The stupendous abundance of unicursal polyhedral triangulations

Since the last post I have been using Gilles Schaeffer's C program planarmap to randomly sample larger polyhedral triangulations (3-vertex-connected triangulations on the sphere.) For example it took less than half an hour for planarmap to output 108,223 random samples of the 256-triangle polyhedral triangulations, and a bit more than 2 hours for SageMath to sort through them and find that 1,416 of these were unicursal: p_unicursal ≈ 1.3%.

That sounds disappointing until you look at the actual totals. The integer sequence A000109 at the Online Encyclopedia of Integer Sequences counts polyhedral triangulations by the number of vertices up to 23 vertices (which is 2v-4 = 42 triangles). But at that size and larger, Tutte's asymptote, Q(n), (n is half the number of triangles), is a close approximation of the number of polyhedral triangulations.

Q(n) = (1÷(64 × SQRT(6×PI))) × (n^(−7÷2)) × (256÷27)^n

For example, the last term available in A000109 is 28615703421545 (2.86 E13) at 23 vertices (42 triangles, n=21), Q(n) = 2.77 E13.

Since we can use Q(n) as an estimate of the total number of polyhedral triangulations of any size, we just need a statistically significant random sample to estimate the total number of those polyhedral triangulations that are unicursal. This approach does come to a stop eventually as the generation of random samples (ever more samples being needed for significance as p_unicursal -> 0) and testing for non-singularity on larger and larger Laplacian minors bogs down.

But look a the largest size I was able to sample from, triangulations with 256 triangles. That is quite a modest-sized triangulation in the shape modeling world, yet there are on the order of 10¹¹⁵ such triangulations (each topologically distinct) of which on the order of 10¹¹³ are unicursal!!