Enumeration of unrooted maps of a given genus

Mednykh, Alexander; Nedela, Roman

doi:10.1016/j.jctb.2006.01.005

Cited by 50 publications

(78 citation statements)

References 35 publications

(42 reference statements)

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“…is related to the orbicyclic (multivariate arithmetic) function ( [21]), which has very interesting combinatorial and topological applications, in particular in counting non-isomorphic maps on orientable surfaces (see [3,21,23,24,37,40]). The problem is also related to Harvey's famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces; see Remark 3.14.…”

Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

“…. , m k ), has very interesting combinatorial and topological applications, in particular, in counting non-isomorphic maps on orientable surfaces, and was investigated in [3,21,23,37]. See also [24,40].…”

Section: Linear Congruences Withmentioning

confidence: 99%

“…Very recently, Bibak et al [3] using Theorem 3.9 as the main ingredient proved an explicit and practical formula for the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group (see also [23]). This problem has important applications in combinatorics, geometry, string theory, and quantum field theory (QFT).…”

Section: Linear Congruences Withmentioning

confidence: 99%

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Restricted linear congruences

Bibak

Kapron

Srinivasan

et al. 2017

Journal of Number Theory

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In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence, where a 1 , t 1 , . . . , a k , t k , b, n (n ≥ 1) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of a i = t i = 1 (1 ≤ i ≤ k). Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [Duke Math. J. 39 (1972), 521-527] gave a nice explicit formula for the number of such solutions when (a 1 , . . . , a k ) = t i = 1 (1 ≤ i ≤ k). The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.

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Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

Section: Linear Congruences Withmentioning

confidence: 99%

Section: Linear Congruences Withmentioning

confidence: 99%

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Restricted linear congruences

Bibak

Kapron

Srinivasan

et al. 2017

Journal of Number Theory

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“…Most of the results on map enumeration in the unrooted case are restricted to planar maps [14,15,30,31,16]. A recent paper [23] presents a breakthrough in the enumeration problem for unrooted maps of genus ≥ 1. In the present paper we apply the methods employed in [23] to solve an analogous problem for hypermaps.…”

Section: Introductionmentioning

confidence: 99%

Enumeration of unrooted hypermaps of a given genus

Mednykh

Nedela

2010

Discrete Mathematics

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a b s t r a c tIn this paper we derive an enumeration formula for the number of hypermaps of a given genus g and given number of darts n in terms of the numbers of rooted hypermaps of genus γ ≤ g with m darts, where m|n. Explicit expressions for the number of rooted hypermaps of genus g with n darts were derived by Walsh [T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory B 18 (2) (1975) 155-163] for g = 0, and by Arquès [D. Arquès, Hypercartes pointées sur le tore: Décompositions et dénombrements, J.Combin. Theory B 43 (1987) 275-286] for g = 1. We apply our general counting formula to derive explicit expressions for the number of unrooted spherical hypermaps and for the number of unrooted toroidal hypermaps with given number of darts. We note that in this paper isomorphism classes of hypermaps of genus g ≥ 0 are distinguished up to the action of orientation-preserving hypermap isomorphisms. The enumeration results can be expressed in terms of Fuchsian groups.

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“…The above definition of a map on an orbifold comes from [23]. The following proposition is a useful extension of Proposition 3.…”

Section: Two-dimensional Orbifolds and Maps On Orbifoldsmentioning

confidence: 95%

Archimedean maps of higher genera

2012

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Abstract. The paper focuses on the classification of vertex-transitive polyhedral maps of genus from 2 to 4. These maps naturally generalise the spherical maps associated with the classical Archimedean solids. Our analysis is based on the fact that each Archimedean map on an orientable surface projects onto a one-or a two-vertex quotient map. For a given genus g ≥ 2 the number of quotients to consider is bounded by a function of g. All Archimedean maps of genus g can be reconstructed from these quotients as regular covers with covering transformation group isomorphic to a group G from a set of g-admissible groups. Since the lists of groups acting on surfaces of genus 2, 3 and 4 are known, the problem can be solved by a computer-aided case-to-case analysis.

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Enumeration of unrooted maps of a given genus

Cited by 50 publications

References 35 publications

Restricted linear congruences

Restricted linear congruences

Enumeration of unrooted hypermaps of a given genus

Archimedean maps of higher genera

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