# Sources and conventions for T483

All maps are rooted at a directed edge, unlabelled, on the oriented sphere,
modulo orientation-preserving homeomorphisms. The exterior face is the face
on a fixed side of the root, not an additional independent marked face.

The six general classes use edges, triangulations use all vertices, and
quadrangulations use all faces. For triangulations v=n+2 and f=2n; shifting
an index by a fixed integer preserves the exponential growth constant.
Face degree counts boundary incidences, including a bridge twice.

* Noy, Requilé and Rué, Enumeration of rooted 3-connected bipartite planar
  maps, C. R. Math. 362 (2024), 143–158, doi:10.5802/crmath.548,
  arXiv:2202.13929v3, introduction pp. 2–4: definition of rooted maps,
  A(z), the 2-core and 3-core equations, and growth constants 12, 27/4,
  4 and 8. The new 3-connected bipartite class is outside this selection.
* Liskovets and Walsh, Counting unrooted maps on the plane, Adv. Appl.
  Math. 36 (2006), 364–387, doi:10.1016/j.aam.2005.03.006,
  https://www.labunix.uqam.ca/~walsh_t/papers/plane%20maps.pdf :
  Section 0 distinguishes spherical maps from maps with a further marked
  face and explains rooted duality; equation (4.1) counts loopless maps.
* Curien and Le Gall, First-passage percolation and local modifications of
  distances in random triangulations, Ann. Sci. ENS 52 (2019), 631–701,
  doi:10.24033/asens.2394, Section 2.1, equation (6) and Figure 2:
  type-I disk formula and bijection from spherical triangulations with v
  vertices to monogon triangulations with v-1 internal vertices.
  The introduction also states the angular bijection between arbitrary
  maps with n edges and quadrangulations with n faces.
* Angel and Schramm, Uniform Infinite Planar Triangulations,
  arXiv:math/0207153, definitions 1.2–1.3 and Section 2: type II and
  type III, their counting formulas and exponential rates.
* Albenque, Fusy and Poulalhon, On symmetric quadrangulations and
  triangulations, arXiv:1103.3657v2, Section 3.1: q=x(a-1)(2-a),
  a=1+x*a^3 for simple quadrangulations, counted by total faces.
  q=x^2+2*x^3+6*x^4+22*x^5+... . Section 3.2 gives the simple
  triangulation count 2(4n-3)!/(n!(3n-1)!) for n+2 vertices.

## Source discrepancies resolved before filling

The printed factorial formula in Corollary 3.3 of arXiv:1103.3657v2 is
4(3n)!/(n!(2n+2)!). This equals the coefficient of x^(n+1), not x^n,
in its own displayed series q(x). The source's displayed series and
parametrisation agree. Coefficient extraction from that parametrisation
instead gives q_n=2(3n-3)!/(n!(2n-1)!) for n>=2. The independent
permutation enumeration checks q_2=1. This is a one-face indexing shift;
both factorial expressions have limiting ratio 27/4. The draft cites the
parametrisation and uses the total-face indexing throughout.

A summary table in the Noy–Requilé–Rué arXiv version prints 125/8 for
2-connected bipartite maps, whereas the preceding asymptotic states
128/25. That class is not included here. No constant is taken from that
summary table; the four included constants are established explicitly in
the surrounding derivations.

## Verification

checks.py derives the rational growth rates from limiting exact count
ratios, checks the type-I two-step ratio and square-root enclosure, and
checks the 3-connected critical equation independently of the constant
returned by the generator. It enumerates planar rotation systems on up
to eight darts to compare actual small-map counts with the factorial
formulas. Its known controls are one edge (one loop and one bridge) and
Catalan enumeration of plane trees. It also tests exact formal series
identities and stored values. These checks supplement the cited counting
theorems; finite-size ratio convergence alone is not a proof of a limit.