Pub Date : 2018-11-29DOI: 10.4310/joc.2021.v12.n4.a1
M. Mazin, Joshua Miller
The original Pak-Stanley labeling was defined by Pak and Stanley as a bijective map from the set of regions of an extended Shi arrangement to the set of parking functions. This map was later generalized to other arrangements associated with graphs and directed multigraphs. In these more general cases the map is no longer bijective. However, it was shown Hopkins and Perkinson and then the first author that it is surjective to the set of the $G$-parking functions, where $G$ is the multigraph associated with the arrangement. This leads to a natural question: when is the generalized Pak-Stanley map bijective? In this paper we answer this question in the special case of centered hyperplane arrangements, i.e. the case when all the hyperplanes of the arrangement pass through a common point.
{"title":"Pak–Stanley labeling for central graphical arrangements","authors":"M. Mazin, Joshua Miller","doi":"10.4310/joc.2021.v12.n4.a1","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n4.a1","url":null,"abstract":"The original Pak-Stanley labeling was defined by Pak and Stanley as a bijective map from the set of regions of an extended Shi arrangement to the set of parking functions. This map was later generalized to other arrangements associated with graphs and directed multigraphs. In these more general cases the map is no longer bijective. However, it was shown Hopkins and Perkinson and then the first author that it is surjective to the set of the $G$-parking functions, where $G$ is the multigraph associated with the arrangement. \u0000This leads to a natural question: when is the generalized Pak-Stanley map bijective? In this paper we answer this question in the special case of centered hyperplane arrangements, i.e. the case when all the hyperplanes of the arrangement pass through a common point.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"117 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74350322","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 : 2018-10-26DOI: 10.4310/joc.2021.v12.n2.a2
P. Alexandersson, J. Haglund, George Wang
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.
{"title":"Some conjectures on the Schur expansion of Jack polynomials","authors":"P. Alexandersson, J. Haglund, George Wang","doi":"10.4310/joc.2021.v12.n2.a2","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n2.a2","url":null,"abstract":"We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"49 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75989712","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 : 2018-10-10DOI: 10.4310/joc.2021.v12.n1.a5
Molly Lynch
We study the combinatorics of crystal graphs given by highest weight representations of types $A_{n}, B_{n}, C_{n}$, and $D_{n}$, uncovering new relations that exist among crystal operators. Much structure in these graphs has been revealed by local relations given by Stembridge and Sternberg. However, there exist relations among crystal operators that are not implied by Stembridge or Sternberg relations. Viewing crystal graphs as edge colored posets, we use poset topology to study them. Using the lexicographic discrete Morse functions of Babson and Hersh, we relate the Mobius function of a given interval in a crystal poset of simply laced or doubly laced type to the types of relations that can occur among crystal operators within this interval. For a crystal of a highest weight representation of finite classical Cartan type, we show that whenever there exists an interval whose Mobius function is not equal to -1, 0, or 1, there must be a relation among crystal operators within this interval not implied by Stembridge or Sternberg relations. As an example of an application, this yields relations among crystal operators in type $C_{n}$ that were not previously known. Additionally, by studying the structure of Sternberg relations in the doubly laced case, we prove that crystals of highest weight representations of types $B_{2}$ and $C_{2}$ are not lattices.
{"title":"Relations in doubly laced crystal graphs via discrete Morse theory","authors":"Molly Lynch","doi":"10.4310/joc.2021.v12.n1.a5","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n1.a5","url":null,"abstract":"We study the combinatorics of crystal graphs given by highest weight representations of types $A_{n}, B_{n}, C_{n}$, and $D_{n}$, uncovering new relations that exist among crystal operators. Much structure in these graphs has been revealed by local relations given by Stembridge and Sternberg. However, there exist relations among crystal operators that are not implied by Stembridge or Sternberg relations. Viewing crystal graphs as edge colored posets, we use poset topology to study them. Using the lexicographic discrete Morse functions of Babson and Hersh, we relate the Mobius function of a given interval in a crystal poset of simply laced or doubly laced type to the types of relations that can occur among crystal operators within this interval. \u0000For a crystal of a highest weight representation of finite classical Cartan type, we show that whenever there exists an interval whose Mobius function is not equal to -1, 0, or 1, there must be a relation among crystal operators within this interval not implied by Stembridge or Sternberg relations. As an example of an application, this yields relations among crystal operators in type $C_{n}$ that were not previously known. Additionally, by studying the structure of Sternberg relations in the doubly laced case, we prove that crystals of highest weight representations of types $B_{2}$ and $C_{2}$ are not lattices.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72836017","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 : 2018-09-11DOI: 10.4310/joc.2020.v11.n3.a6
Colin Defant
A nonnegative integer is called a fertility number if it is equal to the number of preimages of a permutation under West's stack-sorting map. We prove structural results concerning permutations, allowing us to deduce information about the set of fertility numbers. In particular, the set of fertility numbers is closed under multiplication and contains every nonnegative integer that is not congruent to $3$ modulo $4$. We show that the lower asymptotic density of the set of fertility numbers is at least $1954/2565approx 0.7618$. We also exhibit some positive integers that are not fertility numbers and conjecture that there are infinitely many such numbers.
{"title":"Fertility numbers","authors":"Colin Defant","doi":"10.4310/joc.2020.v11.n3.a6","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n3.a6","url":null,"abstract":"A nonnegative integer is called a fertility number if it is equal to the number of preimages of a permutation under West's stack-sorting map. We prove structural results concerning permutations, allowing us to deduce information about the set of fertility numbers. In particular, the set of fertility numbers is closed under multiplication and contains every nonnegative integer that is not congruent to $3$ modulo $4$. We show that the lower asymptotic density of the set of fertility numbers is at least $1954/2565approx 0.7618$. We also exhibit some positive integers that are not fertility numbers and conjecture that there are infinitely many such numbers.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75525071","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 : 2018-09-03DOI: 10.1142/9789813274334_0004
CodesThann WardSeptember
{"title":"Designs and Codes","authors":"CodesThann WardSeptember","doi":"10.1142/9789813274334_0004","DOIUrl":"https://doi.org/10.1142/9789813274334_0004","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"52 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80928024","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 : 2018-07-16DOI: 10.4310/joc.2021.v12.n2.a4
Z. Furedi, A. Kostochka, Ruth Luo
Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $kgeq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $kgeq r+4$.
{"title":"Avoiding long Berge cycles II, exact bounds for all $n$","authors":"Z. Furedi, A. Kostochka, Ruth Luo","doi":"10.4310/joc.2021.v12.n2.a4","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n2.a4","url":null,"abstract":"Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $kgeq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $kgeq r+4$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"58 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84445709","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 : 2018-06-22DOI: 10.4310/joc.2022.v13.n1.a1
Thomas Schweser, M. Stiebitz, B. Toft
For a hypergraph $G$, let $chi(G), Delta(G),$ and $lambda(G)$ denote the chromatic number, the maximum degree, and the maximum local edge connectivity of $G$, respectively. A result of Rhys Price Jones from 1975 says that every connected hypergraph $G$ satisfies $chi(G) leq Delta(G) + 1$ and equality holds if and only if $G$ is a complete graph, an odd cycle, or $G$ has just one (hyper-)edge. By a result of Bjarne Toft from 1970 it follows that every hypergraph $G$ satisfies $chi(G) leq lambda(G) + 1$. In this paper, we show that a hypergraph $G$ with $lambda(G) geq 3$ satisfies $chi(G) = lambda(G) + 1$ if and only if $G$ contains a block which belongs to a family $mathcal{H}_{lambda(G)}$. The class $mathcal{H}_3$ is the smallest family which contains all odd wheels and is closed under taking Haj'os joins. For $k geq 4$, the family $mathcal{H}_k$ is the smallest that contains all complete graphs $K_{k+1}$ and is closed under Haj'os joins. For the proofs of the above results we use critical hypergraphs. A hypergraph $G$ is called $(k+1)$-critical if $chi(G)=k+1$, but $chi(H)leq k$ whenever $H$ is a proper subhypergraph of $G$. We give a characterization of $(k+1)$-critical hypergraphs having a separating edge set of size $k$ as well as a a characterization of $(k+1)$-critical hypergraphs having a separating vertex set of size $2$.
{"title":"Coloring hypergraphs of low connectivity","authors":"Thomas Schweser, M. Stiebitz, B. Toft","doi":"10.4310/joc.2022.v13.n1.a1","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n1.a1","url":null,"abstract":"For a hypergraph $G$, let $chi(G), Delta(G),$ and $lambda(G)$ denote the chromatic number, the maximum degree, and the maximum local edge connectivity of $G$, respectively. A result of Rhys Price Jones from 1975 says that every connected hypergraph $G$ satisfies $chi(G) leq Delta(G) + 1$ and equality holds if and only if $G$ is a complete graph, an odd cycle, or $G$ has just one (hyper-)edge. By a result of Bjarne Toft from 1970 it follows that every hypergraph $G$ satisfies $chi(G) leq lambda(G) + 1$. In this paper, we show that a hypergraph $G$ with $lambda(G) geq 3$ satisfies $chi(G) = lambda(G) + 1$ if and only if $G$ contains a block which belongs to a family $mathcal{H}_{lambda(G)}$. The class $mathcal{H}_3$ is the smallest family which contains all odd wheels and is closed under taking Haj'os joins. For $k geq 4$, the family $mathcal{H}_k$ is the smallest that contains all complete graphs $K_{k+1}$ and is closed under Haj'os joins. For the proofs of the above results we use critical hypergraphs. A hypergraph $G$ is called $(k+1)$-critical if $chi(G)=k+1$, but $chi(H)leq k$ whenever $H$ is a proper subhypergraph of $G$. We give a characterization of $(k+1)$-critical hypergraphs having a separating edge set of size $k$ as well as a a characterization of $(k+1)$-critical hypergraphs having a separating vertex set of size $2$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"100 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85841709","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 : 2018-05-11DOI: 10.4310/JOC.2019.V10.N1.A7
H. Fleischner, G. Chia
This is the second part of joint research in which we show that every $2$-connected graph $G$ has the ${cal F}_4$ property. That is, given distinct $x_iin V(G)$, $1leq ileq 4$, there is an $x_1x_2$-hamiltonian path in $G^2$ containing different edges $x_3y_3, x_4y_4in E(G)$ for some $y_3,y_4in V(G)$. However, it was shown already in cite[Theorem 2]{cf1:refer} that 2-connected DT-graphs have the ${cal F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs. We also show that these results are best possible.
{"title":"Revisiting the Hamiltonian theme in the square of a block: the general case","authors":"H. Fleischner, G. Chia","doi":"10.4310/JOC.2019.V10.N1.A7","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N1.A7","url":null,"abstract":"This is the second part of joint research in which we show that every $2$-connected graph $G$ has the ${cal F}_4$ property. That is, given distinct $x_iin V(G)$, $1leq ileq 4$, there is an $x_1x_2$-hamiltonian path in $G^2$ containing different edges $x_3y_3, x_4y_4in E(G)$ for some $y_3,y_4in V(G)$. However, it was shown already in cite[Theorem 2]{cf1:refer} that 2-connected DT-graphs have the ${cal F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs. We also show that these results are best possible.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"26 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74224650","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 : 2018-05-10DOI: 10.4310/joc.2021.v12.n1.a6
Chang-Pao Chen, Catherine S. Greenhill
The study of substructures in random objects has a long history, beginning with Erdős and Renyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.
{"title":"Threshold functions for substructures in random subsets of finite vector spaces","authors":"Chang-Pao Chen, Catherine S. Greenhill","doi":"10.4310/joc.2021.v12.n1.a6","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n1.a6","url":null,"abstract":"The study of substructures in random objects has a long history, beginning with Erdős and Renyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"46 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87541873","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 : 2018-05-10DOI: 10.4310/JOC.2019.V10.N4.A6
Cesar Ceballos, Rafael S. Gonz'alez D'Le'on
The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting $s$-Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define signature generalizations of parking functions which coincide with the generalized parking functions studied by Pitman-Stanley and Yan, as well as generalizations of permutations which coincide with the notion of Stirling multipermutations introduced by Gessel-Stanley. Some of our constructions differ from the ones of Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain the additional notions of rational permutations and rational trees.
{"title":"Signature Catalan combinatorics","authors":"Cesar Ceballos, Rafael S. Gonz'alez D'Le'on","doi":"10.4310/JOC.2019.V10.N4.A6","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N4.A6","url":null,"abstract":"The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting $s$-Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define signature generalizations of parking functions which coincide with the generalized parking functions studied by Pitman-Stanley and Yan, as well as generalizations of permutations which coincide with the notion of Stirling multipermutations introduced by Gessel-Stanley. Some of our constructions differ from the ones of Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain the additional notions of rational permutations and rational trees.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"67 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82943475","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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