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This paper provides the results on the enumerations of rooted simple outerplanar maps, rooted outerplanar near-triangulations, rooted 2-connected near-triangulations, rooted strict 2-connected near-triangulations ... and rooted simple 2-connected near-triangulations. ... f=O-82 We now construct a function g(x,z) as follows: d--O2 x= 1 +z02(1 -0)' g=o-82. (6.6) (6.7) (6.8) Counting rooted near-triangulations OH the sphere It can be seen that g(x, 1) =f(x). ...doi:10.1016/0012-365x(93)90005-e fatcat:lgfmgfsujbd5dpsxkqybedpsxi
We study 3-connected maps, 3-connected triangulations. and simple triangulations on arbitrary surfaces. ... It is shown that the radii of convergence do not depend on the surface and that the number of maps grows in a smooth manner. ... a canonical way in the root face of a map counted by m,(O). ...doi:10.1016/0095-8956(92)90036-w fatcat:lxawkh3agfhtnl3cv5hcnybg64
Combinatorics, Complexity, and Chance
We survey recent progress on the enumeration of labelled planar graph and on the distribution of several parameters in random planar graphs. ... The emphasis is on presenting the main ideas while keeping technical details to a minimum. ... Let us sketch how Tutte  solved the problem of counting near-triangulations. ...doi:10.1093/acprof:oso/9780198571278.003.0014 fatcat:o2zbivji25gglf5yt2psdsxvri
Haruo Hosoya (J-OCHS-I; Bunkyo) 94j:05064 05C30 05A15 Dong, Feng Ming (PRC-ASBJ-AM; Beijing); Liu, Yan Pei (PRC-ASBJ-AM; Beijing) Counting rooted near-triangulations on the sphere. ... The authors obtain formulas for the number of the following rooted maps enumerated by the num- ber of edges and root face degree: simple outerplanar, outerplanar near-triangulations, 2-connected near-triangulations ...
Summary: “It is shown that constructing generating functions with one even and one odd variable can help in the weighted counting of planar rooted trees. ... A near-triangulation is of type [n,m] if it has n interior vertices, and has m vertices in the boundary of the exterior face; it is rooted if an edge of the exterior face is selected as the root, and oriented ...
A survey is given of the asymptotic enumeration of maps. The asymptotic formulas for both rooted and unrooted maps on surfaces are discussed. ... The techniques used to derive these results are briefly discribed. The survey concludes with open problems and areas for future research. ... The Note added in proof: The conjectured asymptotic formula for the number of maps on a fixed surface has been verified for l-c maps on any surface by Bender and Canlield and for 2-c maps by Bender and ...doi:10.1016/0095-8956(86)90086-9 fatcat:k5lrnkgi55hirgoeifwhpgkkqe
We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. ... This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e. for a boundary component separating the ball from infinity). ... Multi-rooted triangulations One can consider RNT as a triangulation of a disk, or of a sphere with a hole. The disk is obtained from a sphere by cutting the rooted face of a RNT. ...doi:10.1007/s10958-005-0424-4 fatcat:3qkyn2rqq5dv5dlwmkl7dsaqne
Let s0 and pii be the number of rooted loopless near-triangular maps on the sphere and projective plane, respectively, having i vertices and root face valency j. ... Since a single vertex is a map only on the sphere, which is counted by the last term 1 in (2.1), we will assume that our map has edges from now on. ...doi:10.1016/0095-8956(91)90058-r fatcat:hpcdg2s7qjeqfcvc23pdn7cjwe
near-triangulations on the Mobius band. ... In this paper, we count rooted non- separable near-triangulations on the Mdbius band by the root- face valency, the size and the number of inner faces, and give a parametric expression by which an explicit ...
The proof follows and adapts Tutte's solution of properly q-coloured triangulations (1973-1984). ... We present recent results on the enumeration of q-coloured planar maps, where each monochromatic edge carries a weight ν. ... Acknowledgements The author gratefully acknowledges the assistance of Olivier Bernardi in writing this survey. ...arXiv:2004.08792v1 fatcat:3yj3frugm5e7zp6tkxuxbyot7m
In particular using Walkup's theorem we prove that the dominating configurations in the elongated phase are tree-like structures called "stacked spheres". ... Such configurations can be mapped into branched polymers and baby universes arguments are used in order to analyse the critical behaviour of theory in the weak coupling regime. ... an upper bound on the number of , respectively, rooted and unrooted inequivalent stacked spheres. ...arXiv:hep-lat/9711018v2 fatcat:6ntoobaiqvg6rmymkessauxzpq
An ideal triangulation of S defines, for each rank m, a quiver Q(Δ_m), hence a CY_3-category (C,W) for any potential W on Q(Δ_m). ... We show that for ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of (Y,ω) is quasi-isomorphic to (C,W_[ω]) for a certain generic potential W_[ω]. ... Definition 4.5 An eigen-ordering of a generic tuple Φ is a choice of ordering of the roots of Φ(b) = 0 near each point p ∈ D. ...arXiv:2002.10735v1 fatcat:awhzmyik6nd6lgq7fbcyeydq7y
The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. ... Our proofs first follow Tutte's classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration ... Consider a non-separable near-triangulation distinct from L rooted on a digon (i.e. the root-face has degree 2). ...doi:10.1007/s00026-008-0334-5 fatcat:iol5tiajibciheaxlatretspym
The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. ... Our proofs first follow Tutte's classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration ... Consider a non-separable near-triangulation distinct from L rooted on a digon (i.e. the root-face has degree 2). ...arXiv:math/0601678v1 fatcat:5izt573bendurkdiyeye2ocz7u
Near the branch points (the cube roots of 1) one argues as follows. Let p be such a cube root so that p3 = 1. ... Now since our branch points are vertexes in the triangulation, it is easy to see that the inverse image of the triangulation under r is a triangulation of S and simply counting we see that the number d ...doi:10.1016/0001-8708(75)90147-4 fatcat:aw7e4oklyrhfrpdsz2y6etyyxu
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