymptotic enumeration.
{"title":"A Curious Identity Arising From Stirling's Formula and Saddle-Point Method on Two Different Contours","authors":"Hsien-Kuei Hwang","doi":"10.37236/11785","DOIUrl":"https://doi.org/10.37236/11785","url":null,"abstract":"We prove the curious identity in the sense of formal power series:[int_{-infty}^{infty}[y^m]expleft(-frac{t^2}2+sum_{jge3}frac{(it)^j}{j!}, y^{j-2}right)mathrm{d} t= int_{-infty}^{infty}[y^m]expleft(-frac{t^2}2+sum_{jge3}frac{(it)^j}{j}, y^{j-2}right)mathrm{d} t,]for $m=0,1,dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$, which arises from applying the saddle-point method to derive Stirling's formula. The generality of the same approach (saddle-point method over two different contours) is also examined, together with some applications to asymptotic enumeration.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350955","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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