Enumerative Combinatorics and Applications最新文献

{"title":"Interview with Carla D. Savage","authors":"Eça","doi":"10.54550/eca2022v2s3i9","DOIUrl":"https://doi.org/10.54550/eca2022v2s3i9","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125735204","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":"Automatic counting of generalized Latin rectangles and trapezoids","authors":"G. Spahn, D. Zeilberger","doi":"10.54550/eca2022v2s1r8","DOIUrl":"https://doi.org/10.54550/eca2022v2s1r8","url":null,"abstract":"In this case study in “fully automated enumeration”, we illustrate how to take full advantage of symbolic computation by developing (what we call) ‘symbolic-dynamical-programming’ algorithms for computing many terms of ‘hard to compute sequences’, namely the number of Latin trapezoids, generalized derangements, and generalized three-rowed Latin rectangles. At the end we also sketch the proof of a generalization of Ira Gessel’s 1987 theorem that says that for any number of rows, k, the number of Latin rectangles with k rows and n columns is P-recursive in n. Our algorithms are fully implemented in Maple, and generated quite a few terms of such sequences. How it all Started Last year, for a few months, the New York Times magazine published a puzzle, created by Wei-Hwa Huang, called Triangulum. One is given a discrete equilateral triangle consisting of 1+2+3+4+5 = 15 empty circles, where the bottom row has 5 empty circles, the next row has four empty circles, . . . and the top (fifth) row has one empty circle. Intertwined between these circles are 15 mostly empty triangles, but a few of them are filled with integers. The solver has to fill-in the empty circles with the integers 1, 2, 3, 4, 5 in such a way that each of the five horizonal lines, and each of the 10 diagonal lines have distinct entries, and in addition the labels in the entries in the circles around each of the non-empty triangles add-up to the number in that triangle. See (and play!) https://sites.math.rutgers.edu/~zeilberg/EM21/projects/LTgame.html . Of course, from a computational point of view, these puzzles are trivial, and one can do it easily by brute force. It is easy to see that there are only 4 such reduced configurations where the bottom row is 12345. Here there are: 2 3 4 4 5 1 5 1 2 3 1 2 3 4 5 , 3 5 1 2 3 4 4 5 1 2 1 2 3 4 5 , 3 2 4 5 3 1 4 1 5 2 1 2 3 4 5 , 4 2 3 5 1 2 3 4 5 1 1 2 3 4 5 , hence altogether there are 4 · 5! = 480 legal ways, and then even a very stupid, but patient, human, who knows how to add, can pick which of the 480 possibilities meet the extra conditions. 1 Let’s call such equilateral discrete triangles Latin Triangles, and consider the problem of finding the exact number of Latin triangles with side-length n. By brute force, one can easily get the first few terms. Deciding that the bottom row is 1....n, then for n = 3, 4, 5, 6, 7 these numbers are 1, 0, 4, 236, 27820, respectively (to get the total number , multiply by n!). This sequence was not (Aug. 13, 2021) in the OEIS. To see all the reduced Latin triangles of side-length up to 7, see: https://sites.math.rutgers.edu/~zeilberg/tokhniot/oLatinTrapezoids2.txt . We are almost sure that human-kind will never know the exact number of Latin triangles with side-length 30. For the apparently much easier problem of counting Latin squares, given by OEIS sequence A2860 (https://oeis.org/A002860), currently only 11 terms are known. So let’s settle for something easier, but far from trivial: Count Latin trapez","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128802336","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 Ira Gessel","authors":"Eça","doi":"10.54550/eca2022v2s2i8","DOIUrl":"https://doi.org/10.54550/eca2022v2s2i8","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121484901","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 Gil Kalai","authors":"Eça","doi":"10.54550/eca2022v2s2i7","DOIUrl":"https://doi.org/10.54550/eca2022v2s2i7","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127875318","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":"Boolean intersection ideals of permutations in the Bruhat order","authors":"B. E. Tenner","doi":"10.54550/eca2022v2s3r23","DOIUrl":"https://doi.org/10.54550/eca2022v2s3r23","url":null,"abstract":": Motivated by recent work with Mazorchuk, we characterize the conditions under which the intersection of two principal order ideals in the Bruhat order is boolean. That characterization is presented in three versions: in terms of reduced words, in terms of permutation patterns, and in terms of permutation support. The equivalence of these properties follows from an analysis of what it means to have a specific letter repeated in a permutation’s reduced words; namely, that a specific 321-pattern appears.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122586581","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":"Vincular pattern avoidance on cyclic permutations","authors":"Rupert Li","doi":"10.54550/ECA2022V2S4PP3","DOIUrl":"https://doi.org/10.54550/ECA2022V2S4PP3","url":null,"abstract":": Pattern avoidance for permutations has been extensively studied and has been generalized to vincular patterns, where certain elements can be required to be adjacent. In addition, cyclic permutations, i.e., permutations written in a circle rather than a line, have been frequently studied, including in the context of pattern avoidance. We investigate vincular pattern avoidance on cyclic permutations. In particular, we enumerate many avoidance classes of sets of vincular patterns of length 3, including a complete enumeration for all single patterns of length 3. Further, several of the avoidance classes corresponding to a single vincular pattern of length 4 are enumerated by the Catalan numbers. We then study more generally whether sets of vincular patterns of an arbitrary length k can be avoided for arbitrarily long cyclic permutations, in particular investigating the boundary cases of minimal unavoidable sets and maximal avoidable sets.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128798088","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 degree of asymmetry of sequences","authors":"S. Elizalde, Emeric Deutsch","doi":"10.54550/eca2022v2s1r7","DOIUrl":"https://doi.org/10.54550/eca2022v2s1r7","url":null,"abstract":"We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion for compositions, words, matchings, binary trees, and permutations, we find generating functions enumerating these objects with respect to their degree of asymmetry, and we describe the limiting distribution of this statistic in each case.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124123608","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":"Generating functions of Lozenge tilings for hexagonal regions via nonintersecting lattice paths","authors":"Markus Fulmek","doi":"10.54550/eca2021v1s3r24","DOIUrl":"https://doi.org/10.54550/eca2021v1s3r24","url":null,"abstract":"In a recent preprint, Lai showed that the quotient of generating functions of weighted lozenge tilings of two “half hexagons with lateral dents”, which differ only in width, factors nicely, and the same is true for the quotient of generating functions of weighted lozenge tilings of two “quarter hexagons with lateral dents”. Lai achieved this by using “graphical condensation”. The purpose of this paper is to exhibit how some of Lai’s observations can be achieved by the Lindström– Gessel–Viennot method for nonintersecting lattice paths.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131140482","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":"Poly-Cauchy numbers of the second kind - the combinatorics behind","authors":"B. Bényi, J. L. Ramírez","doi":"10.54550/eca2022v2s1r1","DOIUrl":"https://doi.org/10.54550/eca2022v2s1r1","url":null,"abstract":"We introduce poly-Cauchy permutations of the second kind that are enumerated by the polyCauchy numbers of the second kind. We provide combinatorial proofs for several identities involving polyCauchy numbers of the second kind and some of their generalizations. The aim of this work is to demonstrate the power and beauty of the elementary combinatorial approach.","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121194775","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 Karim Adiprasito","authors":"Eça","doi":"10.54550/eca2022v2s2i6","DOIUrl":"https://doi.org/10.54550/eca2022v2s2i6","url":null,"abstract":"","PeriodicalId":340033,"journal":{"name":"Enumerative Combinatorics and Applications","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126056291","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}