{"title":"Graph Theory: Part 2","authors":"P. Mladenovic","doi":"10.1007/978-3-030-00831-4_9","DOIUrl":"https://doi.org/10.1007/978-3-030-00831-4_9","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"4 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82857512","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}
{"title":"Binomial and Multinomial Theorems","authors":"P. Mladenovic","doi":"10.1007/978-3-030-00831-4_3","DOIUrl":"https://doi.org/10.1007/978-3-030-00831-4_3","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"164 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73466615","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}
{"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}
{"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}
{"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}
{"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}
{"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}
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