Pub Date : 2023-09-20DOI: 10.1007/s40598-023-00237-2
David Burrell, William Cocke, Ryan McCulloch
We investigate the question of how many subgroups of a finite group are not in its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu (Bull Aust Math Soc 107(3):451–455, 2023) asked how many subgroups are not in the Chermak–Delgado lattice and classified all groups with two or less subgroups not in the Chermak–Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak–Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak–Delgado lattice is nilpotent. In this vein, we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak–Delgado lattice is (S_3).
{"title":"On Groups with Few Subgroups not in the Chermak–Delgado Lattice","authors":"David Burrell, William Cocke, Ryan McCulloch","doi":"10.1007/s40598-023-00237-2","DOIUrl":"10.1007/s40598-023-00237-2","url":null,"abstract":"<div><p>We investigate the question of how many subgroups of a finite group are not in its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu (Bull Aust Math Soc 107(3):451–455, 2023) asked how many subgroups are not in the Chermak–Delgado lattice and classified all groups with two or less subgroups not in the Chermak–Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak–Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak–Delgado lattice is nilpotent. In this vein, we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak–Delgado lattice is <span>(S_3)</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136309293","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}
Pub Date : 2023-09-13DOI: 10.1007/s40598-023-00236-3
Manish Kumar
In this paper, the normality of a family of meromorphic functions is deduced from the normality of a given family. Precisely, we have proved: Let ({mathcal {F}}) and ({mathcal {G}}) be two families of meromorphic functions on a domain D, and (a, b, c) be three finite complex numbers such that (ane 0) and (bne c). Suppose that ({mathcal {G}}) is normal in D such that no sequence in ({mathcal {G}}) converges locally uniformly to infinity in D. If (nge 3) and for each function (fin {mathcal {F}}) there exists (gin {mathcal {G}}) such that (f^{'}-af^{n}) and (g^{'}-ag^{n}) partially share the values b and c, then ({mathcal {F}}) is normal in D. Further, examples are given to establish the sharpness of the result.
{"title":"Normality of Two Families of Meromorphic Functions Concerning Partially Shared Values","authors":"Manish Kumar","doi":"10.1007/s40598-023-00236-3","DOIUrl":"10.1007/s40598-023-00236-3","url":null,"abstract":"<div><p>In this paper, the normality of a family of meromorphic functions is deduced from the normality of a given family. Precisely, we have proved: Let <span>({mathcal {F}})</span> and <span>({mathcal {G}})</span> be two families of meromorphic functions on a domain <i>D</i>, and <span>(a, b, c)</span> be three finite complex numbers such that <span>(ane 0)</span> and <span>(bne c)</span>. Suppose that <span>({mathcal {G}})</span> is normal in <i>D</i> such that no sequence in <span>({mathcal {G}})</span> converges locally uniformly to infinity in <i>D</i>. If <span>(nge 3)</span> and for each function <span>(fin {mathcal {F}})</span> there exists <span>(gin {mathcal {G}})</span> such that <span>(f^{'}-af^{n})</span> and <span>(g^{'}-ag^{n})</span> partially share the values <i>b</i> and <i>c</i>, then <span>({mathcal {F}})</span> is normal in <i>D</i>. Further, examples are given to establish the sharpness of the result.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135691373","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}
Pub Date : 2023-08-22DOI: 10.1007/s40598-023-00235-4
Alperen A. Ergür, Jesus Rebollo Bueno, Petros Valettas
We investigate the effect of an (varepsilon )-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric d-tensor f, a norm (leftVert cdot rightVert ) on the space of symmetric d-tensors, and (varepsilon >0) are given. What is the smallest symmetric tensor rank in the (varepsilon )-neighborhood of f? In other words, what is the symmetric tensor rank of f after a clever (varepsilon )-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind, we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.
{"title":"Approximate Real Symmetric Tensor Rank","authors":"Alperen A. Ergür, Jesus Rebollo Bueno, Petros Valettas","doi":"10.1007/s40598-023-00235-4","DOIUrl":"10.1007/s40598-023-00235-4","url":null,"abstract":"<div><p>We investigate the effect of an <span>(varepsilon )</span>-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric <i>d</i>-tensor <i>f</i>, a norm <span>(leftVert cdot rightVert )</span> on the space of symmetric <i>d</i>-tensors, and <span>(varepsilon >0)</span> are given. What is the smallest symmetric tensor rank in the <span>(varepsilon )</span>-neighborhood of <i>f</i>? In other words, what is the symmetric tensor rank of <i>f</i> after a clever <span>(varepsilon )</span>-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind, we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45683166","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}
Pub Date : 2023-08-14DOI: 10.1007/s40598-023-00233-6
Johannes Hofscheier, Askold Khovanskii, Leonid Monin
The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over ({mathbb {C}}) can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety G/B. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties G/B previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011).
{"title":"Cohomology Rings of Toric Bundles and the Ring of Conditions","authors":"Johannes Hofscheier, Askold Khovanskii, Leonid Monin","doi":"10.1007/s40598-023-00233-6","DOIUrl":"10.1007/s40598-023-00233-6","url":null,"abstract":"<div><p>The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over <span>({mathbb {C}})</span> can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety <i>G</i>/<i>B</i>. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties <i>G</i>/<i>B</i> previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11265003/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49669473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1007/s40598-023-00234-5
Vladimir Yu. Rovenski, Vladimir A. Sharafutdinov
A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. We describe the local structure of Gavrilov flows in terms of the geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for axisymmetric Gavrilov flows and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus.
{"title":"Steady-State Flows of Ideal Incompressible Fluid with Velocity Pointwise Orthogonal to the Pressure Gradient","authors":"Vladimir Yu. Rovenski, Vladimir A. Sharafutdinov","doi":"10.1007/s40598-023-00234-5","DOIUrl":"10.1007/s40598-023-00234-5","url":null,"abstract":"<div><p>A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. We describe the local structure of Gavrilov flows in terms of the geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for axisymmetric Gavrilov flows and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44985350","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}
Pub Date : 2023-07-25DOI: 10.1007/s40598-023-00231-8
S. Finashin, V. Kharlamov
We prove that the space of affine, transversal at infinity, nonsingular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.
{"title":"On Affine Real Cubic Surfaces","authors":"S. Finashin, V. Kharlamov","doi":"10.1007/s40598-023-00231-8","DOIUrl":"10.1007/s40598-023-00231-8","url":null,"abstract":"<div><p>We prove that the space of affine, transversal at infinity, nonsingular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46485315","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}
Pub Date : 2023-07-06DOI: 10.1007/s40598-023-00232-7
Dimitry Leites, Irina Shchepochkina
We describe supertraces on “queerifications” (see arXiv:2203.06917) of the algebras of matrices of “complex size”, algebras of observables of Calogero–Moser model, Vasiliev higher spin algebras, and (super)algebras of pseudo-differential operators. In the latter case, the supertraces establish complete integrability of the analogs of Euler equations to be written (this is one of several open problems and conjectures offered).
{"title":"Supertraces on Queerified Algebras","authors":"Dimitry Leites, Irina Shchepochkina","doi":"10.1007/s40598-023-00232-7","DOIUrl":"10.1007/s40598-023-00232-7","url":null,"abstract":"<div><p>We describe supertraces on “queerifications” (see arXiv:2203.06917) of the algebras of matrices of “complex size”, algebras of observables of Calogero–Moser model, Vasiliev higher spin algebras, and (super)algebras of pseudo-differential operators. In the latter case, the supertraces establish complete integrability of the analogs of Euler equations to be written (this is one of several open problems and conjectures offered).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48521752","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}
Pub Date : 2023-06-21DOI: 10.1007/s40598-023-00229-2
Daniel Barlet
The aim of this paper is to study in details the regular holonomic (D-)module introduced in Barlet (Math Z 302 (n^03): 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface ({Delta (sigma ).sigma _k = 0 }) are given by the local system generated by the power (lambda ) of the local branches of the multivalued function which is the root of the universal degree k equation (z^k + sum _{h=1}^k (-1)^hsigma _hz^{k-h} = 0 ). We show that for (lambda in mathbb {C} {setminus } mathbb {Z}) this D-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set ({sigma _kDelta (sigma ) not = 0}). We then study the structure of these D-modules in the cases where (lambda = 0, 1, -1) which are a little more complicated, but which are sufficient to determine the structure of all these D-modules when (lambda ) is in (mathbb {Z}). As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near (-1) of the equation:
本文旨在详细研究 Barlet (Math Z 302 (n^03):1627-1655, 2022 arXiv:1911.09347 [math])中引入的正则全局模块,其极性超曲面外的局部解是 {Delta (sigma ).(z^k + sum _{h=1}^k (-1)^hsigma _hz^{k-h} = 0 )的根的多值函数的局部分支的幂(lambda )产生的局部系统给出。)我们证明了对于 ((lambda in mathbb {C} {setminus } mathbb {Z}/))这个 D 模块是全纯向量束的最小扩展,它有一个可积分的全纯连接,这个连接有一个简单极点,是它在开集 ({sigma _kDelta (sigma ) not = 0}) 上的限制。然后我们研究这些D模块在(lambda = 0, 1, -1) 的情况下的结构,这些情况稍微复杂一些,但是当(lambda )在(mathbb {Z})中时,它们足以决定所有这些D模块的结构。作为一个应用,我们展示了这些结果如何允许计算,例如,在方程的(-1)附近根的泰勒展开:$$begin{aligned} z^k + sum _{h=-1}^k (-1)^hsigma _hz^{k-h} - (-1)^k = 0. end{aligned}$$(z^k-(-1)^k=0)附近。
{"title":"On Partial Differential Operators Which Annihilate the Roots of the Universal Equation of Degree k","authors":"Daniel Barlet","doi":"10.1007/s40598-023-00229-2","DOIUrl":"10.1007/s40598-023-00229-2","url":null,"abstract":"<div><p>The aim of this paper is to study in details the regular holonomic <span>(D-)</span>module introduced in Barlet (Math Z 302 <span>(n^03)</span>: 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface <span>({Delta (sigma ).sigma _k = 0 })</span> are given by the local system generated by the power <span>(lambda )</span> of the local branches of the multivalued function which is the root of the universal degree <i>k</i> equation <span>(z^k + sum _{h=1}^k (-1)^hsigma _hz^{k-h} = 0 )</span>. We show that for <span>(lambda in mathbb {C} {setminus } mathbb {Z})</span> this <i>D</i>-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set <span>({sigma _kDelta (sigma ) not = 0})</span>. We then study the structure of these <i>D</i>-modules in the cases where <span>(lambda = 0, 1, -1)</span> which are a little more complicated, but which are sufficient to determine the structure of all these <i>D</i>-modules when <span>(lambda )</span> is in <span>(mathbb {Z})</span>. As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near <span>(-1)</span> of the equation: </p><div><div><span>$$begin{aligned} z^k + sum _{h=-1}^k (-1)^hsigma _hz^{k-h} - (-1)^k = 0. end{aligned}$$</span></div></div><p>near <span>(z^k - (-1)^k = 0)</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46511272","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}
Pub Date : 2023-06-01DOI: 10.1007/s40598-023-00230-9
Sen-Peng Eu, Tung-Shan Fu
For the calculation of Springer numbers (of root systems) of type (B_n) and (D_n), Arnold introduced a signed analogue of alternating permutations, called (beta _n)-snakes, and derived recurrence relations for enumerating the (beta _n)-snakes starting with k. The results are presented in the form of double triangular arrays ((v_{n,k})) of integers, (1le |k|le n). An Arnold family is a sequence of sets of such objects as (beta _n)-snakes that are counted by ((v_{n,k})). As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of (tan x) and (sec x), established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.
{"title":"Springer Numbers and Arnold Families Revisited","authors":"Sen-Peng Eu, Tung-Shan Fu","doi":"10.1007/s40598-023-00230-9","DOIUrl":"10.1007/s40598-023-00230-9","url":null,"abstract":"<div><p>For the calculation of Springer numbers (of root systems) of type <span>(B_n)</span> and <span>(D_n)</span>, Arnold introduced a signed analogue of alternating permutations, called <span>(beta _n)</span>-snakes, and derived recurrence relations for enumerating the <span>(beta _n)</span>-snakes starting with <i>k</i>. The results are presented in the form of double triangular arrays (<span>(v_{n,k})</span>) of integers, <span>(1le |k|le n)</span>. An Arnold family is a sequence of sets of such objects as <span>(beta _n)</span>-snakes that are counted by <span>((v_{n,k}))</span>. As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of <span>(tan x)</span> and <span>(sec x)</span>, established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42461307","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}
Pub Date : 2023-05-11DOI: 10.1007/s40598-023-00227-4
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, Vasilisa Shramchenko
We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.
{"title":"Enumeration of Multi-rooted Plane Trees","authors":"Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, Vasilisa Shramchenko","doi":"10.1007/s40598-023-00227-4","DOIUrl":"10.1007/s40598-023-00227-4","url":null,"abstract":"<div><p>We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41806065","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}