Colin Defant, Valentin Féray, Philippe Nadeau, Nathan Williams
{"title":"Wiener Indices of Minuscule Lattices","authors":"Colin Defant, Valentin Féray, Philippe Nadeau, Nathan Williams","doi":"10.37236/12002","DOIUrl":"https://doi.org/10.37236/12002","url":null,"abstract":"The Wiener index of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.","PeriodicalId":511494,"journal":{"name":"Electron. J. Comb.","volume":"35 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140435741","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}
Tina Li, Suho Oh, Edward Richmond, Grace Yan, Kimberly You
{"title":"Demazure Product of Permutations and Hopping","authors":"Tina Li, Suho Oh, Edward Richmond, Grace Yan, Kimberly You","doi":"10.37236/11983","DOIUrl":"https://doi.org/10.37236/11983","url":null,"abstract":"The Demazure product (also called the 0-Hecke product or greedy product) is an associative operation on Coxeter groups with interesting properties and important applications. In this paper, we study permutation groups and present a way to compute the Demazure product of two permutations using only their one-line notation and not relying on reduced words. The algorithm starts from their usual product and then applies a new operator, which we call the hopping operator. We also give an analogous result for the group of signed permutations.","PeriodicalId":511494,"journal":{"name":"Electron. J. Comb.","volume":"22 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140435771","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":"A Discrete Variation of the Littlewood-Offord Problem","authors":"H. Esmailian, Ebrahim Ghorbani","doi":"10.37236/11956","DOIUrl":"https://doi.org/10.37236/11956","url":null,"abstract":"The Littlewood-Offord problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of the Littlewood-Offord problem where we estimate the number of subsums that are $(0,1)$-vectors. We then utilize this to find the maximum order of graphs with given rank or corank. The rank of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the corank of $G$ is the rank of $A(G)+I$.","PeriodicalId":511494,"journal":{"name":"Electron. J. Comb.","volume":"28 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140437680","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":"Cobiased Graphs: Single-Element Extensions and Elementary Quotients of Graphic Matroids","authors":"Daniel C. Slilaty, Thomas Zaslavsky","doi":"10.37236/12148","DOIUrl":"https://doi.org/10.37236/12148","url":null,"abstract":"Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.","PeriodicalId":511494,"journal":{"name":"Electron. J. Comb.","volume":"417 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140474895","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":"Power of $k$ Choices in the Semi-Random Graph Process","authors":"Pawel Pralat, Harjas Singh","doi":"10.37236/11909","DOIUrl":"https://doi.org/10.37236/11909","url":null,"abstract":"The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. \u0000In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.","PeriodicalId":511494,"journal":{"name":"Electron. J. Comb.","volume":"3 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140509444","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":"An Action of the Cactus Group on Shifted Tableau Crystals","authors":"Ines Rodrigues","doi":"10.37236/9720","DOIUrl":"https://doi.org/10.37236/9720","url":null,"abstract":"Recently, Gillespie, Levinson and Purbhoo introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. We introduce, on this structure, a shifted version of the crystal reflection operators, which coincide with the restrictions of the shifted Schützenberger involution to any primed interval of two adjacent letters. Unlike type $A$ Young tableau crystals, these operators do not realize an action of the symmetric group on the shifted tableau crystal since the braid relations do not need to hold. Following a similar approach as Halacheva, we exhibit a natural internal action of the cactus group on this crystal, realized by the restrictions of the shifted Schützenberger involution to all primed intervals of the underlying crystal alphabet, containing, in particular, the aforesaid action of the shifted crystal reflection operator analogues.","PeriodicalId":511494,"journal":{"name":"Electron. J. Comb.","volume":"115 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141217723","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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