%display latex

load("../specifier.py")

We consider the class $\mathcal T = \mathrm{Av}(2413,3142, 2143,3412)$. It contains no simple permutation, and the non-simple avoided patterns are $2143$ and $3412$.

S = Specification([],[[2,1,4,3],[3,4,1,2]])

S

$$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array} \mathcal T_{0}= &\{ \bullet \} \uplus \oplus[\mathcal T_{1},\mathcal T_{2}]\uplus \oplus[\mathcal T_{1},\mathcal T_{3}]\uplus \oplus[\mathcal T_{4},\mathcal T_{2}]\uplus \ominus[\mathcal T_{1},\mathcal T_{5}]\uplus \ominus[\mathcal T_{1},\mathcal T_{6}]\uplus \ominus[\mathcal T_{7},\mathcal T_{5}]\\ \mathcal T_{1}= &\{ \bullet \} \\ \mathcal T_{2}= &\{ \bullet \} \uplus \oplus[\mathcal T_{1},\mathcal T_{2}]\\ \mathcal T_{3}= &\oplus[\mathcal T_{1},\mathcal T_{3}]\uplus \oplus[\mathcal T_{4},\mathcal T_{2}]\uplus \ominus[\mathcal T_{1},\mathcal T_{5}]\uplus \ominus[\mathcal T_{1},\mathcal T_{6}]\uplus \ominus[\mathcal T_{7},\mathcal T_{5}]\\ \mathcal T_{4}= &\ominus[\mathcal T_{1},\mathcal T_{5}]\uplus \ominus[\mathcal T_{1},\mathcal T_{6}]\uplus \ominus[\mathcal T_{7},\mathcal T_{5}]\\ \mathcal T_{5}= &\{ \bullet \} \uplus \ominus[\mathcal T_{1},\mathcal T_{5}]\\ \mathcal T_{6}= &\oplus[\mathcal T_{1},\mathcal T_{2}]\uplus \oplus[\mathcal T_{1},\mathcal T_{3}]\uplus \oplus[\mathcal T_{4},\mathcal T_{2}]\uplus \ominus[\mathcal T_{1},\mathcal T_{6}]\uplus \ominus[\mathcal T_{7},\mathcal T_{5}]\\ \mathcal T_{7}= &\oplus[\mathcal T_{1},\mathcal T_{2}]\uplus \oplus[\mathcal T_{1},\mathcal T_{3}]\uplus \oplus[\mathcal T_{4},\mathcal T_{2}]\end{array}$$

S.solve()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-\frac{2 \, z^{2} - z}{2 \, z^{2} - 4 \, z + 1}, z, -\frac{z}{z - 1}, -\frac{z^{2}}{2 \, z^{3} - 6 \, z^{2} + 5 \, z - 1}, -\frac{z^{3} - z^{2}}{2 \, z^{2} - 4 \, z + 1}, -\frac{z}{z - 1}, -\frac{z^{2}}{2 \, z^{3} - 6 \, z^{2} + 5 \, z - 1}, -\frac{z^{3} - z^{2}}{2 \, z^{2} - 4 \, z + 1}\right]$$

We can check that critical families are $\mathcal T_0,\mathcal T_3,\mathcal T_6,\mathcal T_7$, with $\mathcal T_3,\mathcal T_6,\mathcal T_7$ forming a strongly connected component, and that we are in the essentially linear case. We see below that the denominator of $T_0$ has two real roots, and $\rho$ is the smallest one.

[ x.n() for x in S.solve()[0].denominator().roots(multiplicities=False)]

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[0.292893218813452, 1.70710678118655\right]$$

C = S.sampler(0.29)

perm = C.run(10)

list_plot(perm,aspect_ratio=1)

It seems that the limiting permuton is the symmetric X-permuton. This is indeed the case thanks to our results.