International audience Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.
{"title":"Oriented Flip Graphs and Noncrossing Tree Partitions","authors":"Alexander Garver, T. McConville","doi":"10.46298/dmtcs.6379","DOIUrl":"https://doi.org/10.46298/dmtcs.6379","url":null,"abstract":"International audience\u0000 \u0000 Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience In their 1987 paper Kraskiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraskiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely Sw Sd(Ki) and Sw Vd(Ki), corresponding to Pieri and dual Pieri rules for Schubert polynomials.
{"title":"Kraskiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials","authors":"Masaki Watanabe","doi":"10.46298/dmtcs.6321","DOIUrl":"https://doi.org/10.46298/dmtcs.6321","url":null,"abstract":"International audience\u0000 \u0000 In their 1987 paper Kraskiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraskiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely Sw Sd(Ki) and Sw Vd(Ki), corresponding to Pieri and dual Pieri rules for Schubert polynomials.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.
{"title":"Schur polynomials and matrix positivity preservers","authors":"A. Belton, D. Guillot, A. Khare, M. Putinar","doi":"10.46298/dmtcs.6408","DOIUrl":"https://doi.org/10.46298/dmtcs.6408","url":null,"abstract":"International audience\u0000 \u0000 A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.
{"title":"Matrix product and sum rule for Macdonald polynomials","authors":"L. Cantini, J. Gier, M. Wheeler","doi":"10.46298/dmtcs.6419","DOIUrl":"https://doi.org/10.46298/dmtcs.6419","url":null,"abstract":"International audience\u0000 \u0000 We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.
{"title":"A lattice point counting generalisation of the Tutte polynomial","authors":"Amanda Cameron, Alex Fink","doi":"10.46298/dmtcs.6331","DOIUrl":"https://doi.org/10.46298/dmtcs.6331","url":null,"abstract":"International audience\u0000 \u0000 The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $psi(x, y, z; t, 1+beta)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of $psi(x, y, z; t, 1+beta)$ in the power-sum basis are polynomials in $beta$ with nonnegative integer coefficients (by construction, these coefficients are rational functions in $beta$). We prove partially this conjecture, nowadays called $b$-conjecture, by showing that coefficients of $psi(x, y, z; t, 1+ beta)$ are polynomials in $beta$ with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter $alpha$ tends to $0$, that may be of independent interest.
Goulden和Jackson(1996)利用Jack对称函数引入了一些多元生成序列$psi(x, y, z; t, 1+beta)$，这些生成序列可以解释为有根超映射生成序列的连续变形。他们做了如下的猜想:$psi(x, y, z; t, 1+beta)$在幂和基中的系数是$beta$中具有非负整数系数的多项式(通过构造，这些系数是$beta$中的有理函数)。我们通过证明$psi(x, y, z; t, 1+ beta)$的系数是$beta$中具有有理系数的多项式，部分地证明了这个猜想，现在称为$b$ -猜想。证明的一个关键步骤是当千斤顶变形参数$alpha$趋于$0$时，千斤顶多项式的强分解性质，这可能是独立的兴趣。
{"title":"Cumulants of Jack symmetric functions and $b$-conjecture","authors":"Maciej Dolkega, Valentin F'eray","doi":"10.1090/tran/7191","DOIUrl":"https://doi.org/10.1090/tran/7191","url":null,"abstract":"Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $psi(x, y, z; t, 1+beta)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of $psi(x, y, z; t, 1+beta)$ in the power-sum basis are polynomials in $beta$ with nonnegative integer coefficients (by construction, these coefficients are rational functions in $beta$). \u0000We prove partially this conjecture, nowadays called $b$-conjecture, by showing that coefficients of $psi(x, y, z; t, 1+ beta)$ are polynomials in $beta$ with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter $alpha$ tends to $0$, that may be of independent interest.","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2016-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/tran/7191","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60695985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.
{"title":"Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)","authors":"Maki Nakasuji, H. Naruse","doi":"10.46298/dmtcs.6370","DOIUrl":"https://doi.org/10.46298/dmtcs.6370","url":null,"abstract":"International audience\u0000 \u0000 We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.
我们给出了最近由于Chapuy和Dołe ā ga对不可定向二部四边形的双射的不同表述，并将其推广到不可定向一般映射的情况。这可以看作是在一般非可进曲面的情况下，对Cori-Vauquelin-Schaeffer双射的一种类似于Bouttier-Di francesco - guitter的推广。在三角剖分的特殊情况下，编码对象采用特别简单的形式，我们恢复了高发现的一个著名的渐近枚举公式。
{"title":"A bijection for nonorientable general maps","authors":"Jérémie Bettinelli","doi":"10.46298/dmtcs.6398","DOIUrl":"https://doi.org/10.46298/dmtcs.6398","url":null,"abstract":"International audience\u0000 \u0000 We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Olivier Bernardi, Mireille Bousquet-M'elou, K. Raschel
Extended abstract presented at the conference FPSAC 2016, Vancouver. International audience In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).
{"title":"Counting quadrant walks via Tutte's invariant method (extended abstract)","authors":"Olivier Bernardi, Mireille Bousquet-M'elou, K. Raschel","doi":"10.46298/dmtcs.6416","DOIUrl":"https://doi.org/10.46298/dmtcs.6416","url":null,"abstract":"Extended abstract presented at the conference FPSAC 2016, Vancouver.\u0000 International audience\u0000 \u0000 In the 1970s, Tutte developed a clever algebraic approach, based on certain \" invariants \" , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
International audience We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.
{"title":"Asymptotics of lattice walks via analytic combinatorics in several variables","authors":"S. Melczer, Mark C. Wilson","doi":"10.46298/dmtcs.6390","DOIUrl":"https://doi.org/10.46298/dmtcs.6390","url":null,"abstract":"International audience\u0000 \u0000 We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.\u0000","PeriodicalId":55175,"journal":{"name":"Discrete Mathematics and Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2015-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70481759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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