Notebook

On the growth of the Kronecker coefficients: Companion Worksheet in Sage.¶

Emmanuel Briand, Departamento Matemática Aplicada 1, Universidad de Sevilla. Version of july 2019.

This notebook implements some of the calculations presented in the preprint On the growth of the Kronecker coefficients (Emmanuel Briand, Amarpreet Rattan, Mercedes Rosas; 2016 (reference [On the growth] below)

References:

[On the growth] On the growth of the Kronecker coefficients. Emmanuel Briand, Amarpreet Rattan, Mercedes Rosas arXiv:1607.02887 [math.RT] (2016).
[appendix] On the growth of Kronecker coefficients: accompanying indices. Emmanuel Briand, Amarpreet Rattan, Mercedes Rosas. arXiv:1611.07348 [math.RT] (2016).

URL for this notebook: http://emmanuel.jean.briand.free.fr/publications/HookStability.ipynb

You may also:

Some functions to deal with partitions and compute stable (=reduced) Kronecker coefficients are provided by the attached files stableKroneckerCoefficients.sage and operations_on_partitions.sage, that are loaded by the commands below, provided that the files have been downloaded with the notebook, and set in the same directory.

In [1]:

load("StableKroneckerCoefficients.sage")
load("operations_on_partitions.sage")

The following functions are loaded:
- partitions_intersection
- stable_Kronecker_product_of_Schur_functions
- stable_Kronecker_coefficient

The following functions are loaded:
- partitions_oplus
- partitions_cut
- partitions_hat
- add_row
- add_column

Here is a command provided by these files. It computes a stable Kronecker coefficient.

In [2]:

stable_Kronecker_coefficient([2, 1],[2, 1], [1, 1, 1])        ## test. Answer must be 5

Out[2]:

Help is available for the functions just loaded.

In [3]:

partitions_oplus?

Signature:      partitions_oplus(lam, i, j)
Docstring:     
   Given a non-empty partition, returns the partition obtained by
   adding to its diagram i boxes in the first row, and j boxes in the
   first column.
Init docstring: x.__init__(...) initializes x; see help(type(x)) for signature
File:           Dynamically generated function. No source code available.
Type:           function

In [4]:

stable_Kronecker_product_of_Schur_functions?

Signature:      stable_Kronecker_product_of_Schur_functions(lam, mu)
Docstring:     
   The stable product of s_{lambda} with , in the Schur basis.

   INPUT:

      * lambda, mu -- two integer partitions, or lists defining
        integer partitions.

   OUTPUT:

   A symmetric function, expanded in the Schur basis.

   EXAMPLES:

      sage: stable_Kronecker_product_of_Schur_functions( [ 1 ], [ 1 ] )
      s[] + s[1] + s[1, 1] + s[2]

   ALGORITHM:

   The stable Kronecker product of Schur functions is computed by
   means of Littlewood's Formula that can be found in [Littlewood] or
   as Theorem 1.1 in [Thibon].

   Note: Littllewood's Formula is not applied recursively.

   REFERENCES:

   [Littlewood] D. E. Littlewood. "The Kronecker product
                of symmetric group representations." J. London Math.
                Soc., 31:89-93, 1956.

   [Thibon] Jean-Yves Thibon. "Hopf algebras of symmetric
            functions and tensor products of symmetric group
            representations." Internat. J. Algebra Comput.,
            1(2):207-221, 1991.
Init docstring: x.__init__(...) initializes x; see help(type(x)) for signature
File:           Dynamically generated function. No source code available.
Type:           function

Theorem 5.2: Bounds for column stability for stable Kronecker coefficients (= reduced Kronecker coefficients).

We start with three partitions $\alpha$, $\beta$ and $\gamma$ and consider the stable Kronecker coefficients indexed by $\alpha + (1^a)$, $\beta + (1^b)$, $\gamma + (1^c)$ with $a$, $b$ and $c$ at least the length of $\alpha$, $\beta$ and $\gamma$ respectively. According to Theorem 5.2, there exist integers $k_1$, $k_2$, $k_3$ such that when

$$b+c-a \geq k_1, \\ a+c-b \geq k_2, \\ a+b -c \geq k_3$$

we have

$$\overline{g}_{\alpha + (1^a),\beta + (1^b),\gamma + (1^c)} = \overline{\overline{g}}_{\alpha, \beta, \gamma}.$$

We will illustrate this on an example.

Here are functions computing values for $k_i$ as given by Theorem A.1 in [appendix].

In [5]:

## The bounds for stability, for the linear forms b+c-a, a-b+c, a+b-c, are given by the following function

def bounds_k(alpha, beta, gamma):
    r"""
    Returns (k1, k2, k3).
    """
    return bound_k1( alpha, beta, gamma), bound_k1(beta, alpha, gamma), bound_k1(gamma, alpha, beta) 
    
## The three bounds k1, k2, k3 are obtained from k1 by permutation of their arguments. Here is the formula for k1.
    
def bound_k1(alpha, beta, gamma):
    r"""
    Returns k1.
    """
    alpha = Partition( alpha )
    beta  = Partition( beta )
    gamma = Partition( gamma )
    if alpha <> Partition([]):
        return alpha.size() + alpha[0] + beta.length() + gamma.length()
    else:
        return 0 + 0 + beta.length() + gamma.length()

Our example will be with

$$\alpha=(2), \quad \beta=(2), \quad \gamma=(1,1).$$

In [6]:

alpha = Partition([2])
beta  = Partition([2])
gamma = Partition([1,1])

(k1, k2, k3 ) = bounds_k(alpha, beta, gamma)
print(k1, k2, k3)

(7, 7, 5)

We compute the $\overline{g}_{\alpha + (1^a),\beta + (1^b),\gamma + (1^c)}$ for $a$, $b$, $c$ between $0$ and $10$.

In [7]:

## WARNING: This takes some time.
M = dict([])
N = 10
for a in [0..N]:
    for b in [0..N]:
        P = stable_Kronecker_product_of_Schur_functions( add_column(alpha, a), add_column(beta, b) )
        for c in [0..N]:
                M[a,b,c] = P.coefficient( add_column(gamma, c) );

The attached file "html_table.sage" contains a function "html_table". This function produces, from a matrix, a html table. The cells fulfilling some condition specified as a parameter of the function, are colored in yellow. We will use it to display the stable Kronecker coefficients indexed by $\alpha+(1^a)$, $\beta+(1^b)$, $\gamma+(1^c)$ that we have just computed.

In [8]:

load("html_table.sage")

In [9]:

html_table?

Signature:      html_table(M, rows_label='', columns_label='', condition=<function <lambda> at 0x7f43eececaa0>)
Docstring:     
   Given a matrix M, builds a html table that represents M. The cells
   that fulfill "condition" are colored in yellow.

   INPUT:

   * M -- a matrix.

   * rows_label -- a character string (default: "").

   * columns_label -- a character string (default: "").

   * condition -- a function of the pair i, j (default: the function
     that returns always False).

   OUTPUT:

   A character string in html.

   EXAMPLES:

      sage: M = matrix([[1,2], [3,4]])
      sage: html_table(M, , condition = lambda i,j: i + j >= 1 )
      <table  cellpadding="5" cellspacing="5"><tr> <th></th>  <th></th><th colspan="2"> </th></tr>
      <tr>  <th></th>  <th></th> <th>0</th><th>1</th></tr><tr> <th rowspan="2"></th> <th>0</th><td>
       1</td><td bgcolor="#ffff00"> 2</td></tr><tr> <th>1</th><td bgcolor="#ffff00">
      3</td><td bgcolor="#ffff00"> 4</td></tr></table>
Init docstring: x.__init__(...) initializes x; see help(type(x)) for signature
File:           Dynamically generated function. No source code available.
Type:           function

We now display the tables of the $\overline{g}_{\alpha + (1^a),\beta + (1^b),\gamma + (1^c)} $, each table corresponding to a fixed $a$.

In [10]:

La = alpha.length()
Lb = beta.length()
Lc = gamma.length()
for a in [0..N]:
    print "For a={a}".format(a=a)
    show(html_table(matrix(N+1,N+1,lambda b,c : M[a,b,c]), 
               rows_label="b", 
               columns_label="c", 
               condition = lambda b, c: (b+c-a >= k1 and a+b-c >= k3 
                                         and a+c-b >= k2 and a>= La 
                                         and b >= Lb and c >= Lc)
               ))

For a=0

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	1	0	0	0	0	0	0	0	0
	2	1	3	2	2	0	0	0	0	0	0	0
	3	0	1	2	3	2	0	0	0	0	0	0
	4	0	0	0	2	3	2	0	0	0	0	0
	5	0	0	0	0	2	3	2	0	0	0	0
	6	0	0	0	0	0	2	3	2	0	0	0
	7	0	0	0	0	0	0	2	3	2	0	0
	8	0	0	0	0	0	0	0	2	3	2	0
	9	0	0	0	0	0	0	0	0	2	3	2
	10	0	0	0	0	0	0	0	0	0	2	3

For a=1

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	2	1	0	0	0	0	0	0	0
	2	2	5	4	4	1	0	0	0	0	0	0
	3	1	3	3	7	5	1	0	0	0	0	0
	4	0	0	1	5	8	5	1	0	0	0	0
	5	0	0	0	1	5	8	5	1	0	0	0
	6	0	0	0	0	1	5	8	5	1	0	0
	7	0	0	0	0	0	1	5	8	5	1	0
	8	0	0	0	0	0	0	1	5	8	5	1
	9	0	0	0	0	0	0	0	1	5	8	5
	10	0	0	0	0	0	0	0	0	1	5	8

For a=2

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	3	2	2	0	0	0	0	0	0	0
	1	2	5	4	4	1	0	0	0	0	0	0
	2	4	13	13	17	8	1	0	0	0	0	0
	3	3	12	15	30	25	9	1	0	0	0	0
	4	1	4	7	26	38	27	9	1	0	0	0
	5	0	0	1	10	29	39	27	9	1	0	0
	6	0	0	0	1	10	29	39	27	9	1	0
	7	0	0	0	0	1	10	29	39	27	9	1
	8	0	0	0	0	0	1	10	29	39	27	9
	9	0	0	0	0	0	0	1	10	29	39	27
	10	0	0	0	0	0	0	0	1	10	29	39

For a=3

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	1	2	3	2	0	0	0	0	0	0
	1	1	3	3	7	5	1	0	0	0	0	0
	2	3	12	15	30	25	9	1	0	0	0	0
	3	4	18	26	55	58	34	10	1	0	0	0
	4	3	13	20	56	80	69	36	10	1	0	0
	5	1	4	7	33	69	86	70	36	10	1	0
	6	0	0	1	10	36	70	86	70	36	10	1
	7	0	0	0	1	10	36	70	86	70	36	10
	8	0	0	0	0	1	10	36	70	86	70	36
	9	0	0	0	0	0	1	10	36	70	86	70
	10	0	0	0	0	0	0	1	10	36	70	86

For a=4

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	2	3	2	0	0	0	0	0
	1	0	0	1	5	8	5	1	0	0	0	0
	2	1	4	7	26	38	27	9	1	0	0	0
	3	3	13	20	56	80	69	36	10	1	0	0
	4	4	18	28	70	104	109	80	38	10	1	0
	5	3	13	20	59	99	121	115	81	38	10	1
	6	1	4	7	33	72	105	122	115	81	38	10
	7	0	0	1	10	36	73	105	122	115	81	38
	8	0	0	0	1	10	36	73	105	122	115	81
	9	0	0	0	0	1	10	36	73	105	122	115
	10	0	0	0	0	0	1	10	36	73	105	122

For a=5

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	2	3	2	0	0	0	0
	1	0	0	0	1	5	8	5	1	0	0	0
	2	0	0	1	10	29	39	27	9	1	0	0
	3	1	4	7	33	69	86	70	36	10	1	0
	4	3	13	20	59	99	121	115	81	38	10	1
	5	4	18	28	70	108	132	138	121	82	38	10
	6	3	13	20	59	99	125	138	139	121	82	38
	7	1	4	7	33	72	105	126	138	139	121	82
	8	0	0	1	10	36	73	105	126	138	139	121
	9	0	0	0	1	10	36	73	105	126	138	139
	10	0	0	0	0	1	10	36	73	105	126	138

For a=6

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	2	3	2	0	0	0
	1	0	0	0	0	1	5	8	5	1	0	0
	2	0	0	0	1	10	29	39	27	9	1	0
	3	0	0	1	10	36	70	86	70	36	10	1
	4	1	4	7	33	72	105	122	115	81	38	10
	5	3	13	20	59	99	125	138	139	121	82	38
	6	4	18	28	70	108	132	142	144	140	121	82
	7	3	13	20	59	99	125	138	143	144	140	121
	8	1	4	7	33	72	105	126	138	143	144	140
	9	0	0	1	10	36	73	105	126	138	143	144
	10	0	0	0	1	10	36	73	105	126	138	143

For a=7

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	2	3	2	0	0
	1	0	0	0	0	0	1	5	8	5	1	0
	2	0	0	0	0	1	10	29	39	27	9	1
	3	0	0	0	1	10	36	70	86	70	36	10
	4	0	0	1	10	36	73	105	122	115	81	38
	5	1	4	7	33	72	105	126	138	139	121	82
	6	3	13	20	59	99	125	138	143	144	140	121
	7	4	18	28	70	108	132	142	144	144	144	140
	8	3	13	20	59	99	125	138	143	144	144	144
	9	1	4	7	33	72	105	126	138	143	144	144
	10	0	0	1	10	36	73	105	126	138	143	144

For a=8

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	2	3	2	0
	1	0	0	0	0	0	0	1	5	8	5	1
	2	0	0	0	0	0	1	10	29	39	27	9
	3	0	0	0	0	1	10	36	70	86	70	36
	4	0	0	0	1	10	36	73	105	122	115	81
	5	0	0	1	10	36	73	105	126	138	139	121
	6	1	4	7	33	72	105	126	138	143	144	140
	7	3	13	20	59	99	125	138	143	144	144	144
	8	4	18	28	70	108	132	142	144	144	144	144
	9	3	13	20	59	99	125	138	143	144	144	144
	10	1	4	7	33	72	105	126	138	143	144	144

For a=9

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	0	2	3	2
	1	0	0	0	0	0	0	0	1	5	8	5
	2	0	0	0	0	0	0	1	10	29	39	27
	3	0	0	0	0	0	1	10	36	70	86	70
	4	0	0	0	0	1	10	36	73	105	122	115
	5	0	0	0	1	10	36	73	105	126	138	139
	6	0	0	1	10	36	73	105	126	138	143	144
	7	1	4	7	33	72	105	126	138	143	144	144
	8	3	13	20	59	99	125	138	143	144	144	144
	9	4	18	28	70	108	132	142	144	144	144	144
	10	3	13	20	59	99	125	138	143	144	144	144

For a=10

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	0	0	2	3
	1	0	0	0	0	0	0	0	0	1	5	8
	2	0	0	0	0	0	0	0	1	10	29	39
	3	0	0	0	0	0	0	1	10	36	70	86
	4	0	0	0	0	0	1	10	36	73	105	122
	5	0	0	0	0	1	10	36	73	105	126	138
	6	0	0	0	1	10	36	73	105	126	138	143
	7	0	0	1	10	36	73	105	126	138	143	144
	8	1	4	7	33	72	105	126	138	143	144	144
	9	3	13	20	59	99	125	138	143	144	144	144
	10	4	18	28	70	108	132	142	144	144	144	144

Example 5.4: the polynomials $Q^-_{(p), (q), (r)}$.

As in Example 5.4, we compute the polynomials $Q^-$ indexed by three one-row shapes $(p)$, $(q)$ and $(r)$. Let $F(x_1,y_1,z_1)$ be their generating series:

$$F=\sum Q_{(p),(q),(r)}^- x_1^p y_1^q z_1^r.$$

Then $F$ has a very simple expression. Below we compute $F$ and extract from it the polynomials $Q^-$ that appear in Example 5.4 of [On the growth].

In [11]:

var('x1 y1 z1 x y z')

numerator =  (1+z*x1*y1)*(1+x*y1*z1)*(1+y*z1*x1) * (1+x1*y)*(1+x*y1)*(1+x1*z)*(1+x*z1)*(1+y*z1)*(1+y1*z)
denominator = (1-x1*y1*z1) * (1-x1*y1)*(1-x1*z1)*(1-y1*z1)*(1-x1*y*z)*(1-x*y1*z)*(1-x*y*z1)*(1+x1/x)*(1+y1/y)*(1+z1/z)
F = numerator / denominator

## For developping in series with "series" I introduce a variable t
var('t')
G = F.subs({ x1 : t*x1, y1 : t*y1, z1 : t*z1 })

def Q_minus(p, q, r):
    global x1, y1, z1, F, G, t, x, y, z;
    N = p + q +r;
    GG = G.series( t, N+1).truncate().subs( {t:1}).expand() 
    return GG.coefficient( x1, p).coefficient( y1, q).coefficient( z1, r)

In [12]:

Q_minus(0,0,0)

Out[12]:

In [13]:

Q_minus(0,0,1)

Out[13]:

x*y + x + y - 1/z

In [14]:

Q_minus(0,0,2)

Out[14]:

x^2*y^2 + x^2*y + x*y^2 + x*y - x*y/z - x/z - y/z + 1/z^2

In [15]:

Q_minus(0,1,1)

Out[15]:

x^2*y*z + x^2*y + x^2*z + 2*x*y*z + x^2 + x*y + x*z + y*z - x - x/y - x/z + 1/(y*z) - 1

Conjecture 5.11 on increasing Kronecker coefficients.

Conjecture 5.11 states that for all triples of partitions $\lambda$, $\mu$, $\nu$ of the same weight,

$$g_{ \lambda , \mu , \nu }\leq g_{\lambda \oplus (1, 1), \mu \oplus (1, 1), \nu \oplus (1, 1)}$$

We checked this for weights up to $16$. Below we reproduce the computation up to weight $10$ (going up to weight $16$ is a bit longer).

In [16]:

def check_conjecture_5_11(N):
    r"""
    Check Conjecture 5.11 of [On the growth] for all triples of partitions with weight at most $N$.
    """
    test = true   ## will be false when a counter example is found
    L = []
    for (i, lam) in enumerate(Partitions(N)):
        for (j, mu) in enumerate(Partitions(N)):
            if j >= i:
                P = s(lam).kronecker_product(s(mu))
                Q = s(partitions_oplus(lam, 1, 1)).kronecker_product(s(partitions_oplus(mu, 1, 1)))
                for (nu, c) in P:
                    nu_oplus_one_one = partitions_oplus(nu, 1, 1)
                    if not ( c <= Q.coefficient(nu_oplus_one_one) ):
                        L.append([lam, mu, nu])
                        print "Counter Example:", lam, mu, nu 
                        test = false
    return test

In [17]:

# Warning: this takes some time for N=8, N=9,N=10
for N in [1..10]: 
    print N, check_conjecture_5_11(N)

1 True
2 True
3 True
4 True
5 True
6 True
7 True
8 True
9 True
10 True

Table 1 in Example 5.5 illustrating Hook Stability.

We reproduce Table 1 in Example 5.5 of [On the growth], about hook stability for the Kronecker coefficients indexed by $(3,3) \oplus(i|j)$ three times.

In [18]:

# WARNING: this takes some time
lam = Partition([3, 3])
Kronecker_square = lambda f : f.kronecker_product(f)
T = matrix(10,10, lambda i, j: 
     Kronecker_square(s(partitions_oplus(lam, i, j))).coefficient( partitions_oplus(lam, i, j) )
     )
     
html_table(T, rows_label="i", columns_label="j", condition = lambda i,j : False)

Out[18]:

		j
		0	1	2	3	4	5	6	7	8	9
i	0	0	1	5	5	1	0	0	0	0	0
	1	1	8	27	40	30	11	1	0	0	0
	2	1	15	53	89	91	64	33	11	1	0
	3	2	19	62	108	129	122	97	64	33	11
	4	2	19	63	112	138	141	135	122	97	64
	5	2	19	63	112	139	145	144	141	135	122
	6	2	19	63	112	139	145	145	145	144	141
	7	2	19	63	112	139	145	145	145	145	145
	8	2	19	63	112	139	145	145	145	145	145
	9	2	19	63	112	139	145	145	145	145	145

We want to color in this table the boxes corresponding to the $(i,j)$ such that the Hook Stability Condition (21) of [On the growth] are fulfilled. First we compute the explicit bounds $d$, $d_1$, $d_2$ and $d_3$ given in the proof of Theorem 5.7.

In [19]:

def first_part(lam):
    r"""
    First part of a partition, or 0 if the partition is empty.
    """
    lam = Partition(lam)
    if lam == Partition([ ]):
        return 0
    else: 
        return lam[0]

def bound_N0(lam, mu, nu):
    r"""
    Bound $N_0$ defined in Formula (9) from [On the growth].
    """
    lam = Partition(lam)
    mu  = Partition(mu)
    nu  = Partition(nu)
    return add(rho.size() + first_part(rho) for rho in (lam, mu, nu) )/2

def bounds_d(lam, mu, nu):
    r"""
    Returns the values of $d$, $d_1$, $d_2$, $d_3$ defined in the proof of Theorem 5.7 in [On the growth].
    """
    lam = Partition(lam)
    mu  = Partition(mu)
    nu  = Partition(nu)
    N = lam.size() ## common weight of all three partitions
    ell = dict([])
    ell[1] = lambda a,b,c : a+b-c
    ell[2] = lambda a,b,c : a+c-b
    ell[3] = lambda a,b,c : b+c-a
    k = dict([])
    (k[1], k[2], k[3]) = bounds_k(partitions_hat(lam), partitions_hat(mu), partitions_hat(nu))
    d = dict([])
    for i in [1,2,3]:
        d[i] = k[i] - ell[i](lam.length(), mu.length(), nu.length()) +1
    dd = ( bound_N0(partitions_hat(lam), partitions_hat(mu), partitions_hat(nu)) 
        + (lam.length() + mu.length()+  nu.length())/2 
        - N)
    return (dd, d[1], d[2], d[3])

In the example under consideration, the bounds $d$, $d_1$, $d_2$ and $d_3$ are

In [20]:

bounds_d([3,3], [3,3], [3,3])

Out[20]:

(3, 5, 5, 5)

In that example, $a=b=c=j$ and $m = i+j$. The condition (21) of [On the growth] writes:

$$ j \geq 5, \text{ and } i -j/2 \geq 3, \text{ and } i \geq 0.$$

The positions in the table that fulfill this condition are represented in yellow below (in gray in [On the growth]).

In [21]:

html_table(T, 
           rows_label="i", 
           columns_label="j", 
           condition = lambda i,j : j >= 5 and i - j/2 >= 3)

Out[21]:

		j
		0	1	2	3	4	5	6	7	8	9
i	0	0	1	5	5	1	0	0	0	0	0
	1	1	8	27	40	30	11	1	0	0	0
	2	1	15	53	89	91	64	33	11	1	0
	3	2	19	62	108	129	122	97	64	33	11
	4	2	19	63	112	138	141	135	122	97	64
	5	2	19	63	112	139	145	144	141	135	122
	6	2	19	63	112	139	145	145	145	144	141
	7	2	19	63	112	139	145	145	145	145	145
	8	2	19	63	112	139	145	145	145	145	145
	9	2	19	63	112	139	145	145	145	145	145

The stability domain (all coefficients $145$) is a little bit bigger than the stable domain guaranteed by Theorem 5.7 (in yellow).

Computation of the coefficients .

Remember the following generating series :

for the Littlewood-richardson coefficients: $$\sigma[XZ+YZ]$$
for the Kronecker coefficients: $$\sigma[XYZ]$$
for the stable Kronecker coefficients: $$\sigma[XYZ + XY + XZ + YZ]$$
for the limits of stable Kronecker coefficients $\overline{\overline{g}}$ (see section 7 in [On the growth].) : $$\sigma[XYZ+(1-\varepsilon)(XY+XZ+YZ)]$$
and there are similar series for the coefficients $A$, $B$, $C$ in the quasipolynomial formulas for the sequences of stable Kronecker coefficients whose first row is increasing, see Section 7 in [On the growth] and below in this notebook.

Here we compute some of these coefficients (small ones) using Sage's construction of tensor products of the algebra of Symmetric functions.

In [22]:

Sym = SymmetricFunctions(QQ)
p = Sym.powersum()
s = Sym.schur()
h = Sym.homogeneous()

The algebra $\operatorname{Sym} \otimes \operatorname{Sym} \otimes \operatorname{Sym}$.

In [23]:

SymSymSym = tensor([p, p, p])

The file "plethystic.sage" contains the definition of a class Specialization whose objects represent the morphisms from $\operatorname{Sym}$ to $\operatorname{Sym} \otimes \operatorname{Sym} \otimes \operatorname{Sym}$.

In [24]:

load("plethystic.sage")

The following class is defined:
- Specialization.

The following functions are loaded:
- pleth
- change_basis
- specialization_binomial
- specialization_rank1

The following constants are defined:
- specialization_X
- specialization_Y
- specialization_Z
- specialization_epsilon

In [25]:

Specialization?

Init signature: Specialization(self, f)
Docstring:     
   Here a specialization is a morphism of QQ-algebras from
   operatorname{Sym} to operatorname{Sym} otimes operatorname{Sym}
   otimes operatorname{Sym}.

   INPUT:

   * f -- a function that associates to each positive integer i an
     element of  operatorname{Sym} otimes operatorname{Sym} otimes
     operatorname{Sym}.

   OUTPUT:

   * the unique Specializationthat maps p_i to f(i).

   EXAMPLES:

   Here is an example of definition of a specialization.

      sage: A = Specialization( lambda i:  )

   Some standard specializations are predefined. specialization_X,
   specialization_Y and specialization_Z are the inclusions in the
   factors of the tensor product.

      sage: specialization_X
      Specialization of symmetric functions in the tensor product of three
      copies of Sym where:
      p[1] |-> p[1] # p[] # p[]
      p[2] |-> p[2] # p[] # p[]
      p[3] |-> p[3] # p[] # p[]
      p[4] |-> p[4] # p[] # p[]
       ...

   The binomial elements are those whose power sums are all equal to
   this element.

      sage: specialization_binomial(2)
      Specialization of symmetric functions in the tensor product of three
      copies of Sym where:
      p[1] |-> 2 p[] # p[] # p[]
      p[2] |-> 2 p[] # p[] # p[]
      p[3] |-> 2 p[] # p[] # p[]
      p[4] |-> 2 p[] # p[] # p[]
       ...

   It is possible to perform operations on specializations.

      sage: X*Z + Y*Z
      Specialization of symmetric functions in the tensor product of three
      copies of Sym where:
      p[1] |-> p[] # p[1] # p[1] + p[1] # p[] # p[1]
      p[2] |-> p[] # p[2] # p[2] + p[2] # p[] # p[2]
      p[3] |-> p[] # p[3] # p[3] + p[3] # p[] # p[3]
      p[4] |-> p[] # p[4] # p[4] + p[4] # p[] # p[4]
      ...

   The function pleth makes the symmetric functions act on the
   morphisms.

      sage: pleth( h[3], X*Z + Y*Z)
      1/6*p[] # p[1, 1, 1] # p[1, 1, 1] + 1/2*p[] # p[2, 1] # p[2, 1] +
      1/3*p[] # p[3] # p[3] + 1/2*p[1] # p[1, 1] # p[1, 1, 1] + 1/2*p[1] #
      p[2] # p[2, 1] + 1/2*p[1, 1] # p[1] # p[1, 1, 1] + 1/6*p[1, 1, 1] # p[]
      # p[1, 1, 1] + 1/2*p[2] # p[1] # p[2, 1] + 1/2*p[2, 1] # p[] # p[2, 1] +
      1/3*p[3] # p[] # p[3
Init docstring:
   Here a specialization is a morphism of QQ-algebras from
   operatorname{Sym} to operatorname{Sym} otimes operatorname{Sym}
   otimes operatorname{Sym}.

   INPUT:

   * f -- a function that associates to each positive integer i an
     element of  operatorname{Sym} otimes operatorname{Sym} otimes
     operatorname{Sym}.

   OUTPUT:

   * the unique Specializationthat maps p_i to f(i).

   EXAMPLES:

   Here is an example of definition of a specialization.

      sage: A = Specialization( lambda i:  )

   Some standard specializations are predefined. specialization_X,
   specialization_Y and specialization_Z are the inclusions in the
   factors of the tensor product.

      sage: specialization_X
      Specialization of symmetric functions in the tensor product of three
      copies of Sym where:
      p[1] |-> p[1] # p[] # p[]
      p[2] |-> p[2] # p[] # p[]
      p[3] |-> p[3] # p[] # p[]
      p[4] |-> p[4] # p[] # p[]
       ...

   The binomial elements are those whose power sums are all equal to
   this element.

      sage: specialization_binomial(2)
      Specialization of symmetric functions in the tensor product of three
      copies of Sym where:
      p[1] |-> 2 p[] # p[] # p[]
      p[2] |-> 2 p[] # p[] # p[]
      p[3] |-> 2 p[] # p[] # p[]
      p[4] |-> 2 p[] # p[] # p[]
       ...

   It is possible to perform operations on specializations.

      sage: X*Z + Y*Z
      Specialization of symmetric functions in the tensor product of three
      copies of Sym where:
      p[1] |-> p[] # p[1] # p[1] + p[1] # p[] # p[1]
      p[2] |-> p[] # p[2] # p[2] + p[2] # p[] # p[2]
      p[3] |-> p[] # p[3] # p[3] + p[3] # p[] # p[3]
      p[4] |-> p[] # p[4] # p[4] + p[4] # p[] # p[4]
      ...

   The function pleth makes the symmetric functions act on the
   morphisms.

      sage: pleth( h[3], X*Z + Y*Z)
      1/6*p[] # p[1, 1, 1] # p[1, 1, 1] + 1/2*p[] # p[2, 1] # p[2, 1] +
      1/3*p[] # p[3] # p[3] + 1/2*p[1] # p[1, 1] # p[1, 1, 1] + 1/2*p[1] #
      p[2] # p[2, 1] + 1/2*p[1, 1] # p[1] # p[1, 1, 1] + 1/6*p[1, 1, 1] # p[]
      # p[1, 1, 1] + 1/2*p[2] # p[1] # p[2, 1] + 1/2*p[2, 1] # p[] # p[2, 1] +
      1/3*p[3] # p[] # p[3
File:           
Type:           classobj

Below $X$, $Y$ and $Z$ represent three independant alphabets, one for each factor of the tensor product of copies of $\operatorname{Sym}$. The file plethystic.sage defines some standard specializations. We give them short names.

In [26]:

X = specialization_X
Y = specialization_Y
Z = specialization_Z
epsilon = specialization_rank1(-1)
b = specialization_binomial

In [27]:

Out[27]:

Specialization of symmetric functions in the tensor product of three copies of Sym where:
p[1] |-> p[1] # p[] # p[]
p[2] |-> p[2] # p[] # p[]
p[3] |-> p[3] # p[] # p[]
p[4] |-> p[4] # p[] # p[]
 ...

In [28]:

epsilon

Out[28]:

Specialization of symmetric functions in the tensor product of three copies of Sym where:
p[1] |-> -p[] # p[] # p[]
p[2] |-> p[] # p[] # p[]
p[3] |-> -p[] # p[] # p[]
p[4] |-> p[] # p[] # p[]
 ...

In [29]:

b(2)

Out[29]:

Specialization of symmetric functions in the tensor product of three copies of Sym where:
p[1] |-> 2*p[] # p[] # p[]
p[2] |-> 2*p[] # p[] # p[]
p[3] |-> 2*p[] # p[] # p[]
p[4] |-> 2*p[] # p[] # p[]
 ...

Below we define

$$ \sigma_k = \sum_{n=0}^k h_n $$

as an approximation for

$$\sigma = \sum_{n=0}^{\infty} h_n.$$

In [30]:

def sigma(k):
    return add( h[i] for i in [0..k])

Example: Littlewood-Richardson coefficients.

In [31]:

import sage.libs.lrcalc.lrcalc as lrcalc

Let us compute all non--zero Littlewood-Richardson coefficients $c_{\mu, \nu}^{\lambda}$ for $|\lambda| \leq 3$.

In [32]:

L = change_basis(pleth(sigma(3), X*Z+Y*Z), s)
for (( mu, nu, lam),c) in L.monomial_coefficients().items():
    print c, lrcalc.lrcoef(lam, mu, nu), (lam, mu, nu)

1 1 ([2], [1], [1])
1 1 ([1, 1, 1], [1, 1], [1])
1 1 ([1, 1, 1], [], [1, 1, 1])
1 1 ([2], [], [2])
1 1 ([1, 1, 1], [1], [1, 1])
1 1 ([1, 1], [1], [1])
1 1 ([1, 1], [], [1, 1])
1 1 ([3], [], [3])
1 1 ([1], [1], [])
1 1 ([3], [3], [])
1 1 ([2, 1], [2], [1])
1 1 ([1, 1, 1], [1, 1, 1], [])
1 1 ([2], [2], [])
1 1 ([2, 1], [], [2, 1])
1 1 ([1], [], [1])
1 1 ([2, 1], [1], [2])
1 1 ([3], [2], [1])
1 1 ([2, 1], [1, 1], [1])
1 1 ([], [], [])
1 1 ([3], [1], [2])
1 1 ([1, 1], [1, 1], [])
1 1 ([2, 1], [2, 1], [])
1 1 ([2, 1], [1], [1, 1])

We now compare the Littlewood-Richardson coefficients $c_{\mu, \nu}^{\lambda}$ for $|\lambda| \leq 6$, with those given by lrcalc. Of course, they should coincide.

In [33]:

L = change_basis(pleth(sigma(6), X*Z+Y*Z), s)
for (( mu, nu, lam),c) in L:
    if c != lrcalc.lrcoef(lam, mu, nu):
        print (lam, mu, nu), c, lrcalc.lrcoef(lam, mu, nu)
        break
else:
    print "All correct."

All correct.

Example: Kronecker coefficients.

We compute the Kronecker coefficients $g_{\lambda, \mu, \nu}$ for $|\lambda| = |\mu| = |\nu| \leq 7$, and check they are correct.

In [34]:

N = 7
L = change_basis(pleth(sigma(N), X*Y*Z), s)
for ((lam, mu, nu),c) in L:
    if c != s(lam).kronecker_product(s(mu)).coefficient(nu):
        print (lam, mu, nu), c, s(lam).kronecker_product(s(mu)).coefficient(nu)
        break
else:
    print "All correct."

All correct.

Stable Kronecker coefficients.

The series

$$ \sum_{k=0}^n h_k[XYZ+XY+XZ+YZ]\text{ and } \sigma[XYZ+XY+XZ+YZ] $$ coincide in all total degrees $ < 2(n+1)$.

In [35]:

# WARNING: it takes some time.
N = 6
L = change_basis(pleth(sigma(N), X*Y*Z + X*Z + Y*Z + X*Y), s)
for ((lam, mu, nu),c) in L:
    if lam.size() + mu.size() + nu.size() < 2*(N+1) and c != stable_Kronecker_coefficient(lam, mu, nu):
        print (lam, mu, nu), c, stable_Kronecker_coefficient(lam, mu, nu)
        break
else:
    print "All correct."

All correct.

Hook-stable Kronecker coefficients: Tables 1 and 2 in [appendix].

The coefficient $\overline{\overline{g}}_{\alpha, \beta, \gamma}$ is a coefficient of

$$\sigma[XYZ+(1-\epsilon)(XY+XZ+YZ+X+Y+Z)].$$

Tables 1 and 2 of [appendix] display all coefficients $\overline{\overline{g}}_{\alpha, \beta, \gamma}$ whose indexing partitions have weight at most $3$. It is too costly to compute them using the same strategy as for the stable Kronecker coefficients above (it would imply expanding $\sigma_9$). Instead we approximate

$$\sigma[XYZ+(1-\epsilon)(XY+XZ+YZ+X+Y+Z)]$$

$$\sigma_3[XYZ] \cdot \sigma_3[(1-\epsilon)XY] \cdot \sigma_3[(1-\epsilon)XZ] \cdot \sigma_3[(1-\epsilon)YZ] \cdot \sigma_3[(1-\epsilon)X] \cdot \sigma_3[(1-\epsilon)Y] \cdot\sigma_3[(1-\epsilon)Z].$$

In [36]:

def projection(f, d):
    r"""
    Projection onto the direct sum of the homogeneous components of Sym of degrees <= d. 
    """
    return add( f.homogeneous_component(k) for k in [0..d])

In [37]:

N = 3
g_barbar=dict([])
series_gbarbar = pleth(sigma(N),X*Y*Z)
for M in [X*Y, X*Z, Y*Z, X, Y, Z]:
    series_gbarbar = series_gbarbar * pleth(sigma(N), (b(1)-epsilon) * M)
    series_gbarbar = series_gbarbar.apply_multilinear_morphism(lambda a, b, c: tensor([projection(a,N),
                                                                                       projection(b,N),
                                                                                       projection(c,N)]) ) 
    ## after each product we simplify by projecting.
series_gbarbar = change_basis(series_gbarbar, s)
for ((alpha, beta, gamma),c) in series_gbarbar:
            g_barbar[(alpha, beta, gamma)] = c
            if  alpha >= beta and beta >= gamma:
                print("{alpha:10} {beta:10} {gamma:10} {c}".format(gamma=vector(gamma), 
                                                                   beta=vector(beta), 
                                                                   alpha=vector(alpha), 
                                                                   c=c)
                     )

(1, 1, 1)  (1, 1)     (1)        84
(2, 1)     ()         ()         2
(2)        (1, 1)     (1)        66
(1)        ()         ()         2
(2, 1)     (2)        (1, 1)     382
(3)        (2)        (1)        84
(3)        (2)        (1, 1, 1)  320
(2)        (2)        (1)        66
(3)        (1, 1, 1)  (1, 1, 1)  565
(3)        (3)        (1)        122
(2, 1)     (1, 1, 1)  (1, 1, 1)  1056
(2)        (1, 1, 1)  ()         16
(2)        (1, 1, 1)  (1, 1, 1)  326
(2, 1)     (1)        (1)        64
(1, 1)     (1, 1)     (1)        66
(1, 1)     (1, 1)     (1, 1)     144
(3)        (3)        (1, 1)     320
()         ()         ()         1
(2)        (2)        ()         14
(3)        (1, 1)     (1)        84
(3)        (1, 1)     (1, 1)     206
(3)        (2, 1)     ()         38
(2, 1)     (2)        (1, 1, 1)  610
(2, 1)     (1, 1)     ()         28
(2, 1)     (1)        ()         12
(2)        (1)        (1)        34
(3)        (3)        (2)        326
(2)        (2)        (1, 1, 1)  204
(2, 1)     (2, 1)     (2, 1)     3933
(1, 1)     ()         ()         2
(2, 1)     (1, 1)     (1)        152
(2, 1)     (2, 1)     (2)        1168
(3)        (1, 1, 1)  ()         22
(3)        (2, 1)     (2)        610
(1, 1, 1)  (1, 1)     (1, 1)     204
(3)        (3)        (3)        565
(3)        (2, 1)     (2, 1)     2037
(3)        (2, 1)     (1, 1)     610
(3)        (1, 1, 1)  (1, 1)     326
(3)        (2, 1)     (1, 1, 1)  1056
(2)        (1, 1)     ()         14
(1, 1)     (1)        (1)        34
(3)        (3)        (2, 1)     1056
(3)        (2)        (2)        206
(2, 1)     (2)        (2)        382
(3)        (1)        (1)        38
(2)        (2)        (1, 1)     144
(1, 1, 1)  (1, 1)     ()         16
(1)        (1)        (1)        21
(2)        (1, 1)     (1, 1)     145
(2, 1)     (1, 1)     (1, 1)     382
(3)        (3)        ()         22
(2)        (2)        (2)        145
(2, 1)     (2)        (1)        152
(3)        (1, 1, 1)  (1)        122
(2, 1)     (1, 1, 1)  (1)        224
(3)        (3)        (1, 1, 1)  544
(1, 1, 1)  (1)        ()         8
(2, 1)     (2, 1)     (1)        428
(1, 1, 1)  ()         ()         2
(2, 1)     (1, 1, 1)  (1, 1)     610
(3)        (1)        ()         8
(2)        (1)        ()         8
(2)        (1, 1, 1)  (1, 1)     206
(1, 1, 1)  (1, 1, 1)  ()         22
(3)        (2, 1)     (1)        224
(3)        (1, 1)     ()         16
(1, 1, 1)  (1, 1, 1)  (1)        122
(2, 1)     (1, 1, 1)  ()         38
(1, 1)     (1)        ()         8
(3)        (2)        (1, 1)     204
(1, 1)     (1, 1)     ()         14
(2, 1)     (2, 1)     ()         74
(1, 1, 1)  (1)        (1)        38
(2, 1)     (2)        ()         28
(1, 1, 1)  (1, 1, 1)  (1, 1, 1)  544
(1, 1, 1)  (1, 1, 1)  (1, 1)     320
(2)        (1, 1, 1)  (1)        84
(3)        ()         ()         2
(2)        ()         ()         2
(2, 1)     (2, 1)     (1, 1)     1168
(3)        (2)        ()         16
(2, 1)     (2, 1)     (1, 1, 1)  2037
(1)        (1)        ()         6

Similarly, we compute the coefficients $A$, $B$ and $C$ appearing in Tables 1 and 2 of [appendix].

First, the coefficients $A$, obtained by coefficient extraction from:

$$\sigma[XYZ+2(XY+XZ+YZ+X+Y+Z)].$$

In [38]:

N = 3
A=dict([])
series_A = 1
for M in [X*Y*Z, b(2)*X*Y, b(2)*X*Z, b(2)*Y*Z, b(2)*X, b(2)*Y, b(2)*Z]:
    series_A = series_A*pleth(sigma(N), M)
    series_A = series_A.apply_multilinear_morphism(lambda a, b, c: tensor([projection(a,N),
                                                                           projection(b,N),
                                                                           projection(c,N)]) )
series_A = change_basis(series_A, s)
for ((alpha, beta, gamma),c) in series_A:
            A[(alpha, beta, gamma)] = c
            if alpha >= beta and beta >= gamma:
                print "{alpha:10} {beta:10} {gamma:10} {c}".format(gamma=vector(gamma), 
                                                                   beta=vector(beta), 
                                                                   alpha=vector(alpha), 
                                                                   c=c)

(1, 1, 1)  (1, 1)     (1)        54
(3)        (1, 1, 1)  (1)        80
(2, 1)     ()         ()         2
(1)        ()         ()         2
(2, 1)     (2)        (1, 1)     378
(3)        (2)        (1, 1, 1)  250
(1, 1)     (1, 1)     (1)        54
(2)        (2)        (1)        86
(3)        (3)        (1)        240
(2, 1)     (1, 1, 1)  (1, 1, 1)  736
(2)        (1, 1, 1)  (1, 1, 1)  240
(2, 1)     (1)        (1)        64
(3)        (1)        (1)        59
(3)        (2)        (1)        138
(1, 1)     (1, 1)     (1, 1)     110
(1, 1, 1)  (1, 1)     ()         10
(3)        (3)        (1, 1)     478
(2)        (2)        ()         20
(3)        (1, 1)     (1)        98
(1, 1)     ()         ()         1
(3)        (1, 1)     (1, 1)     220
(3)        (2, 1)     ()         50
(2, 1)     (1, 1)     ()         26
(2, 1)     (1)        ()         12
(2)        (1)        (1)        40
(3)        (3)        (2)        640
(2)        (2)        (1, 1, 1)  134
(2, 1)     (2, 1)     (2, 1)     3933
(2, 1)     (1, 1)     (1)        140
(3)        ()         ()         4
(2, 1)     (2, 1)     (2)        1242
(3)        (1, 1, 1)  ()         10
(2)        (1, 1, 1)  ()         6
(3)        (2, 1)     (2)        824
(1, 1, 1)  (1, 1)     (1, 1)     134
(3)        (3)        (3)        1243
(3)        (2, 1)     (2, 1)     2465
(3)        (2, 1)     (1, 1)     700
(3)        (1, 1, 1)  (1, 1)     260
(3)        (2, 1)     (1, 1, 1)  928
(2)        (1, 1)     ()         12
(1, 1)     (1)        (1)        28
(3)        (3)        (2, 1)     1632
(3)        (2)        (2)        348
(2, 1)     (2)        (2)        442
(2)        ()         ()         3
(3)        (1, 1, 1)  (1, 1, 1)  451
(2)        (2)        (1, 1)     150
(1)        (1)        (1)        21
(2)        (1, 1)     (1, 1)     131
(2, 1)     (1, 1)     (1, 1)     330
(3)        (3)        ()         50
(2)        (2)        (2)        203
(2, 1)     (2)        (1)        164
(2, 1)     (1, 1, 1)  (1)        160
(3)        (3)        (1, 1, 1)  506
(1, 1, 1)  (1)        ()         2
(2, 1)     (2, 1)     (1)        428
(3)        (1, 1)     ()         18
(2, 1)     (1, 1, 1)  (1, 1)     444
(3)        (1)        ()         14
(2)        (1)        ()         10
(2)        (1, 1, 1)  (1, 1)     144
(1, 1, 1)  (1, 1, 1)  ()         18
(3)        (2, 1)     (1)        288
(1, 1, 1)  (1, 1, 1)  (1)        88
(2, 1)     (1, 1, 1)  ()         26
(1, 1)     (1)        ()         6
(3)        (2)        (1, 1)     258
(2, 1)     (2, 1)     ()         74
(2)        (1, 1)     (1)        62
(1, 1, 1)  (1)        (1)        17
()         ()         ()         1
(2, 1)     (2)        ()         30
(2, 1)     (2)        (1, 1, 1)  472
(1, 1, 1)  (1, 1, 1)  (1, 1, 1)  322
(1, 1, 1)  (1, 1, 1)  (1, 1)     206
(2)        (1, 1, 1)  (1)        46
(2, 1)     (2, 1)     (1, 1)     1094
(1, 1)     (1, 1)     ()         12
(3)        (2)        ()         30
(2, 1)     (2, 1)     (1, 1, 1)  1609
(1)        (1)        ()         6

Now the coefficients $C$, from

$$\sigma[XYZ+(1+\epsilon)(XY+XZ+YZ+X+Y+Z)].$$

In [39]:

N = 3
C =dict([])
series_C = 1
for M in [X*Y*Z, (b(1)+epsilon)*X*Y, (b(1)+epsilon)*X*Z, (b(1)+epsilon)*Y*Z, 
          (b(1)+epsilon)*X, (b(1)+epsilon)*Y, (b(1)+epsilon)*Z]:
    series_C = series_C*pleth(sigma(N), M)
    series_C = series_C.apply_multilinear_morphism(lambda a, b, c: tensor([projection(a,N),
                                                                           projection(b,N),
                                                                           projection(c,N)]) )
series_C = change_basis(series_C, s)
for ((alpha, beta, gamma),c) in series_C:
            C[(alpha, beta, gamma)] = c
            if alpha >= beta and beta >= gamma:
                print "{alpha:10} {beta:10} {gamma:10} {c}".format(gamma=vector(gamma), 
                                                                   beta=vector(beta), 
                                                                   alpha=vector(alpha), 
                                                                   c=c)

(1, 1)     (1, 1)     ()         2
(3)        (3)        (1, 1, 1)  -4
(1, 1)     (1, 1)     (1, 1)     -4
(2)        (2)        ()         2
(1, 1, 1)  (1)        (1)        -1
(3)        (3)        (3)        5
(1, 1)     ()         ()         -1
(3)        (2, 1)     (2, 1)     1
(2)        (2)        (1, 1)     -4
(2, 1)     (2, 1)     (2, 1)     1
(1, 1, 1)  (1, 1, 1)  (1)        2
(3)        (1, 1, 1)  (1, 1, 1)  5
(2)        (1, 1)     (1, 1)     5
(1)        (1)        (1)        1
(2)        ()         ()         1
(3)        (1, 1, 1)  (1)        -2
(2, 1)     (2, 1)     (1, 1, 1)  1
(3)        (1)        (1)        1
(3)        (3)        (1)        2
(2)        (2)        (2)        5
()         ()         ()         1
(2)        (1, 1)     ()         -2
(1, 1, 1)  (1, 1, 1)  (1, 1, 1)  -4

For the $B$ coefficients we have the following form of the generating series:

$$\frac{3}{4} \sigma[XYZ+2W]+\frac{1}{4} \sigma[XYZ+(1+\varepsilon) W]+\sigma[XYZ+2W] \left(\chi[YZ-X]-\frac{1}{2}\chi[W] \right)$$

In [40]:

def chi(k):
    return add(p[i] for i in [1..k])

N = 3
B =dict([])
W = X*Y + X*Z + Y*Z + X + Y + Z
series_B = pleth(chi(N),Y*Z-X)-1/2*pleth(chi(N),W)
series_B = series_A*series_B
series_B = series_B.apply_multilinear_morphism(lambda a, b, c: tensor([projection(a,N),
                                                                       projection(b,N),
                                                                       projection(c,N)]) )
series_B = 3/4*series_A + 1/4*series_C + series_B
series_B = change_basis(series_B, s)
for ((alpha, beta, gamma),c) in series_B:
            B[(alpha, beta, gamma)] = c
            if  beta >= gamma:
                print "{alpha:10} {beta:10} {gamma:10} {c}".format(gamma=vector(gamma), beta=vector(beta), alpha=vector(alpha), c=c)

(2)        (1)        (1)        -25
(1, 1)     (1, 1, 1)  (1, 1, 1)  -27
(3)        (2, 1)     (1, 1)     -738
(1, 1, 1)  (1, 1)     (1)        -42
(3)        (1, 1)     (1)        -118
(3)        (1, 1, 1)  (1)        -85
(3)        (3)        (2)        -820
(1, 1)     (2, 1)     (1, 1)     -128
(1, 1)     (2, 1)     (1, 1, 1)  -116
()         (1)        ()         1
(2, 1)     ()         ()         -3
(1, 1)     (2, 1)     (2, 1)     -435
(1, 1)     (3)        (2)        -170
(3)        (2)        (2)        -435
(1, 1, 1)  (3)        (2)        -250
(2, 1)     (2)        (1, 1)     -371
(1, 1, 1)  (1)        (1)        -15
(1, 1)     (3)        (1, 1, 1)  -110
(1, 1)     (2, 1)     ()         -15
(1)        (3)        ()         -6
(3)        (2, 1)     ()         -66
()         (2)        ()         1
(2, 1)     (3)        (1)        -344
(2, 1)     (2)        (1, 1, 1)  -394
(1, 1, 1)  (3)        (2, 1)     -832
(3)        (1)        (1)        -78
()         (3)        (1, 1, 1)  3
(2)        (3)        (3)        -500
(1, 1, 1)  (1, 1)     ()         -8
(3)        (2)        (1, 1, 1)  -250
(2, 1)     (2, 1)     (2, 1)     -3470
(3)        (1, 1)     ()         -24
()         (3)        (1)        1
(2)        (2, 1)     ()         -21
(1, 1, 1)  (1, 1, 1)  (1)        -59
()         (2)        (1, 1, 1)  3
(2, 1)     (1)        ()         -15
(1, 1)     (1, 1)     (1)        -21
(2, 1)     (1, 1)     (1, 1)     -293
(1, 1, 1)  (3)        ()         -12
(2)        (3)        (1)        -109
(1, 1)     (2)        ()         -8
(1, 1)     ()         ()         -1
(2, 1)     (1, 1, 1)  ()         -24
(1, 1)     (1)        ()         -3
(2, 1)     (3)        (1, 1, 1)  -832
(1, 1, 1)  (2, 1)     (2)        -394
()         (2, 1)     (1, 1)     11
(2, 1)     (3)        (3)        -1888
(2, 1)     (1, 1)     (1)        -139
(1, 1, 1)  (3)        (1)        -85
(2)        (2)        (1, 1)     -84
(2)        (2)        (1)        -57
(3)        (2)        ()         -42
(2)        (3)        (1, 1, 1)  -125
(1, 1)     (1, 1, 1)  (1)        -15
()         (3)        (3)        3
()         (2)        (2)        6
(1, 1)     (1, 1, 1)  (1, 1)     -30
(2)        (2)        (1, 1, 1)  -54
(1, 1, 1)  (2)        ()         -6
(1, 1)     (2)        (2)        -84
(1, 1, 1)  (1, 1)     (1, 1)     -97
(1, 1)     (1)        (1)        -13
()         (2, 1)     (1)        3
(3)        (1, 1, 1)  ()         -12
(2, 1)     (1, 1)     ()         -28
(2)        (2, 1)     (1, 1)     -182
(1)        (3)        (1)        -19
(1)        (1, 1, 1)  (1)        2
(3)        (2)        (1, 1)     -299
(1, 1, 1)  (2)        (2)        -121
(2)        (1, 1)     ()         -8
(1)        (1, 1)     (1)        1
(2, 1)     (3)        (2)        -938
()         (1, 1)     (1)        3
(1, 1)     (3)        ()         -15
(3)        (2)        (1)        -178
(1, 1, 1)  (2, 1)     (1)        -136
()         (1, 1)     (1, 1)     8
(1, 1, 1)  (1, 1, 1)  (1, 1, 1)  -175
(2, 1)     (3)        (1, 1)     -738
(2)        (3)        (2)        -261
()         (2, 1)     (1, 1, 1)  15
(1)        (2, 1)     (2)        -17
(1)        (2)        (1, 1, 1)  4
(1)        (1, 1)     (1, 1)     6
(3)        (3)        (1)        -321
(2)        (2, 1)     (2, 1)     -597
()         (3)        (2)        3
(3)        (2, 1)     (2, 1)     -2515
(1, 1)     (2, 1)     (1)        -69
()         (3)        (2, 1)     9
()         (1, 1, 1)  (1, 1)     7
(2)        (2, 1)     (2)        -256
(2)        (1, 1, 1)  (1)        -19
()         (3)        (1, 1)     3
(1)        (3)        (1, 1)     -20
(2, 1)     (1, 1, 1)  (1, 1, 1)  -480
(1, 1)     (3)        (3)        -335
(1, 1, 1)  (3)        (3)        -521
(1, 1, 1)  (2, 1)     ()         -24
(1)        (2, 1)     (1)        -8
(2, 1)     (3)        (2, 1)     -2515
(3)        (3)        ()         -72
(1, 1, 1)  (1, 1, 1)  (1, 1)     -130
(1, 1, 1)  (2)        (1, 1)     -117
(3)        (3)        (3)        -1597
(1)        (3)        (2, 1)     -56
(1)        (2)        (2)        -14
(2)        (1, 1, 1)  (1, 1, 1)  -48
(1)        (2, 1)     (1, 1)     1
(2, 1)     (1)        (1)        -72
(1)        (3)        (2)        -40
(2, 1)     (2, 1)     (1, 1)     -982
(1)        (2, 1)     (2, 1)     -5
(1, 1)     (3)        (2, 1)     -388
(3)        ()         ()         -6
(3)        (2, 1)     (1)        -344
(2)        (2)        (2)        -133
(1, 1)     (2)        (1)        -33
(2)        ()         ()         -2
()         (2, 1)     (2)        9
(2)        (3)        ()         -27
(1)        (1, 1, 1)  (1, 1)     12
(3)        (1, 1, 1)  (1, 1, 1)  -355
(2)        (2, 1)     (1)        -99
(1, 1, 1)  (2)        (1, 1, 1)  -168
()         (2)        (1)        3
(3)        (1, 1, 1)  (1, 1)     -240
(1, 1, 1)  (3)        (1, 1)     -240
(3)        (2, 1)     (1, 1, 1)  -832
(1)        (3)        (3)        -81
(1, 1)     (1, 1, 1)  ()         -3
(2, 1)     (1, 1, 1)  (1)        -136
(2, 1)     (2, 1)     (2)        -1218
(1, 1)     (3)        (1, 1)     -125
(3)        (2, 1)     (2)        -938
(1)        (2, 1)     (1, 1, 1)  24
()         (1, 1, 1)  (1, 1, 1)  15
(1, 1, 1)  (1)        ()         -2
(1, 1)     (3)        (1)        -69
(2)        (3)        (2, 1)     -526
(3)        (3)        (1, 1, 1)  -521
(1, 1)     (1, 1)     (1, 1)     -38
()         (2)        (1, 1)     4
(2, 1)     (2, 1)     (1)        -433
(2, 1)     (2)        (2)        -477
(3)        (3)        (1, 1)     -574
()         ()         ()         1
(1)        (2, 1)     ()         -3
(2, 1)     (2)        ()         -36
(2)        (2)        ()         -14
(1)        (2)        ()         -2
(1, 1)     (2)        (1, 1)     -55
(2)        (2, 1)     (1, 1, 1)  -158
(1, 1, 1)  (2, 1)     (1, 1, 1)  -480
()         (2, 1)     (2, 1)     30
(1, 1)     (1, 1)     ()         -4
(2)        (1, 1, 1)  ()         -3
(2)        (1, 1)     (1)        -33
()         (1)        (1)        3
(1)        (2)        (1, 1)     -2
(2, 1)     (2)        (1)        -181
(1)        (1, 1, 1)  (1, 1, 1)  29
(1, 1)     (2, 1)     (2)        -182
(1, 1, 1)  (3)        (1, 1, 1)  -355
(2, 1)     (1, 1, 1)  (1, 1)     -338
(3)        (3)        (2, 1)     -1888
(1)        (3)        (1, 1, 1)  -5
(2)        (3)        (1, 1)     -170
(2, 1)     (2, 1)     (1, 1, 1)  -1221
(3)        (1, 1)     (1, 1)     -235
(2, 1)     (2, 1)     ()         -81
(1, 1, 1)  (2, 1)     (2, 1)     -1221
()         (1, 1, 1)  (1)        1
(3)        (1)        ()         -20
(2)        (1, 1)     (1, 1)     -55
(1)        (2)        (1)        -5
(1, 1, 1)  (2)        (1)        -42
(2)        (1)        ()         -7
(2, 1)     (3)        ()         -66
(2)        (1, 1, 1)  (1, 1)     -45
(1, 1, 1)  (2, 1)     (1, 1)     -338
(1, 1, 1)  (1, 1, 1)  ()         -12
(1, 1)     (2)        (1, 1, 1)  -45

Production of Tables 1 and 2 of [appendix].

We use the results of the previous computatiions to provide a table LaTeX of the coefficients indexed by partitions of weight at most $3$. Note that the coefficients $A$, $C$ and $\overline{\overline{g}}$ are symmetric in their three arguments. The coefficients $B$ are only symmetric in their last two arguments.

In [41]:

def phi(S,lam, mu, nu):
    if (lam, mu, nu) in S:
        return S[lam, mu, nu]
    else:
        return 0

res=("""\\begin{array}[ccccccccc]\\\\
         \\alpha 
         & \\beta 
         & \\gamma 
         & \\overline{\\overline{g}}_{\\alpha, \\beta, \\gamma} 
         & A_{\\alpha, \\beta, \\gamma} 
         & B_{\\alpha, \\beta, \\gamma} 
         & B_{\\beta, \\alpha, \\gamma} 
         & B_{\\gamma, \\alpha, \\beta} 
         & C_{\\alpha, \\beta, \\gamma}  
         \\\\\n""")
L = []
for i in [0..3]:
    L.extend(Partitions(i))
for alpha in L:
    for beta in L:
        if alpha >= beta:
            for gamma in L:
                if beta >= gamma:
                    res += ("{alpha:9}&{beta:9}&{gamma:9}&{gbarbar:5}&{A:5}&{B1:5}&{B2:5}&{B3:5}&{C:5} \\\\ \n".
                            format(alpha=vector(alpha), 
                                   beta=vector(beta), 
                                   gamma=vector(gamma), 
                                   gbarbar=phi(g_barbar,alpha, beta, gamma), 
                                   A=phi(A,alpha, beta, gamma), 
                                   B1=phi(B,alpha,beta,gamma), 
                                   B2=phi(B, beta, alpha, gamma), 
                                   B3=phi(B,gamma, alpha, beta), 
                                   C=phi(C,alpha, beta, gamma))
                           )
res += "\\end{array}"

show(LatexExpr(res))

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}[ccccccccc]\\ \alpha & \beta & \gamma & \overline{\overline{g}}_{\alpha, \beta, \gamma} & A_{\alpha, \beta, \gamma} & B_{\alpha, \beta, \gamma} & B_{\beta, \alpha, \gamma} & B_{\gamma, \alpha, \beta} & C_{\alpha, \beta, \gamma} \\ () &() &() &1 &1 &1 &1 &1 &1 \\ (1) &() &() &2 &2 & 0&1 &1 & 0 \\ (1) &(1) &() &6 &6 & 0& 0&3 & 0 \\ (1) &(1) &(1) &21 &21 & 0& 0& 0&1 \\ (2) &() &() &2 &3 &-2 &1 &1 &1 \\ (2) &(1) &() &8 &10 &-7 &-2 &3 & 0 \\ (2) &(1) &(1) &34 &40 &-25 &-5 &-5 & 0 \\ (2) &(2) &() &14 &20 &-14 &-14 &6 &2 \\ (2) &(2) &(1) &66 &86 &-57 &-57 &-14 & 0 \\ (2) &(2) &(2) &145 &203 &-133 &-133 &-133 &5 \\ (2) &(2) &(1, 1) &144 &150 &-84 &-84 &-84 &-4 \\ (2) &(2) &(1, 1, 1)&204 &134 &-54 &-54 &-121 & 0 \\ (2) &(1, 1) &() &14 &12 &-8 &-8 &4 &-2 \\ (2) &(1, 1) &(1) &66 &62 &-33 &-33 &-2 & 0 \\ (2) &(1, 1) &(1, 1) &145 &131 &-55 &-55 &-55 &5 \\ (2) &(1, 1, 1)&() &16 &6 &-3 &-6 &3 & 0 \\ (2) &(1, 1, 1)&(1) &84 &46 &-19 &-42 &4 & 0 \\ (2) &(1, 1, 1)&(1, 1) &206 &144 &-45 &-117 &-45 & 0 \\ (2) &(1, 1, 1)&(1, 1, 1)&326 &240 &-48 &-168 &-168 & 0 \\ (1, 1) &() &() &2 &1 &-1 & 0& 0&-1 \\ (1, 1) &(1) &() &8 &6 &-3 & 0&3 & 0 \\ (1, 1) &(1) &(1) &34 &28 &-13 &1 &1 & 0 \\ (1, 1) &(1, 1) &() &14 &12 &-4 &-4 &8 &2 \\ (1, 1) &(1, 1) &(1) &66 &54 &-21 &-21 &6 & 0 \\ (1, 1) &(1, 1) &(1, 1) &144 &110 &-38 &-38 &-38 &-4 \\ (3) &() &() &2 &4 &-6 & 0& 0& 0 \\ (3) &(1) &() &8 &14 &-20 &-6 &1 & 0 \\ (3) &(1) &(1) &38 &59 &-78 &-19 &-19 &1 \\ (3) &(2) &() &16 &30 &-42 &-27 &3 & 0 \\ (3) &(2) &(1) &84 &138 &-178 &-109 &-40 & 0 \\ (3) &(2) &(2) &206 &348 &-435 &-261 &-261 & 0 \\ (3) &(2) &(1, 1) &204 &258 &-299 &-170 &-170 & 0 \\ (3) &(2) &(1, 1, 1)&320 &250 &-250 &-125 &-250 & 0 \\ (3) &(1, 1) &() &16 &18 &-24 &-15 &3 & 0 \\ (3) &(1, 1) &(1) &84 &98 &-118 &-69 &-20 & 0 \\ (3) &(1, 1) &(1, 1) &206 &220 &-235 &-125 &-125 & 0 \\ (3) &(3) &() &22 &50 &-72 &-72 &3 & 0 \\ (3) &(3) &(1) &122 &240 &-321 &-321 &-81 &2 \\ (3) &(3) &(2) &326 &640 &-820 &-820 &-500 & 0 \\ (3) &(3) &(1, 1) &320 &478 &-574 &-574 &-335 & 0 \\ (3) &(3) &(3) &565 &1243 &-1597&-1597&-1597&5 \\ (3) &(3) &(2, 1) &1056 &1632 &-1888&-1888&-1888& 0 \\ (3) &(3) &(1, 1, 1)&544 &506 &-521 &-521 &-521 &-4 \\ (3) &(2, 1) &() &38 &50 &-66 &-66 &9 & 0 \\ (3) &(2, 1) &(1) &224 &288 &-344 &-344 &-56 & 0 \\ (3) &(2, 1) &(2) &610 &824 &-938 &-938 &-526 & 0 \\ (3) &(2, 1) &(1, 1) &610 &700 &-738 &-738 &-388 & 0 \\ (3) &(2, 1) &(2, 1) &2037 &2465 &-2515&-2515&-2515&1 \\ (3) &(2, 1) &(1, 1, 1)&1056 &928 &-832 &-832 &-832 & 0 \\ (3) &(1, 1, 1)&() &22 &10 &-12 &-12 &3 & 0 \\ (3) &(1, 1, 1)&(1) &122 &80 &-85 &-85 &-5 &-2 \\ (3) &(1, 1, 1)&(1, 1) &326 &260 &-240 &-240 &-110 & 0 \\ (3) &(1, 1, 1)&(1, 1, 1)&565 &451 &-355 &-355 &-355 &5 \\ (2, 1) &() &() &2 &2 &-3 & 0& 0& 0 \\ (2, 1) &(1) &() &12 &12 &-15 &-3 &3 & 0 \\ (2, 1) &(1) &(1) &64 &64 &-72 &-8 &-8 & 0 \\ (2, 1) &(2) &() &28 &30 &-36 &-21 &9 & 0 \\ (2, 1) &(2) &(1) &152 &164 &-181 &-99 &-17 & 0 \\ (2, 1) &(2) &(2) &382 &442 &-477 &-256 &-256 & 0 \\ (2, 1) &(2) &(1, 1) &382 &378 &-371 &-182 &-182 & 0 \\ (2, 1) &(2) &(1, 1, 1)&610 &472 &-394 &-158 &-394 & 0 \\ (2, 1) &(1, 1) &() &28 &26 &-28 &-15 &11 & 0 \\ (2, 1) &(1, 1) &(1) &152 &140 &-139 &-69 &1 & 0 \\ (2, 1) &(1, 1) &(1, 1) &382 &330 &-293 &-128 &-128 & 0 \\ (2, 1) &(2, 1) &() &74 &74 &-81 &-81 &30 & 0 \\ (2, 1) &(2, 1) &(1) &428 &428 &-433 &-433 &-5 & 0 \\ (2, 1) &(2, 1) &(2) &1168 &1242 &-1218&-1218&-597 & 0 \\ (2, 1) &(2, 1) &(1, 1) &1168 &1094 &-982 &-982 &-435 & 0 \\ (2, 1) &(2, 1) &(2, 1) &3933 &3933 &-3470&-3470&-3470&1 \\ (2, 1) &(2, 1) &(1, 1, 1)&2037 &1609 &-1221&-1221&-1221&1 \\ (2, 1) &(1, 1, 1)&() &38 &26 &-24 &-24 &15 & 0 \\ (2, 1) &(1, 1, 1)&(1) &224 &160 &-136 &-136 &24 & 0 \\ (2, 1) &(1, 1, 1)&(1, 1) &610 &444 &-338 &-338 &-116 & 0 \\ (2, 1) &(1, 1, 1)&(1, 1, 1)&1056 &736 &-480 &-480 &-480 & 0 \\ (1, 1, 1)&() &() &2 & 0& 0& 0& 0& 0 \\ (1, 1, 1)&(1) &() &8 &2 &-2 & 0&1 & 0 \\ (1, 1, 1)&(1) &(1) &38 &17 &-15 &2 &2 &-1 \\ (1, 1, 1)&(1, 1) &() &16 &10 &-8 &-3 &7 & 0 \\ (1, 1, 1)&(1, 1) &(1) &84 &54 &-42 &-15 &12 & 0 \\ (1, 1, 1)&(1, 1) &(1, 1) &204 &134 &-97 &-30 &-30 & 0 \\ (1, 1, 1)&(1, 1, 1)&() &22 &18 &-12 &-12 &15 & 0 \\ (1, 1, 1)&(1, 1, 1)&(1) &122 &88 &-59 &-59 &29 &2 \\ (1, 1, 1)&(1, 1, 1)&(1, 1) &320 &206 &-130 &-130 &-27 & 0 \\ (1, 1, 1)&(1, 1, 1)&(1, 1, 1)&544 &322 &-175 &-175 &-175 &-4 \\ \end{array}$

End of the worksheet.

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	1	0	0	0	0	0	0	0	0
	2	1	3	2	2	0	0	0	0	0	0	0
	3	0	1	2	3	2	0	0	0	0	0	0
	4	0	0	0	2	3	2	0	0	0	0	0
	5	0	0	0	0	2	3	2	0	0	0	0
	6	0	0	0	0	0	2	3	2	0	0	0
	7	0	0	0	0	0	0	2	3	2	0	0
	8	0	0	0	0	0	0	0	2	3	2	0
	9	0	0	0	0	0	0	0	0	2	3	2
	10	0	0	0	0	0	0	0	0	0	2	3

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	2	1	0	0	0	0	0	0	0
	2	2	5	4	4	1	0	0	0	0	0	0
	3	1	3	3	7	5	1	0	0	0	0	0
	4	0	0	1	5	8	5	1	0	0	0	0
	5	0	0	0	1	5	8	5	1	0	0	0
	6	0	0	0	0	1	5	8	5	1	0	0
	7	0	0	0	0	0	1	5	8	5	1	0
	8	0	0	0	0	0	0	1	5	8	5	1
	9	0	0	0	0	0	0	0	1	5	8	5
	10	0	0	0	0	0	0	0	0	1	5	8

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	3	2	2	0	0	0	0	0	0	0
	1	2	5	4	4	1	0	0	0	0	0	0
	2	4	13	13	17	8	1	0	0	0	0	0
	3	3	12	15	30	25	9	1	0	0	0	0
	4	1	4	7	26	38	27	9	1	0	0	0
	5	0	0	1	10	29	39	27	9	1	0	0
	6	0	0	0	1	10	29	39	27	9	1	0
	7	0	0	0	0	1	10	29	39	27	9	1
	8	0	0	0	0	0	1	10	29	39	27	9
	9	0	0	0	0	0	0	1	10	29	39	27
	10	0	0	0	0	0	0	0	1	10	29	39

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	1	2	3	2	0	0	0	0	0	0
	1	1	3	3	7	5	1	0	0	0	0	0
	2	3	12	15	30	25	9	1	0	0	0	0
	3	4	18	26	55	58	34	10	1	0	0	0
	4	3	13	20	56	80	69	36	10	1	0	0
	5	1	4	7	33	69	86	70	36	10	1	0
	6	0	0	1	10	36	70	86	70	36	10	1
	7	0	0	0	1	10	36	70	86	70	36	10
	8	0	0	0	0	1	10	36	70	86	70	36
	9	0	0	0	0	0	1	10	36	70	86	70
	10	0	0	0	0	0	0	1	10	36	70	86

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	2	3	2	0	0	0	0	0
	1	0	0	1	5	8	5	1	0	0	0	0
	2	1	4	7	26	38	27	9	1	0	0	0
	3	3	13	20	56	80	69	36	10	1	0	0
	4	4	18	28	70	104	109	80	38	10	1	0
	5	3	13	20	59	99	121	115	81	38	10	1
	6	1	4	7	33	72	105	122	115	81	38	10
	7	0	0	1	10	36	73	105	122	115	81	38
	8	0	0	0	1	10	36	73	105	122	115	81
	9	0	0	0	0	1	10	36	73	105	122	115
	10	0	0	0	0	0	1	10	36	73	105	122

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	2	3	2	0	0	0	0
	1	0	0	0	1	5	8	5	1	0	0	0
	2	0	0	1	10	29	39	27	9	1	0	0
	3	1	4	7	33	69	86	70	36	10	1	0
	4	3	13	20	59	99	121	115	81	38	10	1
	5	4	18	28	70	108	132	138	121	82	38	10
	6	3	13	20	59	99	125	138	139	121	82	38
	7	1	4	7	33	72	105	126	138	139	121	82
	8	0	0	1	10	36	73	105	126	138	139	121
	9	0	0	0	1	10	36	73	105	126	138	139
	10	0	0	0	0	1	10	36	73	105	126	138

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	2	3	2	0	0	0
	1	0	0	0	0	1	5	8	5	1	0	0
	2	0	0	0	1	10	29	39	27	9	1	0
	3	0	0	1	10	36	70	86	70	36	10	1
	4	1	4	7	33	72	105	122	115	81	38	10
	5	3	13	20	59	99	125	138	139	121	82	38
	6	4	18	28	70	108	132	142	144	140	121	82
	7	3	13	20	59	99	125	138	143	144	140	121
	8	1	4	7	33	72	105	126	138	143	144	140
	9	0	0	1	10	36	73	105	126	138	143	144
	10	0	0	0	1	10	36	73	105	126	138	143

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	2	3	2	0	0
	1	0	0	0	0	0	1	5	8	5	1	0
	2	0	0	0	0	1	10	29	39	27	9	1
	3	0	0	0	1	10	36	70	86	70	36	10
	4	0	0	1	10	36	73	105	122	115	81	38
	5	1	4	7	33	72	105	126	138	139	121	82
	6	3	13	20	59	99	125	138	143	144	140	121
	7	4	18	28	70	108	132	142	144	144	144	140
	8	3	13	20	59	99	125	138	143	144	144	144
	9	1	4	7	33	72	105	126	138	143	144	144
	10	0	0	1	10	36	73	105	126	138	143	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	2	3	2	0
	1	0	0	0	0	0	0	1	5	8	5	1
	2	0	0	0	0	0	1	10	29	39	27	9
	3	0	0	0	0	1	10	36	70	86	70	36
	4	0	0	0	1	10	36	73	105	122	115	81
	5	0	0	1	10	36	73	105	126	138	139	121
	6	1	4	7	33	72	105	126	138	143	144	140
	7	3	13	20	59	99	125	138	143	144	144	144
	8	4	18	28	70	108	132	142	144	144	144	144
	9	3	13	20	59	99	125	138	143	144	144	144
	10	1	4	7	33	72	105	126	138	143	144	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	0	2	3	2
	1	0	0	0	0	0	0	0	1	5	8	5
	2	0	0	0	0	0	0	1	10	29	39	27
	3	0	0	0	0	0	1	10	36	70	86	70
	4	0	0	0	0	1	10	36	73	105	122	115
	5	0	0	0	1	10	36	73	105	126	138	139
	6	0	0	1	10	36	73	105	126	138	143	144
	7	1	4	7	33	72	105	126	138	143	144	144
	8	3	13	20	59	99	125	138	143	144	144	144
	9	4	18	28	70	108	132	142	144	144	144	144
	10	3	13	20	59	99	125	138	143	144	144	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	1	0	0	0	0	0	0	0	0
	2	1	3	2	2	0	0	0	0	0	0	0
	3	0	1	2	3	2	0	0	0	0	0	0
	4	0	0	0	2	3	2	0	0	0	0	0
	5	0	0	0	0	2	3	2	0	0	0	0
	6	0	0	0	0	0	2	3	2	0	0	0
	7	0	0	0	0	0	0	2	3	2	0	0
	8	0	0	0	0	0	0	0	2	3	2	0
	9	0	0	0	0	0	0	0	0	2	3	2
	10	0	0	0	0	0	0	0	0	0	2	3

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	2	1	0	0	0	0	0	0	0
	2	2	5	4	4	1	0	0	0	0	0	0
	3	1	3	3	7	5	1	0	0	0	0	0
	4	0	0	1	5	8	5	1	0	0	0	0
	5	0	0	0	1	5	8	5	1	0	0	0
	6	0	0	0	0	1	5	8	5	1	0	0
	7	0	0	0	0	0	1	5	8	5	1	0
	8	0	0	0	0	0	0	1	5	8	5	1
	9	0	0	0	0	0	0	0	1	5	8	5
	10	0	0	0	0	0	0	0	0	1	5	8

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	3	2	2	0	0	0	0	0	0	0
	1	2	5	4	4	1	0	0	0	0	0	0
	2	4	13	13	17	8	1	0	0	0	0	0
	3	3	12	15	30	25	9	1	0	0	0	0
	4	1	4	7	26	38	27	9	1	0	0	0
	5	0	0	1	10	29	39	27	9	1	0	0
	6	0	0	0	1	10	29	39	27	9	1	0
	7	0	0	0	0	1	10	29	39	27	9	1
	8	0	0	0	0	0	1	10	29	39	27	9
	9	0	0	0	0	0	0	1	10	29	39	27
	10	0	0	0	0	0	0	0	1	10	29	39

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	1	2	3	2	0	0	0	0	0	0
	1	1	3	3	7	5	1	0	0	0	0	0
	2	3	12	15	30	25	9	1	0	0	0	0
	3	4	18	26	55	58	34	10	1	0	0	0
	4	3	13	20	56	80	69	36	10	1	0	0
	5	1	4	7	33	69	86	70	36	10	1	0
	6	0	0	1	10	36	70	86	70	36	10	1
	7	0	0	0	1	10	36	70	86	70	36	10
	8	0	0	0	0	1	10	36	70	86	70	36
	9	0	0	0	0	0	1	10	36	70	86	70
	10	0	0	0	0	0	0	1	10	36	70	86

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	2	3	2	0	0	0	0	0
	1	0	0	1	5	8	5	1	0	0	0	0
	2	1	4	7	26	38	27	9	1	0	0	0
	3	3	13	20	56	80	69	36	10	1	0	0
	4	4	18	28	70	104	109	80	38	10	1	0
	5	3	13	20	59	99	121	115	81	38	10	1
	6	1	4	7	33	72	105	122	115	81	38	10
	7	0	0	1	10	36	73	105	122	115	81	38
	8	0	0	0	1	10	36	73	105	122	115	81
	9	0	0	0	0	1	10	36	73	105	122	115
	10	0	0	0	0	0	1	10	36	73	105	122

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	2	3	2	0	0	0	0
	1	0	0	0	1	5	8	5	1	0	0	0
	2	0	0	1	10	29	39	27	9	1	0	0
	3	1	4	7	33	69	86	70	36	10	1	0
	4	3	13	20	59	99	121	115	81	38	10	1
	5	4	18	28	70	108	132	138	121	82	38	10
	6	3	13	20	59	99	125	138	139	121	82	38
	7	1	4	7	33	72	105	126	138	139	121	82
	8	0	0	1	10	36	73	105	126	138	139	121
	9	0	0	0	1	10	36	73	105	126	138	139
	10	0	0	0	0	1	10	36	73	105	126	138

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	2	3	2	0	0	0
	1	0	0	0	0	1	5	8	5	1	0	0
	2	0	0	0	1	10	29	39	27	9	1	0
	3	0	0	1	10	36	70	86	70	36	10	1
	4	1	4	7	33	72	105	122	115	81	38	10
	5	3	13	20	59	99	125	138	139	121	82	38
	6	4	18	28	70	108	132	142	144	140	121	82
	7	3	13	20	59	99	125	138	143	144	140	121
	8	1	4	7	33	72	105	126	138	143	144	140
	9	0	0	1	10	36	73	105	126	138	143	144
	10	0	0	0	1	10	36	73	105	126	138	143

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	2	3	2	0	0
	1	0	0	0	0	0	1	5	8	5	1	0
	2	0	0	0	0	1	10	29	39	27	9	1
	3	0	0	0	1	10	36	70	86	70	36	10
	4	0	0	1	10	36	73	105	122	115	81	38
	5	1	4	7	33	72	105	126	138	139	121	82
	6	3	13	20	59	99	125	138	143	144	140	121
	7	4	18	28	70	108	132	142	144	144	144	140
	8	3	13	20	59	99	125	138	143	144	144	144
	9	1	4	7	33	72	105	126	138	143	144	144
	10	0	0	1	10	36	73	105	126	138	143	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	2	3	2	0
	1	0	0	0	0	0	0	1	5	8	5	1
	2	0	0	0	0	0	1	10	29	39	27	9
	3	0	0	0	0	1	10	36	70	86	70	36
	4	0	0	0	1	10	36	73	105	122	115	81
	5	0	0	1	10	36	73	105	126	138	139	121
	6	1	4	7	33	72	105	126	138	143	144	140
	7	3	13	20	59	99	125	138	143	144	144	144
	8	4	18	28	70	108	132	142	144	144	144	144
	9	3	13	20	59	99	125	138	143	144	144	144
	10	1	4	7	33	72	105	126	138	143	144	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	0	2	3	2
	1	0	0	0	0	0	0	0	1	5	8	5
	2	0	0	0	0	0	0	1	10	29	39	27
	3	0	0	0	0	0	1	10	36	70	86	70
	4	0	0	0	0	1	10	36	73	105	122	115
	5	0	0	0	1	10	36	73	105	126	138	139
	6	0	0	1	10	36	73	105	126	138	143	144
	7	1	4	7	33	72	105	126	138	143	144	144
	8	3	13	20	59	99	125	138	143	144	144	144
	9	4	18	28	70	108	132	142	144	144	144	144
	10	3	13	20	59	99	125	138	143	144	144	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	1	0	0	0	0	0	0	0	0
	2	1	3	2	2	0	0	0	0	0	0	0
	3	0	1	2	3	2	0	0	0	0	0	0
	4	0	0	0	2	3	2	0	0	0	0	0
	5	0	0	0	0	2	3	2	0	0	0	0
	6	0	0	0	0	0	2	3	2	0	0	0
	7	0	0	0	0	0	0	2	3	2	0	0
	8	0	0	0	0	0	0	0	2	3	2	0
	9	0	0	0	0	0	0	0	0	2	3	2
	10	0	0	0	0	0	0	0	0	0	2	3

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	2	1	0	0	0	0	0	0	0	0
	1	1	2	2	1	0	0	0	0	0	0	0
	2	2	5	4	4	1	0	0	0	0	0	0
	3	1	3	3	7	5	1	0	0	0	0	0
	4	0	0	1	5	8	5	1	0	0	0	0
	5	0	0	0	1	5	8	5	1	0	0	0
	6	0	0	0	0	1	5	8	5	1	0	0
	7	0	0	0	0	0	1	5	8	5	1	0
	8	0	0	0	0	0	0	1	5	8	5	1
	9	0	0	0	0	0	0	0	1	5	8	5
	10	0	0	0	0	0	0	0	0	1	5	8

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	1	3	2	2	0	0	0	0	0	0	0
	1	2	5	4	4	1	0	0	0	0	0	0
	2	4	13	13	17	8	1	0	0	0	0	0
	3	3	12	15	30	25	9	1	0	0	0	0
	4	1	4	7	26	38	27	9	1	0	0	0
	5	0	0	1	10	29	39	27	9	1	0	0
	6	0	0	0	1	10	29	39	27	9	1	0
	7	0	0	0	0	1	10	29	39	27	9	1
	8	0	0	0	0	0	1	10	29	39	27	9
	9	0	0	0	0	0	0	1	10	29	39	27
	10	0	0	0	0	0	0	0	1	10	29	39

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	1	2	3	2	0	0	0	0	0	0
	1	1	3	3	7	5	1	0	0	0	0	0
	2	3	12	15	30	25	9	1	0	0	0	0
	3	4	18	26	55	58	34	10	1	0	0	0
	4	3	13	20	56	80	69	36	10	1	0	0
	5	1	4	7	33	69	86	70	36	10	1	0
	6	0	0	1	10	36	70	86	70	36	10	1
	7	0	0	0	1	10	36	70	86	70	36	10
	8	0	0	0	0	1	10	36	70	86	70	36
	9	0	0	0	0	0	1	10	36	70	86	70
	10	0	0	0	0	0	0	1	10	36	70	86

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	2	3	2	0	0	0	0	0
	1	0	0	1	5	8	5	1	0	0	0	0
	2	1	4	7	26	38	27	9	1	0	0	0
	3	3	13	20	56	80	69	36	10	1	0	0
	4	4	18	28	70	104	109	80	38	10	1	0
	5	3	13	20	59	99	121	115	81	38	10	1
	6	1	4	7	33	72	105	122	115	81	38	10
	7	0	0	1	10	36	73	105	122	115	81	38
	8	0	0	0	1	10	36	73	105	122	115	81
	9	0	0	0	0	1	10	36	73	105	122	115
	10	0	0	0	0	0	1	10	36	73	105	122

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	2	3	2	0	0	0	0
	1	0	0	0	1	5	8	5	1	0	0	0
	2	0	0	1	10	29	39	27	9	1	0	0
	3	1	4	7	33	69	86	70	36	10	1	0
	4	3	13	20	59	99	121	115	81	38	10	1
	5	4	18	28	70	108	132	138	121	82	38	10
	6	3	13	20	59	99	125	138	139	121	82	38
	7	1	4	7	33	72	105	126	138	139	121	82
	8	0	0	1	10	36	73	105	126	138	139	121
	9	0	0	0	1	10	36	73	105	126	138	139
	10	0	0	0	0	1	10	36	73	105	126	138

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	2	3	2	0	0	0
	1	0	0	0	0	1	5	8	5	1	0	0
	2	0	0	0	1	10	29	39	27	9	1	0
	3	0	0	1	10	36	70	86	70	36	10	1
	4	1	4	7	33	72	105	122	115	81	38	10
	5	3	13	20	59	99	125	138	139	121	82	38
	6	4	18	28	70	108	132	142	144	140	121	82
	7	3	13	20	59	99	125	138	143	144	140	121
	8	1	4	7	33	72	105	126	138	143	144	140
	9	0	0	1	10	36	73	105	126	138	143	144
	10	0	0	0	1	10	36	73	105	126	138	143

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	2	3	2	0	0
	1	0	0	0	0	0	1	5	8	5	1	0
	2	0	0	0	0	1	10	29	39	27	9	1
	3	0	0	0	1	10	36	70	86	70	36	10
	4	0	0	1	10	36	73	105	122	115	81	38
	5	1	4	7	33	72	105	126	138	139	121	82
	6	3	13	20	59	99	125	138	143	144	140	121
	7	4	18	28	70	108	132	142	144	144	144	140
	8	3	13	20	59	99	125	138	143	144	144	144
	9	1	4	7	33	72	105	126	138	143	144	144
	10	0	0	1	10	36	73	105	126	138	143	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	2	3	2	0
	1	0	0	0	0	0	0	1	5	8	5	1
	2	0	0	0	0	0	1	10	29	39	27	9
	3	0	0	0	0	1	10	36	70	86	70	36
	4	0	0	0	1	10	36	73	105	122	115	81
	5	0	0	1	10	36	73	105	126	138	139	121
	6	1	4	7	33	72	105	126	138	143	144	140
	7	3	13	20	59	99	125	138	143	144	144	144
	8	4	18	28	70	108	132	142	144	144	144	144
	9	3	13	20	59	99	125	138	143	144	144	144
	10	1	4	7	33	72	105	126	138	143	144	144

		c
		0	1	2	3	4	5	6	7	8	9	10
b	0	0	0	0	0	0	0	0	0	2	3	2
	1	0	0	0	0	0	0	0	1	5	8	5
	2	0	0	0	0	0	0	1	10	29	39	27
	3	0	0	0	0	0	1	10	36	70	86	70
	4	0	0	0	0	1	10	36	73	105	122	115
	5	0	0	0	1	10	36	73	105	126	138	139
	6	0	0	1	10	36	73	105	126	138	143	144
	7	1	4	7	33	72	105	126	138	143	144	144
	8	3	13	20	59	99	125	138	143	144	144	144
	9	4	18	28	70	108	132	142	144	144	144	144
	10	3	13	20	59	99	125	138	143	144	144	144