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$n$ inner points in a triangular convex hull

$C$ = Catalan numbers

$A$ = abstract stacked triangulations = ternary trees with $n$ inner
nodes and $2n-1$ leaves = $\binom{3n+1}{n}/(3n+1)\approx 6.75^n$
% stmin.py

$D$ = 3 chains, of lengths $_{i,j,k}$. The limit exponent of 4.5 has
been established by Marc.
% 3chains.py

$T_{\min},T_{\max}$ all triangulations of a point set (Oswin, from the
database)

$ST_{\min},ST_{\max}$ stacked triangulations of a point set (Oswin, from the
database)

$ST_{\mathrm{avg}}$
Average over all realizable order types of the given
cardinality (Oswin)

$E_{\min}$ %,E_{\max}$ 
lower %/upper
 bounds on stacked triangulations (G\"unter)
% bestperm.py
It seems that the lower-bound examples (for $n$ a multiple of 3) look
like a series of nested concentric
triangles, where successive levels are rotated by 180 degrees.
The points lying close to a line through the center (they lie on both
sides of the center) are probably not uniformly ``curved'' but they
lie in such a way that a line through two such points cuts the points
between them evenly. It remains to define such a family precisely and
count
the stacked triangulations for this family.

$U=$ something random
defined by the recursion
$U_n = \sum_{i=1}^n U_{n-i}\cdot
   \frac{\sum_{j=1}^{i-1} U_{j-1}U_{i-1-j}}{i-1}  
$.
(For $i=1$, the value of the fraction is taken as 1.)
%     ( sum(f[j-1]*f[i-1-j] for j in range(1,i))/float(i-1) if i>1 else 1)
%     * f[n-i] for i in range(1,n+1))
% rnadperm.py
Maybe this is something related to the degree-3-vertices?


$R$ = average number of stacked triangulations on a random set, according to
Emo's recursion
$R_n = \sum_{i+j+k=n-1}R_iR_jR_k\cdot \frac 2{n+1}$.
% rnadperm.py


Prob[the balanced stacked triangulation with $n$ inner vertices can
be embedded on a random point set] $\approx 0.61886974^n$ (when $n$ is
of the form $(3^k-1)/2$).
\\
This is probably $>$
Prob[any other fixed stacked triangulation with $n$ inner vertices can
be embedded on a random point set].

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