In this paper we show that if θ is a T-design of an association scheme (Ω,ℛ), and the Krein parameters q i,j h vanish for some h∉T and all i,j∉T (i,j,h≠0), then θ consists of precisely half of the vertices of (Ω,ℛ) or it is a T ′ -design, where |T ′ |>|T|. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial m-ovoids of generalised octagons of order (s,s 2 ) do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order (s,s 2 ); (iii) the dual polar spaces DQ(2d,q), DW(2d-1,q) and DH(2d-1,q 2 ), for d≥3; (iv) the Penttila–Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila–Williford scheme in Q - (2n-1,q), n⩾3.
{"title":"Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry","authors":"John Bamberg, Jesse Lansdown","doi":"10.5802/alco.246","DOIUrl":"https://doi.org/10.5802/alco.246","url":null,"abstract":"In this paper we show that if θ is a T-design of an association scheme (Ω,ℛ), and the Krein parameters q i,j h vanish for some h∉T and all i,j∉T (i,j,h≠0), then θ consists of precisely half of the vertices of (Ω,ℛ) or it is a T ′ -design, where |T ′ |>|T|. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial m-ovoids of generalised octagons of order (s,s 2 ) do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order (s,s 2 ); (iii) the dual polar spaces DQ(2d,q), DW(2d-1,q) and DH(2d-1,q 2 ), for d≥3; (iv) the Penttila–Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila–Williford scheme in Q - (2n-1,q), n⩾3.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136122093","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}
We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus g, n-marked curve C to ℙ r , sending the marked points on C to specified general points in ℙ r , is equal to (r+1) g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the (r+1)-ary sequences of length g, and we explore our bijection’s combinatorial properties.
{"title":"A Generalized RSK for Enumerating Linear Series on n-pointed Curves","authors":"Maria Gillespie, Andrew Reimer-Berg","doi":"10.5802/alco.250","DOIUrl":"https://doi.org/10.5802/alco.250","url":null,"abstract":"We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus g, n-marked curve C to ℙ r , sending the marked points on C to specified general points in ℙ r , is equal to (r+1) g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the (r+1)-ary sequences of length g, and we explore our bijection’s combinatorial properties.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136146307","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}
In this note, we show that certain dual canonical basis elements of C[SLm] are positive when evaluated on k-positive matrices, matrices whose minors of size k × k and smaller are positive. Skandera showed that all dual canonical basis elements of C[SLm] can be written in terms ofKazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324and 2143-avoiding permutations. This extends previous work of the authors on KazhdanLusztig immanants and uses similar tools, namely Lewis Carroll’s identity (also known as the Desnanot-Jacobi identity).
{"title":"k-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan–Lusztig immanants","authors":"Sunita Chepuri, M. Sherman-Bennett","doi":"10.5802/alco.257","DOIUrl":"https://doi.org/10.5802/alco.257","url":null,"abstract":"In this note, we show that certain dual canonical basis elements of C[SLm] are positive when evaluated on k-positive matrices, matrices whose minors of size k × k and smaller are positive. Skandera showed that all dual canonical basis elements of C[SLm] can be written in terms ofKazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324and 2143-avoiding permutations. This extends previous work of the authors on KazhdanLusztig immanants and uses similar tools, namely Lewis Carroll’s identity (also known as the Desnanot-Jacobi identity).","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49008405","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}
Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator, uniquely defined up to conjugacy. The main result of this article is the fact that this operator behaves in a controlled way under graph covering maps. When such an operator can be used to enumerate objects, or compute a partition function, this has concrete implications on the corresponding enumeration problem, or statistical mechanics model. For example, we show that if Γ ˜ is a finite covering graph of a connected graph Γ endowed with edge-weights x={x e } e , then the spanning tree partition function of Γ divides the one of Γ ˜ in the ring ℤ[x]. Several other consequences are obtained, some known, others new.
{"title":"Graph coverings and twisted operators","authors":"David Cimasoni, Adrien Kassel","doi":"10.5802/alco.258","DOIUrl":"https://doi.org/10.5802/alco.258","url":null,"abstract":"Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator, uniquely defined up to conjugacy. The main result of this article is the fact that this operator behaves in a controlled way under graph covering maps. When such an operator can be used to enumerate objects, or compute a partition function, this has concrete implications on the corresponding enumeration problem, or statistical mechanics model. For example, we show that if Γ ˜ is a finite covering graph of a connected graph Γ endowed with edge-weights x={x e } e , then the spanning tree partition function of Γ divides the one of Γ ˜ in the ring ℤ[x]. Several other consequences are obtained, some known, others new.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136122155","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}
In this paper we prove a new combinatorial formula for the modified Macdonald polynomials H ˜ λ (X;q,t), motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials P λ (X;q,t). In the case q=1 and X=(1,1,⋯,1), that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary X=(x 1 ,⋯,x n ); the x i play the role of site-dependent jump rates for the mTAZRP.
{"title":"Modified Macdonald polynomials and the multispecies zero-range process: I","authors":"Arvind Ayyer, Olya Mandelshtam, James B. Martin","doi":"10.5802/alco.248","DOIUrl":"https://doi.org/10.5802/alco.248","url":null,"abstract":"In this paper we prove a new combinatorial formula for the modified Macdonald polynomials H ˜ λ (X;q,t), motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials P λ (X;q,t). In the case q=1 and X=(1,1,⋯,1), that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary X=(x 1 ,⋯,x n ); the x i play the role of site-dependent jump rates for the mTAZRP.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136122156","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}
Vincent Pilaud, Vivane Pons, Daniel Tamayo Jimenez
We introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. On the combinatorial side, we study the rotation lattices on permutrees and their lattice homomorphisms, unifying the weak order, Tamari, Cambrian and boolean lattices and the classical maps between them. On the geometric side, we provide both the vertex and facet descriptions of a polytope realizing the rotation lattice, specializing to the permutahedron, the associahedra, and certain graphical zonotopes. On the algebraic side, we construct a Hopf algebra on permutrees containing the known Hopf algebraic structures on permutations, binary trees, Cambrian trees, and binary sequences.
{"title":"Permutree sorting","authors":"Vincent Pilaud, Vivane Pons, Daniel Tamayo Jimenez","doi":"10.5802/alco.249","DOIUrl":"https://doi.org/10.5802/alco.249","url":null,"abstract":"We introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. On the combinatorial side, we study the rotation lattices on permutrees and their lattice homomorphisms, unifying the weak order, Tamari, Cambrian and boolean lattices and the classical maps between them. On the geometric side, we provide both the vertex and facet descriptions of a polytope realizing the rotation lattice, specializing to the permutahedron, the associahedra, and certain graphical zonotopes. On the algebraic side, we construct a Hopf algebra on permutrees containing the known Hopf algebraic structures on permutations, binary trees, Cambrian trees, and binary sequences.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136146494","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":"Cycle type factorizations in GL n 𝔽 q ","authors":"Graham Gordon","doi":"10.5802/alco.259","DOIUrl":"https://doi.org/10.5802/alco.259","url":null,"abstract":"","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42952285","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}
Joanna A. Ellis-Monaghan, A. Goodall, I. Moffatt, S. Noble, Lluís Vena
We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.
{"title":"Irreducibility of the Tutte polynomial of an embedded graph","authors":"Joanna A. Ellis-Monaghan, A. Goodall, I. Moffatt, S. Noble, Lluís Vena","doi":"10.5802/alco.252","DOIUrl":"https://doi.org/10.5802/alco.252","url":null,"abstract":"We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49040743","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":"On the k-measure of partitions and distinct partitions","authors":"G. Andrews, Shane Chern, Zhitai Li","doi":"10.5802/alco.253","DOIUrl":"https://doi.org/10.5802/alco.253","url":null,"abstract":"","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43558929","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}
Angèle M. Foley, Alejandro H. Morales, Amarpreet Rattan, Karen Yeats
In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collaborative work.
{"title":"Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson","authors":"Angèle M. Foley, Alejandro H. Morales, Amarpreet Rattan, Karen Yeats","doi":"10.5802/alco.269","DOIUrl":"https://doi.org/10.5802/alco.269","url":null,"abstract":"In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collaborative work.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42848422","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：