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### Google ''permutation, Sage'' and you will get

### Permutations - SageMath Documentation
### http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/permutation.html
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### Permutations in Sage

print Permutation([2, 1, 4, 5, 3])
print Permutation( [ (1,2), (3,4,5)] )
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### Use ? to find the possible ways of input
### Permutation is a function convert the input to a Permutation object

Permutation?
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### An class in Python defines a data type
### Elements in a class is called an object.

type(1)
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### Permutation is a class
### each element created by Permutation(l) is a Permutation object

p = Permutation( [ (1,2), (3,4,5)] )
type(p)
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### Each object is associated with lots of attributes
### list them by dir

dir(p)
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### Attributes are set up when the class was defined
### but you can set attribute by setattr

setattr(Permutation,'self_introduction','Hello, I am a permutation on 1,...,n')
p.self_introduction
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### An attribute that is a function is called a method

p.size
p.size()
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### A method always take self as its first argument

def send(self,i):
    return self[i-1];

setattr(Permutation,'send',send)

print 'p =',p
for i in range(1,p.size()+1):
    print i,p.send(i)
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### Find associated attributes and functions by tab
### move the pointer after . and press tab

p = Permutation( [ (1,2), (3,4,5)] )
p.
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### the computer needs to know what it is first
### in order to tell you the associated attributes

reset() ### clean all variables
ppp = Permutation([3,1,2])
ppp.
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### Use ? to read to documentation
### Use ?? to read the source code

ppp.size?
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### Exercise 1:
###     Read the documentation of "length" and "inversions" and find their relations

ppp.inversions?
ppp.length?
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### Exercise 2:
###     Find a function that return the inverse of a permutation.

ppp.inv...
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### Exercise 3:
###     Explore other functions:  to_inversion_vector, action, fixed_points, sign, ...

ppp.action...
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%md

## Project 1

Define a function `perm_to_inv(p)` that send a permutation `p` to its inversion table.\
`Permutation([a1,...,an])` will be sent to `[b1,...,bn]`\
where `bk` is the number of `a1,...,a{i-1}` that is greater than `ai=k`.

Again, do not use the function `to_inversion_vector`.
︡1491331a-ff0e-4231-ae95-9c18acf6be1f︡{"done":true,"md":"\n## Project 1\n\nDefine a function `perm_to_inv(p)` that send a permutation `p` to its inversion table.\\\n`Permutation([a1,...,an])` will be sent to `[b1,...,bn]`\\\nwhere `bk` is the number of `a1,...,a{i-1}` that is greater than `ai=k`.\n\nAgain, do not use the function `to_inversion_vector`."}
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def perm_to_inv(p):

    ### Your answers:
    inv_table = [];
    for k in range(1,p.size()+1):
        ind = p.index(k);
        inv_table.append(len([i for i in range(ind) if p[i] > k]));
    return inv_table;

### test
for per in Permutations(4):
    print "per =", per
    print "Your answer:", perm_to_inv(per);
    print "True answer:", per.to_inversion_vector();
    print "Correct?", perm_to_inv(per) == per.to_inversion_vector();
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%md

## Project 2

Define a function `inv_to_perm(t)` that send an inversion table `t` to its permutation.
︡0eb7d90f-1194-46f1-84f4-f874b498e416︡{"done":true,"md":"\n## Project 2\n\nDefine a function `inv_to_perm(t)` that send an inversion table `t` to its permutation."}
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def inv_to_perm(t):

    ### You answers:
    nn = len(t);
    perm = [nn+1] * nn;
    for k in range(1,nn+1):
        left = t[k-1];
        counter = 0;
        for i in range(nn):
            if counter == left and perm[i] == nn+1:
                perm[i] = k;
                break;
            if perm[i] > k:
                counter += 1;
    return Permutation(perm);

### test
### your functions should have inv_to_perm(perm_to_inv(p)) == p
for per in Permutations(4):
    print "per =", per
    #print perm_to_inv(per)
    print "Correct?", inv_to_perm(perm_to_inv(per)) == per;

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