{{{id=35|

///
}}}

<p>The function Subsets(n, k) creates all subsets of range(1, n+1) of length k</p>
<p>However, to list them you should use Subsets(n,k).list() or list(Subsets(n,k))</p>

{{{id=2|
Subsets(4,2).list()
///
[{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}]
}}}

<p>The following function is one half of the answer to Exercise 1, in order to compete the problem you should write the function "subset_to_monomial" which takes a subset $SS = \{ a_1 &lt; a_2 &lt; ... &lt; a_{n-1} \}$ and converts it into the monomial</p>
<p>$$x_1^{a_1 - 1} x_2^{a_2-a_1-1} \cdots x_{n-1}^{a_{n-1}-a_{n-2}-1} x_n^{n+d-1-a_{n-1}}$$</p>

{{{id=1|
def list_all_monomials(n, d):
    return [subset_to_monomial(SS, d) for SS in Subsets(n+d-1,d)]
///
}}}

{{{id=3|
list_all_monomials(3,2)
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_7.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bGlzdF9hbGxfbW9ub21pYWxzKDMsMik="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/private/var/folders/_3/qq_ptsxd0bd645cgmgt_ssrm0000gn/T/tmpxKSUg9/___code___.py", line 3, in <module>
    exec compile(u'list_all_monomials(_sage_const_3 ,_sage_const_2 )' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/private/var/folders/_3/qq_ptsxd0bd645cgmgt_ssrm0000gn/T/tmp14LB_Z/___code___.py", line 4, in list_all_monomials
    return [subset_to_monomial(SS, n) for SS in Subsets(n+d-_sage_const_1 ,d)]
NameError: global name 'subset_to_monomial' is not defined
}}}

<p>The following command make a list of 10 variables x0 through x9</p>

{{{id=5|
vx = [var("x"+str(i)) for i in range(10)]
///
}}}

{{{id=6|
vx[3]
///
x3
}}}

<p>The following is an expample of a product over a set</p>
<p>syntax:</p>
<p>mul( expr for var in aset )</p>

{{{id=7|
mul( vx[i]^i for i in range(10) )
///
x1*x2^2*x3^3*x4^4*x5^5*x6^6*x7^7*x8^8*x9^9
}}}

{{{id=42|

///
}}}

<p>You can also create a monomial with words instead of subsets, but since they are in bijection, they should be equivalent.</p>

{{{id=11|
def num_0_eq_2( w ):
    return w.count('0')==2 # true if the number of 0's is equal to 2
///
}}}

{{{id=8|
Words(alphabet=['0','1'], length=4).filter( num_0_eq_2 ).list()
///
[word: 0011, word: 0101, word: 0110, word: 1001, word: 1010, word: 1100]
}}}

{{{id=25|
for w in Words(alphabet=[0,1], length=4):
    if count_zeros(w)==4:
        print w
///
0000
}}}

{{{id=10|
w = Word(['0','1','0','1'])
///
}}}

{{{id=26|
LL = [[1,2],[3,4],[5,6,7]]
///
}}}

{{{id=27|
LL[0]
///
[1, 2]
}}}

{{{id=28|
LL[0][0]
///
1
}}}

{{{id=29|
[mul(var("x"+str(v)) for v in L if v>3) for L in LL] 
///
[1, x4, x5*x6*x7]
}}}

{{{id=30|
SS = [3,5,7]
///
}}}

{{{id=34|
deg = 7
///
}}}

{{{id=33|
SSp = [0]+SS+[deg+len(SS)+1]
///
}}}

{{{id=31|
mul(var("x"+str(j))^(SSp[j+1]-SSp[j]-1) for j in range(len(SSp)-1))
///
x0^2*x1*x2*x3^3
}}}

{{{id=24|
def subset_to_monomial( SS, n ):
    
///
}}}

{{{id=32|
[0]+[1,2,3]+[8]
///
[0, 1, 2, 3, 8]
}}}

{{{id=15|
w.count('0')
///
2
}}}

{{{id=17|
w = Word([1,0,0,0,1,0,0,0,0,1])
///
}}}

{{{id=18|
for i in range(len(w)):
    print w[i]
///
1
0
0
0
1
0
0
0
0
1
}}}

{{{id=16|
out = 0
for i in range(len(w)):
    if w[i]==0:
        out=out+1
out
///
7
}}}

{{{id=19|
def count_zeros( w ):
    out = 0
    for i in range(len(w)):
        if w[i]==0:
            out=out+1
    return out
///
}}}

{{{id=20|
count_zeros(w)
///
7
}}}

{{{id=21|
w.count(0)
///
7
}}}

{{{id=22|

///
}}}