spp_matrices.sagews

Computation of Matrices for the SPP Point Process in $B(n,n,n)$

Consider the Pfaffian point process $\mathscr Y_{n,m}$ for symmetric plane partitions fitting in $B(n,n,m)$. The code below constructs the matrices $M_{n,m}$, $M_{n,m}^{-1}$, and $K_{n,m}$ associated with this process, in the special case $m = n$. We note, however, that the code can be adapted to other choices of $n$ and $m$. Recall that $M_{n,m} = (m_{i,j})_{i,j = \delta}^{n}$, where $\delta$ is the residue of $n+1$ modulo $2$, and where $$m_{i,j} = \sum_{x = \delta}^{n+m} \sum_{y = \delta}^{n+m} \epsilon(x,y) \, \phi_i(x) \, \phi_j(y) .$$ Here, we set $\epsilon(x,y) = \operatorname{sgn}(y-x)$ and $$\phi_k(y) = \begin{cases} \binom{y-1}{y-k} & \text{if } k>0 \\ \delta_0(y) & \text{if } k=0 . \end{cases}$$ Define the matrix $K_{n,m} = (K_{n,m}(x,y))_{x,y = 1}^{n+m}$, where the matrix blocks $K_{n,m}(x,y)$ are given by $$K_{n,m}(x,y) = \begin{pmatrix} \sum_{i,j = \delta}^{n} \phi_i(x) \, \phi_j(y) \, \tilde m_{j,i} & \sum_{i,j = \delta}^{n} \phi_i(x) \, (\epsilon \phi_j)(y) \, \tilde m_{j,i} \\ \sum_{i,j = \delta}^{n} (\epsilon \phi_i)(x) \, \phi_j(y) \, \tilde m_{j,i} & -\epsilon(x,y) + \sum_{i,j = \delta}^{n} (\epsilon \phi_i)(x) \, (\epsilon \phi_j)(y) \, \tilde m_{j,i} \end{pmatrix} .$$ Here, $\tilde m_{j,i}$ denotes entry $(j,i)$ of matrix $M_{n,m}^{-1}$, and $$(\epsilon \phi_k)(y) = \sum_{z = \delta}^{n+m} \epsilon(y,z) \, \phi_k(z) .$$

1. Computing and Saving the Values $\phi_k(y)$

Let $\Phi = (\phi_k(y))_{k \geq 0, y \geq 0}$ be the array containing all values $\phi_k(y)$, where $k$ indexes rows and $y$ indexes columns. Note that this array contains Pascal's triangle: $$\Phi = \begin{pmatrix} 1 & & & & & & \\ & 1 & 1 & 1 & 1 & 1 & \cdots \\ & & 1 & 2 & 3 & 4 & \cdots \\ & & & 1 & 3 & 6 & \cdots \\ & & & & 1 & 4 & \cdots \\ & & & & & 1 & \cdots \\ & & & & & & \ddots \end{pmatrix} .$$ Let $\Phi_{n,m} = (\phi_k(y))_{0 \leq k \leq n, 0 \leq y \leq n+m}$ denote a truncation of $\Phi$. Below, we create and save the array $\Phi_{N,N} = (\phi_k(y))_{0 \leq k \leq N, 0 \leq y \leq 2N}$, where $N$ is the index of the largest correlation kernel $K_{N,N}$ we wish to compute.

def phi(y, k):
    if k == 0 and y == 0:
        return(1)
    elif k == 0 and y != 0:
        return(0)
    else:
        return(binomial(y-1, y-k))

def eps(x, y):
    return(sgn(y-x))

def epsphi(y, k, n, m):
    delta = (n+1) % 2
    return(sum(eps(y, z) * phi(z, k) for z in [delta .. n+m]))

# define functions phi and epsphi

print('Done')

Done

import os
load('mat_to_list.sage')

N = 100

n = N
m = N
Phi = Matrix(ZZ, [[phi(x, k) for x in [0 .. n+m]] for k in [0 .. n]])

path = 'spp_data/'
file_name = 'Phi_({0},{1})'.format(n, n+m)
text_file = open(os.path.join(path, file_name), 'w')
text_file.write(str(mat_to_list(Phi)))
text_file.close()

# creates a text file 'Phi_(100,100).txt' in the directory 'spp_data/'

print('Done')

Done

2. Computing and Saving the Values $(\epsilon \phi_k)(y)$

With the array $\Phi_{N,N}$ saved, we now create arrays $E \Phi_{n,m} = ((\epsilon \phi_k)(y))_{\delta \leq k \leq n, 1 \leq y \leq n+m}$, which are needed for computation of the matrices $K_{n,m}$. Note that $$(\epsilon \phi_k)(y) = \sum_{z = \delta}^{m+n} \epsilon(y,z) \, \phi_k(z)$$ Depends on the dimensions $m$ and $n$ of the SPP point process, and that $$E \Phi_{n,m} = (\phi_k(z))_{\delta \leq k \leq n, \delta \leq z \leq n+m} (\epsilon(y,z))_{1 \leq y \leq n+m, \delta \leq z \leq n+m}^T$$ Below, we compute all arrays $E \Phi_{n,m}$ for $1 \leq n,m \leq N$, where $N = 100$ was chosen previously. Note that only the arrays $E \Phi_{n,n}$ will be needed in computing the kernels $K_{n,n}$.

N = 100

Phi_text = open(os.path.join('spp_data/', 'Phi_({0},{1})'.format(n, n+m)), 'r')
Phi_data = sage_eval(Phi_text.readline())
Phi = Matrix(ZZ, Phi_data)
Phi_text.close()

# loads phi values stored in 'Phi_(100,100).txt'

path = 'spp_data/spp_Epsphi'
for n in [1 .. N]:
    delta = (n+1) % 2
    for m in [1 .. N]:
        Eps = Matrix(ZZ, [[eps(y, z) for z in [delta .. n+m]] for y in [1 .. n+m]])
        Epsphi = Phi[delta:n+1, delta:n+m+1] * Eps.transpose()

        file_name = 'Epsphi_({0},{1})'.format(n, m)
        text_file = open(os.path.join(path, file_name), 'w')
        text_file.write(str(mat_to_list(Epsphi)))
        text_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

# creates a series of files 'Epsphi_(n,m).txt' in the directory 'spp_data/spp_Epsphi'

print('Done')

n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

3. Computing and Saving the Matrices $M_{n,n}$

Recall that $$M_{m,n} = (m_{i,j})_{i,j = \delta}^n = \left( \sum_{x = \delta}^{n+m} \sum_{y = \delta}^{n+m} \epsilon(x,y) \, \phi_i(x) \, \phi_j(y) \right)_{i,j = \delta}^n = (\phi_i(x))_{i,x} (\epsilon(x,y))_{x,y} (\phi_j(y))_{y,j} .$$ Note that on the right-hand side, $i$ and $j$ run from $\delta$ to $n$, and where $x$ and $y$ run from $\delta$ to $n+m$. We now compute and save the matrices $M_{n,n}$ for $1 \leq n \leq N = 100$.

def make_M(n, m, eps, Phi_arr):
    delta = (n+1) % 2
    Eps = Matrix(ZZ, [[eps(x, y) for y in [delta .. n+m]] for x in [delta .. n+m]])
    Phi = Phi_arr[delta:n+1, delta:n+m+1]
    return(Phi * Eps * Phi.transpose())

# creates M_(m,n) using the function eps and the complete array of phi values Phi_arr

N = 100
n = N
m = N

path = 'spp_data/'
file_name = 'Phi_({0},{1})'.format(n, n+m)

text_file = open(os.path.join(path, file_name), 'r')
Phi_arr_data = sage_eval(text_file.readline())
Phi_arr = Matrix(ZZ, Phi_arr_data)
text_file.close()

# loads phi values from 'Phi_(100,100).txt'

path = 'spp_data/spp_M/'
for n in [1 .. N]:
    M = make_M(n, n, eps, Phi_arr)
    file_name = 'M_({0},{0})'.format(n)
    text_file = open(os.path.join(path, file_name), 'w')
    text_file.write(str(mat_to_list(M)))
    text_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

print('Done')

n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

4. Computing and Saving the Matrices $M_{n,n}^{-1}$

By simply loading the matrices $M_{n,n}$ saved in the last step, we now compute the matrices $M_{n,n}^{-1}$ for $1 \leq n \leq N = 100$.

N = 100

for n in [1 .. N]:
    path = 'spp_data/spp_M/'
    file_name = 'M_({0},{0})'.format(n)
    text_file = open(os.path.join(path, file_name), 'r')
    M_data = sage_eval(text_file.readline())
    M = Matrix(ZZ, M_data)
    text_file.close()

    path = 'spp_data/spp_M_inv'
    file_name = 'M_inv_({0},{0})'.format(n)
    text_file = open(os.path.join(path, file_name), 'w')
    text_file.write(str(mat_to_list(M.inverse())))
    text_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

print('Done')

n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

5. Computing and Saving the Kernels $K_{n,n}$

Our final step is to compute the kernels $K_{n,n}$ for $1 \leq n \leq N = 100$. Recalling that $K_{n,n} = (K_{n,n}(x,y))_{x,y = 1}^{n+m}$, we construct four separate arrays $(K_{n,n}(x,y)_{\alpha, \beta})_{x,y = 1}^{n+m}$, where $1 \leq \alpha, \beta \leq 2$. Notice that $$(K_{n,n}(x,y)_{1,1})_{x,y = 1}^{n+m} = \left( \sum_{i=\delta}^{n} \sum_{j=\delta}^{n} \phi_i(x) \, \phi_j(y) \, \tilde m_{j,i} \right)_{x,y = 1}^{n+m} = (\phi_i(x))_{i,x}^T M_{n,m}^{-T} (\phi_j(y))_{j,y}$$ where $M_{n,m}^{-T}$ denotes the transpose of $M_{n,m}^{-1}$. On the right-hand side, $i$ and $j$ run from $\delta$ to $n$, and $x$ and $y$ run from $1$ to $n+m$. Similarly, we have $$\begin{align*} (K_{n,n}(x,y)_{1,2})_{x,y = 1}^{n+m} &= (\phi_i(x))_{i,x}^T M_{n,m}^{-T} ((\epsilon \phi_j)(y))_{j,y} \\ (K_{n,n}(x,y)_{2,1})_{x,y = 1}^{n+m} &= ((\epsilon \phi_i)(x))_{i,x}^T M_{n,m}^{-T} (\phi_j(y))_{j,y} \\ (K_{n,n}(x,y)_{2,2})_{x,y = 1}^{n+m} &= -(\epsilon(x,y))_{x,y = 1}^{n+m} + ((\epsilon \phi_i)(x))_{i,x}^T M_{n,m}^{-T} ((\epsilon \phi_j)(y))_{j,y} \end{align*}$$ These four arrays are then combined to form $K_{n,n}$.

def make_K(n, m, N):
    delta = (n+1) % 2

    Phi_text = open(os.path.join('spp_data/', 'Phi_({0},{1})'.format(N, N+N)), 'r')
    Phi_data = sage_eval(Phi_text.readline())
    Phi_arr = Matrix(ZZ, Phi_data)
    Phi = Phi_arr[delta:n+1, 1:n+m+1]
    Phi_text.close()

    Epsphi_text = open(os.path.join('spp_data/spp_Epsphi/', 'Epsphi_({0},{1})'.format(n, m)), 'r')
    Epsphi_data = sage_eval(Epsphi_text.readline())
    Epsphi = Matrix(ZZ, Epsphi_data)
    Epsphi_text.close()

    Eps = Matrix(ZZ, [[eps(x, y) for y in [1 .. n+m]] for x in [1 .. n+m]])

    M_inv_text = open(os.path.join('spp_data/spp_M_inv/', 'M_inv_({0},{1})'.format(n, m)))
    M_inv_data = sage_eval(M_inv_text.readline())
    M_inv = Matrix(QQ, M_inv_data)
    M_inv_text.close()

    # loads values of phi, epsphi, eps, and the inverse of matrix M_(n,m)

    K_11 = Phi.transpose() * M_inv.transpose() * Phi
    K_12 = Phi.transpose() * M_inv.transpose() * Epsphi
    K_21 = Epsphi.transpose() * M_inv.transpose() * Phi
    K_22 = -Eps + Epsphi.transpose() * M_inv.transpose() * Epsphi

    # computes the four arrays of entries of K_(n,m)

    L1 = [0 .. 2*(n+m-1), step=2]
    L2 = [1 .. 2*(n+m-1) + 1, step=2]
    K = Matrix(QQ, 2*(n+m))

    K[L1,L1] = K_11
    K[L1,L2] = K_12
    K[L2,L1] = K_21
    K[L2,L2] = K_22

    # combines the four arrays into K_(n,m)

    return(K)

print('Done')

Done

N = 100

for n in [1 .. N]:
    K = make_K(n, n, N)
    K_file = open(os.path.join('spp_data/spp_K/', 'K_({0},{0})').format(n), 'w')
    K_file.write(str(mat_to_list(K)))
    K_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

print('Done')

n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

Loading and Viewing Matrix Data

Below, we access and print a few of the matrices $M_{n,n}$, $M_{n,n}^{-1}$, and $K_{n,n}$:

def retrieve_mat(name, n, m):
    if name == 'M' or name == 'M_inv' or name == 'K':
        mat_file = open('spp_data/spp_{0}/{0}_({1},{2})'.format(name, n, m), 'r')
        mat_data = sage_eval(mat_file.readline())
        mat = Matrix(QQ, mat_data)
        mat_file.close()
        return(mat)
    else:
        print("Error: first argument must be 'M', 'M_inv', or 'K'")
        return

# reads in the matrix M_(n,m), M_inv_(n,m), or K_(n,m)

print('Done')

Done

a = 1
b = 7

print('Matrices M_({0},{0}) through M_({1},{1}) are as follows:\n'.format(a, b))

for n in [a .. b]:
    M = retrieve_mat('M', n, n)
    print('M_({0},{0}) ='.format(n))
    show(M)

Matrices M_(1,1) through M_(7,7) are as follows: M_(1,1) =

$\displaystyle \left(\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right)$

M_(2,2) =

$\displaystyle \left(\begin{array}{rr} 0 & 10 \\ -10 & 0 \end{array}\right)$

M_(3,3) =

$\displaystyle \left(\begin{array}{rrrr} 0 & 6 & 15 & 20 \\ -6 & 0 & 35 & 70 \\ -15 & -35 & 0 & 77 \\ -20 & -70 & -77 & 0 \end{array}\right)$

M_(4,4) =

$\displaystyle \left(\begin{array}{rrrr} 0 & 84 & 252 & 378 \\ -84 & 0 & 378 & 798 \\ -252 & -378 & 0 & 726 \\ -378 & -798 & -726 & 0 \end{array}\right)$

M_(5,5) =

$\displaystyle \left(\begin{array}{rrrrrr} 0 & 10 & 45 & 120 & 210 & 252 \\ -10 & 0 & 165 & 660 & 1386 & 1848 \\ -45 & -165 & 0 & 1254 & 3696 & 5742 \\ -120 & -660 & -1254 & 0 & 4422 & 9438 \\ -210 & -1386 & -3696 & -4422 & 0 & 7667 \\ -252 & -1848 & -5742 & -9438 & -7667 & 0 \end{array}\right)$

M_(6,6) =

$\displaystyle \left(\begin{array}{rrrrrr} 0 & 286 & 1430 & 3861 & 6864 & 8580 \\ -286 & 0 & 3289 & 12441 & 25740 & 35178 \\ -1430 & -3289 & 0 & 18447 & 52338 & 82225 \\ -3861 & -12441 & -18447 & 0 & 54197 & 115687 \\ -6864 & -25740 & -52338 & -54197 & 0 & 87074 \\ -8580 & -35178 & -82225 & -115687 & -87074 & 0 \end{array}\right)$

M_(7,7) =

$\displaystyle \left(\begin{array}{rrrrrrrr} 0 & 14 & 91 & 364 & 1001 & 2002 & 3003 & 3432 \\ -14 & 0 & 455 & 2730 & 9009 & 20020 & 32175 & 38610 \\ -91 & -455 & 0 & 7371 & 34034 & 87945 & 154440 & 196625 \\ -364 & -2730 & -7371 & 0 & 60203 & 213070 & 429715 & 593164 \\ -1001 & -9009 & -34034 & -60203 & 0 & 268125 & 733876 & 1156311 \\ -2002 & -20020 & -87945 & -213070 & -268125 & 0 & 686387 & 1460758 \\ -3003 & -32175 & -154440 & -429715 & -733876 & -686387 & 0 & 1040405 \\ -3432 & -38610 & -196625 & -593164 & -1156311 & -1460758 & -1040405 & 0 \end{array}\right)$

a = 1
b = 7

print('Matrices M_inv_({0},{0}) through M_inv_({1},{1}) are as follows:\n'.format(a, b))

for n in [a .. b]:
    M_inv = retrieve_mat('M_inv', n, n)
    print('M_inv_({0},{0}) ='.format(n))
    show(M_inv)

Matrices M_inv_(1,1) through M_inv_(7,7) are as follows: M_inv_(1,1) =

$\displaystyle \left(\begin{array}{rr} 0 & -\frac{1}{2} \\ \frac{1}{2} & 0 \end{array}\right)$

M_inv_(2,2) =

$\displaystyle \left(\begin{array}{rr} 0 & -\frac{1}{10} \\ \frac{1}{10} & 0 \end{array}\right)$

M_inv_(3,3) =

$\displaystyle \left(\begin{array}{rrrr} 0 & -\frac{11}{16} & \frac{5}{8} & -\frac{5}{16} \\ \frac{11}{16} & 0 & -\frac{5}{28} & \frac{15}{112} \\ -\frac{5}{8} & \frac{5}{28} & 0 & -\frac{3}{56} \\ \frac{5}{16} & -\frac{15}{112} & \frac{3}{56} & 0 \end{array}\right)$

M_inv_(4,4) =

$\displaystyle \left(\begin{array}{rrrr} 0 & -\frac{11}{42} & \frac{19}{66} & -\frac{3}{22} \\ \frac{11}{42} & 0 & -\frac{3}{22} & \frac{1}{11} \\ -\frac{19}{66} & \frac{3}{22} & 0 & -\frac{1}{33} \\ \frac{3}{22} & -\frac{1}{11} & \frac{1}{33} & 0 \end{array}\right)$

M_inv_(5,5) =

$\displaystyle \left(\begin{array}{rrrrrr} 0 & -\frac{13}{16} & 1 & -\frac{29}{32} & \frac{9}{16} & -\frac{3}{16} \\ \frac{13}{16} & 0 & -\frac{397}{1144} & \frac{2139}{4576} & -\frac{783}{2288} & \frac{285}{2288} \\ -1 & \frac{397}{1144} & 0 & -\frac{1721}{6864} & \frac{295}{1144} & -\frac{125}{1144} \\ \frac{29}{32} & -\frac{2139}{4576} & \frac{1721}{6864} & 0 & -\frac{15}{143} & \frac{35}{572} \\ -\frac{9}{16} & \frac{783}{2288} & -\frac{295}{1144} & \frac{15}{143} & 0 & -\frac{5}{286} \\ \frac{3}{16} & -\frac{285}{2288} & \frac{125}{1144} & -\frac{35}{572} & \frac{5}{286} & 0 \end{array}\right)$

M_inv_(6,6) =

$\displaystyle \left(\begin{array}{rrrrrr} 0 & -\frac{4183}{9724} & \frac{1620}{2431} & -\frac{175}{286} & \frac{79}{221} & -\frac{95}{884} \\ \frac{4183}{9724} & 0 & -\frac{1917}{4862} & \frac{95}{187} & -\frac{305}{884} & \frac{25}{221} \\ -\frac{1620}{2431} & \frac{1917}{4862} & 0 & -\frac{577}{2431} & \frac{49}{221} & -\frac{37}{442} \\ \frac{175}{286} & -\frac{95}{187} & \frac{577}{2431} & 0 & -\frac{35}{442} & \frac{9}{221} \\ -\frac{79}{221} & \frac{305}{884} & -\frac{49}{221} & \frac{35}{442} & 0 & -\frac{9}{884} \\ \frac{95}{884} & -\frac{25}{221} & \frac{37}{442} & -\frac{9}{221} & \frac{9}{884} & 0 \end{array}\right)$

M_inv_(7,7) =

$\displaystyle \left(\begin{array}{rrrrrrrr} 0 & -\frac{1137}{1280} & \frac{829}{640} & -\frac{193}{128} & \frac{11}{8} & -\frac{121}{128} & \frac{143}{320} & -\frac{143}{1280} \\ \frac{1137}{1280} & 0 & -\frac{26281}{51680} & \frac{181161}{206720} & -\frac{2849}{3040} & \frac{144639}{206720} & -\frac{35893}{103360} & \frac{37037}{413440} \\ -\frac{829}{640} & \frac{26281}{51680} & 0 & -\frac{754701}{1343680} & \frac{282953}{335920} & -\frac{981299}{1343680} & \frac{4081}{10336} & -\frac{1309}{12160} \\ \frac{193}{128} & -\frac{181161}{206720} & \frac{754701}{1343680} & 0 & -\frac{29425}{67184} & \frac{70147}{134368} & -\frac{8393}{25840} & \frac{9933}{103360} \\ -\frac{11}{8} & \frac{2849}{3040} & -\frac{282953}{335920} & \frac{29425}{67184} & 0 & -\frac{14567}{67184} & \frac{2373}{12920} & -\frac{189}{3040} \\ \frac{121}{128} & -\frac{144639}{206720} & \frac{981299}{1343680} & -\frac{70147}{134368} & \frac{14567}{67184} & 0 & -\frac{189}{3230} & \frac{693}{25840} \\ -\frac{143}{320} & \frac{35893}{103360} & -\frac{4081}{10336} & \frac{8393}{25840} & -\frac{2373}{12920} & \frac{189}{3230} & 0 & -\frac{77}{12920} \\ \frac{143}{1280} & -\frac{37037}{413440} & \frac{1309}{12160} & -\frac{9933}{103360} & \frac{189}{3040} & -\frac{693}{25840} & \frac{77}{12920} & 0 \end{array}\right)$

a = 1
b = 4

print('Kernels K_({0},{0}) through K_({1},{1}) are as follows:\n'.format(a, b))

for n in [a .. b]:
    K = retrieve_mat('K', n, n)
    print('K_{{{0},{0}}} ='.format(n))
    show(K)

Kernels K_(1,1) through K_(4,4) are as follows: K_{1,1} =

$\displaystyle \left(\begin{array}{rrrr} 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ -\frac{1}{2} & 0 & -\frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ -\frac{1}{2} & 0 & -\frac{1}{2} & 0 \end{array}\right)$

K_{2,2} =

$\displaystyle \left(\begin{array}{rrrrrrrr} 0 & \frac{3}{5} & \frac{1}{10} & \frac{1}{2} & \frac{1}{5} & \frac{1}{5} & \frac{3}{10} & -\frac{3}{10} \\ -\frac{3}{5} & 0 & -\frac{3}{10} & -\frac{1}{10} & 0 & \frac{1}{5} & \frac{3}{10} & -\frac{1}{10} \\ -\frac{1}{10} & \frac{3}{10} & 0 & \frac{2}{5} & \frac{1}{10} & \frac{3}{10} & \frac{1}{5} & 0 \\ -\frac{1}{2} & \frac{1}{10} & -\frac{2}{5} & 0 & -\frac{3}{10} & -\frac{3}{10} & -\frac{1}{5} & \frac{1}{5} \\ -\frac{1}{5} & 0 & -\frac{1}{10} & \frac{3}{10} & 0 & \frac{2}{5} & \frac{1}{10} & \frac{3}{10} \\ -\frac{1}{5} & -\frac{1}{5} & -\frac{3}{10} & \frac{3}{10} & -\frac{2}{5} & 0 & -\frac{1}{2} & -\frac{1}{10} \\ -\frac{3}{10} & -\frac{3}{10} & -\frac{1}{5} & \frac{1}{5} & -\frac{1}{10} & \frac{1}{2} & 0 & \frac{3}{5} \\ \frac{3}{10} & \frac{1}{10} & 0 & -\frac{1}{5} & -\frac{3}{10} & \frac{1}{10} & -\frac{3}{5} & 0 \end{array}\right)$

K_{3,3} =

$\displaystyle \left(\begin{array}{rrrrrrrrrrrr} 0 & \frac{11}{16} & \frac{5}{28} & \frac{57}{112} & \frac{25}{112} & \frac{3}{28} & \frac{15}{112} & -\frac{1}{4} & -\frac{5}{56} & -\frac{33}{112} & -\frac{25}{56} & \frac{27}{112} \\ -\frac{11}{16} & 0 & -\frac{27}{112} & -\frac{1}{14} & \frac{3}{112} & \frac{1}{7} & \frac{13}{112} & 0 & \frac{3}{112} & -\frac{1}{7} & -\frac{27}{112} & \frac{1}{14} \\ -\frac{5}{28} & \frac{27}{112} & 0 & \frac{47}{112} & \frac{11}{112} & \frac{9}{28} & \frac{13}{112} & \frac{3}{28} & \frac{3}{56} & -\frac{1}{16} & -\frac{5}{56} & -\frac{3}{112} \\ -\frac{57}{112} & \frac{1}{14} & -\frac{47}{112} & 0 & -\frac{33}{112} & -\frac{2}{7} & -\frac{15}{112} & \frac{1}{7} & \frac{1}{16} & \frac{3}{14} & \frac{33}{112} & -\frac{1}{7} \\ -\frac{25}{112} & -\frac{3}{112} & -\frac{11}{112} & \frac{33}{112} & 0 & \frac{11}{28} & \frac{1}{14} & \frac{9}{28} & \frac{13}{112} & \frac{15}{112} & \frac{15}{112} & -\frac{13}{112} \\ -\frac{3}{28} & -\frac{1}{7} & -\frac{9}{28} & \frac{2}{7} & -\frac{11}{28} & 0 & -\frac{9}{28} & -\frac{2}{7} & -\frac{3}{28} & \frac{1}{7} & \frac{1}{4} & 0 \\ -\frac{15}{112} & -\frac{13}{112} & -\frac{13}{112} & \frac{15}{112} & -\frac{1}{14} & \frac{9}{28} & 0 & \frac{11}{28} & \frac{11}{112} & \frac{33}{112} & \frac{25}{112} & -\frac{3}{112} \\ \frac{1}{4} & 0 & -\frac{3}{28} & -\frac{1}{7} & -\frac{9}{28} & \frac{2}{7} & -\frac{11}{28} & 0 & -\frac{9}{28} & -\frac{2}{7} & -\frac{3}{28} & \frac{1}{7} \\ \frac{5}{56} & -\frac{3}{112} & -\frac{3}{56} & -\frac{1}{16} & -\frac{13}{112} & \frac{3}{28} & -\frac{11}{112} & \frac{9}{28} & 0 & \frac{47}{112} & \frac{5}{28} & \frac{27}{112} \\ \frac{33}{112} & \frac{1}{7} & \frac{1}{16} & -\frac{3}{14} & -\frac{15}{112} & -\frac{1}{7} & -\frac{33}{112} & \frac{2}{7} & -\frac{47}{112} & 0 & -\frac{57}{112} & -\frac{1}{14} \\ \frac{25}{56} & \frac{27}{112} & \frac{5}{56} & -\frac{33}{112} & -\frac{15}{112} & -\frac{1}{4} & -\frac{25}{112} & \frac{3}{28} & -\frac{5}{28} & \frac{57}{112} & 0 & \frac{11}{16} \\ -\frac{27}{112} & -\frac{1}{14} & \frac{3}{112} & \frac{1}{7} & \frac{13}{112} & 0 & \frac{3}{112} & -\frac{1}{7} & -\frac{27}{112} & \frac{1}{14} & -\frac{11}{16} & 0 \end{array}\right)$

K_{4,4} =

$\displaystyle \left(\begin{array}{rrrrrrrrrrrrrrrr} 0 & \frac{25}{33} & \frac{11}{42} & \frac{229}{462} & \frac{109}{462} & -\frac{1}{462} & \frac{9}{154} & -\frac{137}{462} & -\frac{31}{231} & -\frac{17}{77} & -\frac{95}{462} & \frac{5}{42} & -\frac{3}{154} & \frac{53}{154} & \frac{37}{66} & -\frac{13}{66} \\ -\frac{25}{33} & 0 & -\frac{13}{66} & -\frac{1}{22} & \frac{1}{22} & \frac{7}{66} & \frac{5}{66} & -\frac{1}{66} & 0 & -\frac{1}{11} & -\frac{5}{66} & -\frac{1}{66} & -\frac{1}{22} & \frac{7}{66} & \frac{13}{66} & -\frac{1}{22} \\ -\frac{11}{42} & \frac{13}{66} & 0 & \frac{106}{231} & \frac{17}{154} & \frac{23}{66} & \frac{53}{462} & \frac{19}{154} & \frac{9}{154} & -\frac{23}{462} & -\frac{1}{77} & -\frac{2}{21} & -\frac{25}{462} & -\frac{13}{462} & -\frac{3}{154} & \frac{1}{22} \\ -\frac{229}{462} & \frac{1}{22} & -\frac{106}{231} & 0 & -\frac{139}{462} & -\frac{37}{154} & -\frac{15}{154} & \frac{73}{462} & \frac{5}{66} & \frac{83}{462} & \frac{1}{7} & -\frac{3}{77} & \frac{13}{462} & -\frac{97}{462} & -\frac{53}{154} & \frac{7}{66} \\ -\frac{109}{462} & -\frac{1}{22} & -\frac{17}{154} & \frac{139}{462} & 0 & \frac{95}{231} & \frac{37}{462} & \frac{51}{154} & \frac{53}{462} & \frac{3}{22} & \frac{41}{462} & -\frac{31}{462} & -\frac{1}{77} & -\frac{1}{7} & -\frac{95}{462} & \frac{5}{66} \\ \frac{1}{462} & -\frac{7}{66} & -\frac{23}{66} & \frac{37}{154} & -\frac{95}{231} & 0 & -\frac{45}{154} & -\frac{137}{462} & -\frac{15}{154} & \frac{43}{462} & \frac{31}{462} & \frac{19}{154} & \frac{2}{21} & -\frac{3}{77} & -\frac{5}{42} & -\frac{1}{66} \\ -\frac{9}{154} & -\frac{5}{66} & -\frac{53}{462} & \frac{15}{154} & -\frac{37}{462} & \frac{45}{154} & 0 & \frac{86}{231} & \frac{37}{462} & \frac{45}{154} & \frac{53}{462} & \frac{15}{154} & \frac{9}{154} & -\frac{5}{66} & -\frac{31}{231} & 0 \\ \frac{137}{462} & \frac{1}{66} & -\frac{19}{154} & -\frac{73}{462} & -\frac{51}{154} & \frac{137}{462} & -\frac{86}{231} & 0 & -\frac{45}{154} & -\frac{155}{462} & -\frac{3}{22} & \frac{43}{462} & \frac{23}{462} & \frac{83}{462} & \frac{17}{77} & -\frac{1}{11} \\ \frac{31}{231} & 0 & -\frac{9}{154} & -\frac{5}{66} & -\frac{53}{462} & \frac{15}{154} & -\frac{37}{462} & \frac{45}{154} & 0 & \frac{86}{231} & \frac{37}{462} & \frac{45}{154} & \frac{53}{462} & \frac{15}{154} & \frac{9}{154} & -\frac{5}{66} \\ \frac{17}{77} & \frac{1}{11} & \frac{23}{462} & -\frac{83}{462} & -\frac{3}{22} & -\frac{43}{462} & -\frac{45}{154} & \frac{155}{462} & -\frac{86}{231} & 0 & -\frac{51}{154} & -\frac{137}{462} & -\frac{19}{154} & \frac{73}{462} & \frac{137}{462} & -\frac{1}{66} \\ \frac{95}{462} & \frac{5}{66} & \frac{1}{77} & -\frac{1}{7} & -\frac{41}{462} & -\frac{31}{462} & -\frac{53}{462} & \frac{3}{22} & -\frac{37}{462} & \frac{51}{154} & 0 & \frac{95}{231} & \frac{17}{154} & \frac{139}{462} & \frac{109}{462} & -\frac{1}{22} \\ -\frac{5}{42} & \frac{1}{66} & \frac{2}{21} & \frac{3}{77} & \frac{31}{462} & -\frac{19}{154} & -\frac{15}{154} & -\frac{43}{462} & -\frac{45}{154} & \frac{137}{462} & -\frac{95}{231} & 0 & -\frac{23}{66} & -\frac{37}{154} & \frac{1}{462} & \frac{7}{66} \\ \frac{3}{154} & \frac{1}{22} & \frac{25}{462} & -\frac{13}{462} & \frac{1}{77} & -\frac{2}{21} & -\frac{9}{154} & -\frac{23}{462} & -\frac{53}{462} & \frac{19}{154} & -\frac{17}{154} & \frac{23}{66} & 0 & \frac{106}{231} & \frac{11}{42} & \frac{13}{66} \\ -\frac{53}{154} & -\frac{7}{66} & \frac{13}{462} & \frac{97}{462} & \frac{1}{7} & \frac{3}{77} & \frac{5}{66} & -\frac{83}{462} & -\frac{15}{154} & -\frac{73}{462} & -\frac{139}{462} & \frac{37}{154} & -\frac{106}{231} & 0 & -\frac{229}{462} & -\frac{1}{22} \\ -\frac{37}{66} & -\frac{13}{66} & \frac{3}{154} & \frac{53}{154} & \frac{95}{462} & \frac{5}{42} & \frac{31}{231} & -\frac{17}{77} & -\frac{9}{154} & -\frac{137}{462} & -\frac{109}{462} & -\frac{1}{462} & -\frac{11}{42} & \frac{229}{462} & 0 & \frac{25}{33} \\ \frac{13}{66} & \frac{1}{22} & -\frac{1}{22} & -\frac{7}{66} & -\frac{5}{66} & \frac{1}{66} & 0 & \frac{1}{11} & \frac{5}{66} & \frac{1}{66} & \frac{1}{22} & -\frac{7}{66} & -\frac{13}{66} & \frac{1}{22} & -\frac{25}{33} & 0 \end{array}\right)$