{"title":"The Gini index in the representation theory of the general linear group","authors":"Grant Kopitzke","doi":"10.54550/eca2023v3s3r21","DOIUrl":"https://doi.org/10.54550/eca2023v3s3r21","url":null,"abstract":"The Gini index is a function that attempts to measure the amount of inequality in the distribution of a finite resource throughout a population. It is commonly used in economics as a measure of inequality of income or wealth. We define a discrete Gini index on the set of integer partitions with at most $n$ parts and show how this function emerges in the representation theory of the complex general linear group.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126377890","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}
Spencer J. Franks, P. Harris, Kimberly Harry, Jan Kretschmann, Megan Vance
{"title":"Counting parking sequences and parking assortments through permutations","authors":"Spencer J. Franks, P. Harris, Kimberly Harry, Jan Kretschmann, Megan Vance","doi":"10.54550/eca2024v4s1r2","DOIUrl":"https://doi.org/10.54550/eca2024v4s1r2","url":null,"abstract":"Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs and the car fails to park. In contrast, parking assortments generalize parking sequences (and parking functions) by allowing cars (also of assorted lengths) to seek forward from their preference to identify a set of contiguous unoccupied spots in which they fit. We consider both parking sequences and parking assortments and establish that the number of preferences resulting in a fixed parking order $sigma$ is related to the lengths of cars indexed by certain subsequences in $sigma$. The sum of these numbers over all parking orders (i.e. permutations of $[n]$) yields new formulas for the total number of parking sequences and of parking assortments.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130097538","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":"Interview with Stuart Whittington","authors":"T. Mansour","doi":"10.54550/eca2023v3s2i6","DOIUrl":"https://doi.org/10.54550/eca2023v3s2i6","url":null,"abstract":"Photo by Ann Whittington Stuart Whittington was educated at Queens’ College Cambridge and has spent most of his working life at the University of Toronto. He spent a post-doctoral year at the University of California San Diego as a Fulbright fellow, working with Fred Wall on self-avoiding walks, and a second post-doctoral year at the University of Toronto, working with John Valleau, mainly estimating the numbers of embeddings of some classes of graphs in lattices. His primary areas of study are statistical mechanics, especially problems with a combinatorial or topological flavour, self-avoiding walks and related objects like lattice trees and lattice animals, self-averaging in quenched random systems, and random knotting and linking. Most of his research interests are in the statistical mechanics of lattice models, especially those related to the configurational and statistical properties of polymers, as well as phase transitions and critical phenomena. He is also interested in the theory and application of Markov chain Monte Carlo methods.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117272476","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":"Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach","authors":"Koustav Banerjee","doi":"10.54550/eca2023v3s2r12","DOIUrl":"https://doi.org/10.54550/eca2023v3s2r12","url":null,"abstract":". This note is devoted to the study of inequalities related to the second shifted difference of the number of integer partitions p ( n ) and of overpartitions p ( n ) by an elemen-tary combinatorial approach. Recently Gomez, Males, and Rolen proved the positivity of ∆ 2 j ( p ( n )) = p ( n ) − 2 p ( n − j )+ p ( n − 2 j ) by employing the Hardy-Ramanujan-Rademacher formula for p ( n ) and Lehmer’s error bound. Our goal is to prove ∆ 2 j ( p ( n )) ≥ 0 (resp. ∆ 2 j ( p ( n )) > 0) by an explicit description of a non-empty subset, say X 2 p ( n, j ) of the set of integer partitions P ( n ) (resp. X 2 p ( n, j ) and the set of overpartitions P ( n )) with | X 2 p ( n, j ) | = ∆ 2 j ( p ( n )) (resp. | X 2 p ( n, j ) | = ∆ 2 j ( p ( n ))) .","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133944769","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":"Interview with Anne Schilling","authors":"T. Mansour","doi":"10.54550/eca2023v3s2i5","DOIUrl":"https://doi.org/10.54550/eca2023v3s2i5","url":null,"abstract":"Photo by Andrew Waldron Anne Schilling studied mathematics and physics at the University of Bonn. She went to the State University of New York at Stony Brook as a Fulbright scholar and completed her Ph.D. in 1997 under the supervision of Barry M. McCoy. From 1997 until 1999 she was a postdoctoral fellow at Institute for Theoretical Physics at Amsterdam University. From 1999 until 2001 she was a C.L.E. Moore Instructor in the Mathematics Department at the Massachusetts Institute of Technology. After that she joined the faculty of the Department of Mathematics at the University of California at Davis, where she is a Full Professor. Anne Schilling was a Humboldt Fellow in 2002, a Simons Research Fellow during 2012/13, and was elected to the 2019 class of fellows of the American Mathematical Society. Her research interests include Algebraic Combinatorics, Representation Theory, and Mathematical Physics. She is a member of the editorial board for various journals, notably the open-access journals Combinatorial Theory and Algebraic Combinatorics.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114175189","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":"Snake paths in king and knight graphs","authors":"N. Beluhov","doi":"10.54550/ECA2023V3S2R16","DOIUrl":"https://doi.org/10.54550/ECA2023V3S2R16","url":null,"abstract":"A snake path in a graph $G$ is a path in $G$ which is also an induced subgraph of $G$. For all $n$, we find the greatest length of a snake path in the $n times n$ king graph and we give a complete description of the paths which attain this greatest length. The even and odd cases behave very differently. We also estimate the greatest length of a snake path or cycle in the $m times n$ knight graph, for all $m$ and $n$.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131032108","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":"Combinatorial reciprocity for non-intersecting paths","authors":"S. Hopkins, Gjergji Zaimi","doi":"10.54550/eca2023v3s2r13","DOIUrl":"https://doi.org/10.54550/eca2023v3s2r13","url":null,"abstract":"We prove a combinatorial reciprocity theorem for the enumeration of non-intersecting paths in a linearly growing sequence of acyclic planar networks. We explain two applications of this theorem: reciprocity for fans of bounded Dyck paths, and reciprocity for Schur function evaluations with repeated values.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132191277","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 symmetric chain decomposition of L(5,n)","authors":"Xiangdong Wen","doi":"10.54550/eca2024v4s1r5","DOIUrl":"https://doi.org/10.54550/eca2024v4s1r5","url":null,"abstract":"We give a constructive proof that Young's lattice $L(5, n)$ has a partition into saturated symmetric chains.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126160848","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":"Partitions with constrained ranks and lattice paths","authors":"S. Corteel, S. Elizalde, C. Savage","doi":"10.54550/eca2023v3s3r18","DOIUrl":"https://doi.org/10.54550/eca2023v3s3r18","url":null,"abstract":"In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee rectangle. We also obtain a new bijective proof of a result of Andrews and Bressoud that the number of partitions of $N$ with all ranks at least $1-ell$ equals the number of partitions of $N$ with no parts equal to $ell+1$, for $ellge0$, which allows us to refine it by the above statistics. Combining Foata's second fundamental transformation for words with Greene and Kleitman's mapping for subsets, interpreted in terms of lattice paths, we obtain enumeration formulas for partitions whose successive ranks satisfy certain constraints, such as being bounded by a constant.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127396533","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":"The expected embedding dimension, type and weight of a numerical semigroup","authors":"N. Kaplan, Deepesh Singhal","doi":"10.54550/eca2023v3s2r14","DOIUrl":"https://doi.org/10.54550/eca2023v3s2r14","url":null,"abstract":"We study statistical properties of numerical semigroups of genus $g$ as $g$ goes to infinity. More specifically, we answer a question of Eliahou by showing that as $g$ goes to infinity, the proportion of numerical semigroups of genus $g$ with embedding dimension close to $g/sqrt{5}$ approaches $1$. We prove similar results for the type and weight of a numerical semigroup of genus $g$.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123174682","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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