%display latex

load("../specifier.py")

We consider the class $\mathcal T = \mathrm{Av}(2413,1243,2341,531642,41352)$. Its only simple permutation is $3142$,and the non-simple avoided patterns are $2143$ and $34512$.

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

S

$$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array} \mathcal T_{0}= &\{ \bullet \} \uplus \oplus[\mathcal T_{1},\mathcal T_{2}]\uplus \oplus[\mathcal T_{1},\mathcal T_{3}]\uplus \oplus[\mathcal T_{4},\mathcal T_{2}]\uplus \ominus[\mathcal T_{5},\mathcal T_{0}]\uplus 3142[\mathcal T_{1},\mathcal T_{1},\mathcal T_{1},\mathcal T_{6}]\\ \mathcal T_{1}= &\{ \bullet \} \uplus \ominus[\mathcal T_{7},\mathcal T_{1}]\\ \mathcal T_{2}= &\{ \bullet \} \uplus \oplus[\mathcal T_{7},\mathcal T_{2}]\\ \mathcal T_{3}= &\oplus[\mathcal T_{8},\mathcal T_{2}]\uplus \ominus[\mathcal T_{9},\mathcal T_{6}]\\ \mathcal T_{4}= &\ominus[\mathcal T_{10},\mathcal T_{11}]\uplus \ominus[\mathcal T_{10},\mathcal T_{1}]\uplus \ominus[\mathcal T_{7},\mathcal T_{11}]\uplus 3142[\mathcal T_{1},\mathcal T_{1},\mathcal T_{1},\mathcal T_{6}]\\ \mathcal T_{5}= &\{ \bullet \} \uplus \oplus[\mathcal T_{1},\mathcal T_{1}]\uplus 3142[\mathcal T_{1},\mathcal T_{1},\mathcal T_{1},\mathcal T_{1}]\\ \mathcal T_{6}= &\{ \bullet \} \uplus \oplus[\mathcal T_{12},\mathcal T_{2}]\uplus \ominus[\mathcal T_{9},\mathcal T_{6}]\\ \mathcal T_{7}= &\{ \bullet \} \\ \mathcal T_{8}= &\ominus[\mathcal T_{9},\mathcal T_{6}]\\ \mathcal T_{9}= &\{ \bullet \} \uplus \oplus[\mathcal T_{1},\mathcal T_{7}]\\ \mathcal T_{10}= &\oplus[\mathcal T_{1},\mathcal T_{1}]\uplus 3142[\mathcal T_{1},\mathcal T_{1},\mathcal T_{1},\mathcal T_{1}]\\ \mathcal T_{11}= &\oplus[\mathcal T_{1},\mathcal T_{2}]\uplus \oplus[\mathcal T_{1},\mathcal T_{3}]\uplus \oplus[\mathcal T_{4},\mathcal T_{2}]\uplus \ominus[\mathcal T_{10},\mathcal T_{11}]\uplus \ominus[\mathcal T_{10},\mathcal T_{1}]\uplus \ominus[\mathcal T_{7},\mathcal T_{11}]\uplus 3142[\mathcal T_{1},\mathcal T_{1},\mathcal T_{1},\mathcal T_{6}]\\ \mathcal T_{12}= &\{ \bullet \} \uplus \ominus[\mathcal T_{9},\mathcal T_{6}]\end{array}$$

S.solve()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-\frac{z^{7} - 7 \, z^{6} + 20 \, z^{5} - 28 \, z^{4} + 20 \, z^{3} - 7 \, z^{2} + z}{2 \, z^{7} - 13 \, z^{6} + 37 \, z^{5} - 62 \, z^{4} + 59 \, z^{3} - 32 \, z^{2} + 9 \, z - 1}, -\frac{z}{z - 1}, -\frac{z}{z - 1}, -\frac{z^{2}}{z^{3} - 4 \, z^{2} + 4 \, z - 1}, \frac{z^{8} - 4 \, z^{7} + 11 \, z^{6} - 13 \, z^{5} + 8 \, z^{4} - 2 \, z^{3}}{2 \, z^{7} - 13 \, z^{6} + 37 \, z^{5} - 62 \, z^{4} + 59 \, z^{3} - 32 \, z^{2} + 9 \, z - 1}, \frac{z^{5} - 2 \, z^{4} + 4 \, z^{3} - 3 \, z^{2} + z}{z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1}, -\frac{z^{2} - z}{z^{2} - 3 \, z + 1}, z, \frac{z^{2}}{z^{2} - 3 \, z + 1}, -\frac{z}{z - 1}, \frac{2 \, z^{4} - 2 \, z^{3} + z^{2}}{z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1}, \frac{z^{8} - 5 \, z^{7} + 10 \, z^{6} - 14 \, z^{5} + 11 \, z^{4} - 5 \, z^{3} + z^{2}}{2 \, z^{8} - 15 \, z^{7} + 50 \, z^{6} - 99 \, z^{5} + 121 \, z^{4} - 91 \, z^{3} + 41 \, z^{2} - 10 \, z + 1}, \frac{z^{3} - 2 \, z^{2} + z}{z^{2} - 3 \, z + 1}\right]$$

We can check that critical families are $\mathcal T_0,\mathcal T_4,\mathcal T_{11}$, with $\{\mathcal T_4,\mathcal T_{11}\}$ forming a strongly connected component, and that we are in the essentially linear case. We find below that smallest root $\rho$ of the denominator of $T_0$. Hence $\rho$ is the radius of convergence of our class. We store a numerical approximation along with its minimal polynomial.

S.solve()[0].denominator().factor()

$$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(2 \, z^{5} - 7 \, z^{4} + 14 \, z^{3} - 13 \, z^{2} + 6 \, z - 1\right)} {\left(z^{2} - 3 \, z + 1\right)}$$

poly = 2*z^5-7*z^4+14*z^3-13*z^2+6*z-1

rho_approx = poly.find_root(0,1)

rho_approx

$$\newcommand{\Bold}[1]{\mathbf{#1}}0.337160300927$$

#plot(S.solve()[0].denominator().factor(),0,rho_approx) #uncomment to check that there are no other roots before rho

C = S.sampler(0.335)

perm = C.run(10)

list_plot(perm)

It seems that the limit is a X-permuton with $p_{\mathrm{left}}^+ = p_{\mathrm{right}}^-$. Indeed we are in the essentially linear case. We need to perform the analysis on a specification where critical families are strongly connected, that is $\{\mathcal T_{4}, \mathcal T_{11}\}$. As $\mathcal T_{11} \subset \mathcal T_{0}$ and $\mathcal T_{0} \setminus \mathcal T_{11}$ has a smaller growth rate than $\mathcal T_{0}$ (see below), these classes have the same asymptotics.

(S.solve()[0] - S.solve()[11]).simplify_full()

$$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{z}{z - 1}$$

To compute the parameters of the limit permuton, we retrieve from the specification the matrices $\mathbb M^\star, \mathbb D^{\mathrm{left},+},\mathbb D^{\mathrm{right},+},\mathbb D^{\mathrm{left},-},\mathbb D^{\mathrm{right},-}$.

M,Dlp,Drp,Dlm,Drm = S.linear_analysis([4,11])

M,Dlp,Drp,Dlm,Drm

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rr} 0 & z + \frac{2 \, z^{4} - 2 \, z^{3} + z^{2}}{z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1} \\ -\frac{z}{z - 1} & z + \frac{2 \, z^{4} - 2 \, z^{3} + z^{2}}{z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1} \end{array}\right), \left(\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}\right), \left(\begin{array}{rr} 0 & 0 \\ -\frac{1}{z - 1} + \frac{z}{{\left(z - 1\right)}^{2}} & 0 \end{array}\right), \left(\begin{array}{rr} 0 & \frac{2 \, {\left(4 \, z^{3} - 3 \, z^{2} + z\right)}}{z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1} - \frac{4 \, {\left(2 \, z^{4} - 2 \, z^{3} + z^{2}\right)} {\left(z^{3} - 3 \, z^{2} + 3 \, z - 1\right)}}{{\left(z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1\right)}^{2}} + 1 \\ 0 & \frac{2 \, {\left(4 \, z^{3} - 3 \, z^{2} + z\right)}}{z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1} - \frac{4 \, {\left(2 \, z^{4} - 2 \, z^{3} + z^{2}\right)} {\left(z^{3} - 3 \, z^{2} + 3 \, z - 1\right)}}{{\left(z^{4} - 4 \, z^{3} + 6 \, z^{2} - 4 \, z + 1\right)}^{2}} + 1 \end{array}\right), \left(\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}\right)\right)$$

We see that $\mathbb D^{\mathrm{right},-}$ and $\mathbb D^{\mathrm{left},+}$ are identically $0$. Hence we know immediately that $p_{\mathrm{left}}^+ = p_{\mathrm{right}}^-$.

plp,prm = 0,0

We create the number field $K = \mathbb Q(\rho)$ in which we will perform our computations. We specify the approximate value of $\rho$ for numerical evaluation purposes.

K.<rho> = NumberField(2*z^5 - 7*z^4 + 14*z^3 - 13*z^2 + 6*z - 1,embedding = RR(rho_approx))

We apply the matrices of interest in $z=\rho$.

Mrho = apply_symbolic_matrix(M,QQ,z,rho)
Drprho = apply_symbolic_matrix(Drp,QQ,z,rho)
Dlmrho = apply_symbolic_matrix(Dlm,QQ,z,rho)

Mrho,Dlmrho,Drprho

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rr} 0 & -\rho + 1 \\ 2 \rho^{4} - 5 \rho^{3} + 9 \rho^{2} - 4 \rho + 1 & -\rho + 1 \end{array}\right), \left(\begin{array}{rr} 0 & -20 \rho^{4} + 54 \rho^{3} - 92 \rho^{2} + 42 \rho - 1 \\ 0 & -20 \rho^{4} + 54 \rho^{3} - 92 \rho^{2} + 42 \rho - 1 \end{array}\right), \left(\begin{array}{rr} 0 & 0 \\ 8 \rho^{4} - 22 \rho^{3} + 39 \rho^{2} - 22 \rho + 6 & 0 \end{array}\right)\right)$$

The CAS has simplified the expression of $\mathbb M^\star(\rho), \mathbb D^{\mathrm{left},-}(\rho),\mathbb D^{\mathrm{right,+}}(\rho)$ thanks to the knoweldge of the minimal polynomial of $\rho$.

Let us now compute the left and right eigenvectors of $\mathbb M^\star(\rho)$.

index_one = Mrho.eigenvalues().index(1)

u = Matrix(Mrho.left_eigenvectors()[index_one][1][0])

v = transpose(Matrix(Mrho.right_eigenvectors()[index_one][1][0]))

We are now able to compute $p^{\mathrm{left},-}$ and $p^{\mathrm{right},+}$. Since the other two probabilities are $0$, we have $p^{\mathrm{right},+} = 1-p^{\mathrm{left},-}$. We then compute a numerical approximation and the minimal polynomial of $p^{\mathrm{left},-}$.

plm,prp = (u*Dlmrho*v)[0,0]/((u*Dlmrho*v)[0,0]+(u*Drprho*v)[0,0]),(u*Drprho*v)[0,0]/((u*Dlmrho*v)[0,0]+(u*Drprho*v)[0,0])

plm

$$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{192}{599} \rho^{4} + \frac{600}{599} \rho^{3} - \frac{1119}{599} \rho^{2} + \frac{1507}{1198} \rho + \frac{343}{599}$$

plm.n()

$$\newcommand{\Bold}[1]{\mathbf{#1}}0.818632668576995$$

19168*minimal_polynomial(plm)

$$\newcommand{\Bold}[1]{\mathbf{#1}}19168 x^{5} - 86256 x^{4} + 155880 x^{3} - 141412 x^{2} + 64394 x - 11773$$

We plot the shape of the limiting permuton.

show(plot_X_permuton(plp,plm,prp,prm),aspect_ratio=1, axes = False)