Stretched exponential asymptotics for bases of triangular bootstrap percolation

In this paper, we study a bootstrap percolation process on the finite triangular grid ${\mathfrak{T}}_n$ of side length $n$. We say that a subset $\eta$ of points in ${\mathfrak{T}}_n$ percolates if the final configuration, starting from $\eta$, is the whole grid ${\mathfrak{T}}_n$. A basis of size $n$ is then a subset of points of ${\mathfrak{T}}_n$ of minimum cardinality which percolates. In this paper, we first prove that the generating function counting bases satisfies an algebraic differential equation. Then, by analysing a modified version of this equation, we prove that the number $t_n$ of bases of size $n$ exhibits a stretched exponential asymptotic behaviour. More precisely, we show that $t_n\sim cn!e^{\sqrt{12n}}{n}^{5\mathrm{/}12}$, for some constant $c>0$. These bases were recently shown by the second author to be in bijection with $3$-permutations avoiding the patterns $(12,12)$ and $(231,312)$, so this represents to our knowledge the first proven example of an asymptotic stretched exponential appearing in the study of pattern avoiding permutations.