1 | initial version |

The image of 1 is 4, the image of 6 is 2:

```
sage: P(1)
4
sage: P(6)
2
sage: P(6) < P(1)
True
```

Hence, (1,6) should be part of the inversions.

If you want to decompose a permutations into transpositions, note that there are many ways, none of them is canonical. However, you can decompose your premutation into disjoint cycles and then each cycle can be decomposed into transpositions.

```
sage: P.cycle_tuples()
[(1, 4), (2, 8, 7, 6), (3,), (5, 9)]
```

2 | No.2 Revision |

The image of 1 is 4, the image of 6 is 2:

```
sage: P(1)
4
sage: P(6)
2
sage: P(6) < P(1)
True
```

Hence, ~~ (1,6) ~~`(1,6)`

should be part of the inversions.

If you want to decompose a permutations into transpositions, note that there are many ways, none of them is canonical. However, you can decompose your premutation into disjoint cycles and then each cycle can be decomposed into transpositions.

```
sage: P.cycle_tuples()
[(1, 4), (2, 8, 7, 6), (3,), (5, 9)]
```

From such a decomposition, you can easily get a decomposition of the permutation into tuples (because `(a1,a2,a3,...,an) = (a1,a2)(a1,a3)...(a1,an)`

) :

```
sage: L = []
....: for c in C:
....: if len(C) >= 2:
....: a = c[0]
....: for b in c[1:]:
....: L.append((a,b))
sage: L
[(1, 4), (2, 8), (2, 7), (2, 6), (5, 9)]
```

You can check:

```
sage: prod(Permutation(t) for t in L)
[4, 8, 3, 1, 9, 2, 6, 7, 5]
sage: prod(Permutation(t) for t in L) == P
True
```

3 | No.3 Revision |

~~The ~~First, note that the image of 1 is 4, the image of 6 is 2:

```
sage: P(1)
4
sage: P(6)
2
sage: P(6) < P(1)
True
```

Hence, `(1,6)`

should be part of the ~~inversions.~~

If inversions (an inversion is a pair `(i,j)`

such that `i<j`

and `P(i)>P(j)`

).

Now, if you want to decompose a permutations into transpositions, note that there are many ways, none of them is canonical. However, you can decompose your premutation into disjoint cycles and then each cycle can be decomposed into transpositions.

```
sage: P.cycle_tuples()
[(1, 4), (2, 8, 7, 6), (3,), (5, 9)]
```

From such a decomposition, you can easily get a decomposition of the permutation into tuples (because `(a1,a2,a3,...,an) = (a1,a2)(a1,a3)...(a1,an)`

) :

```
sage: L = []
....: for c in C:
....: if len(C) >= 2:
....: a = c[0]
....: for b in c[1:]:
....: L.append((a,b))
sage: L
[(1, 4), (2, 8), (2, 7), (2, 6), (5, 9)]
```

You can check:

```
sage: prod(Permutation(t) for t in L)
[4, 8, 3, 1, 9, 2, 6, 7, 5]
sage: prod(Permutation(t) for t in L) == P
True
```

4 | No.4 Revision |

First, note that the image of 1 is 4, the image of 6 is 2:

```
sage: P(1)
4
sage: P(6)
2
sage: P(6) < P(1)
True
```

Hence, `(1,6)`

should be part of the inversions (an inversion is a pair `(i,j)`

such that `i<j`

and `P(i)>P(j)`

).

Now, if you want to decompose a permutations into transpositions, note that there are many ways, none of them is canonical. However, you can decompose your ~~premutation ~~permutation into disjoint cycles and then each cycle can be decomposed into transpositions.

```
sage: P.cycle_tuples()
[(1, 4), (2, 8, 7, 6), (3,), (5, 9)]
```

From such a decomposition, you can easily get a decomposition of the permutation into tuples (because `(a1,a2,a3,...,an) = (a1,a2)(a1,a3)...(a1,an)`

) :

```
sage: L = []
....: for c in C:
....: if len(C) >= 2:
....: a = c[0]
....: for b in c[1:]:
....: L.append((a,b))
sage: L
[(1, 4), (2, 8), (2, 7), (2, 6), (5, 9)]
```

You can check:

```
sage: prod(Permutation(t) for t in L)
[4, 8, 3, 1, 9, 2, 6, 7, 5]
sage: prod(Permutation(t) for t in L) == P
True
```

5 | No.5 Revision |

First, note that the image of 1 is 4, the image of 6 is 2:

```
sage: P(1)
4
sage: P(6)
2
sage: P(6) < P(1)
True
```

Hence, `(1,6)`

should be part of the inversions (an inversion is a pair `(i,j)`

such that `i<j`

and `P(i)>P(j)`

).

Now, if you want to decompose a permutations into transpositions, note that there are many ways, none of them is canonical. However, you can decompose your permutation into disjoint cycles and then each cycle can be decomposed into transpositions.

`sage: `~~P.cycle_tuples()
~~C = P.cycle_tuples() ; C
[(1, 4), (2, 8, 7, 6), (3,), (5, 9)]

`(a1,a2,a3,...,an) = (a1,a2)(a1,a3)...(a1,an)`

) :

```
sage: L = []
....: for c in C:
....: if len(C) >= 2:
....: a = c[0]
....: for b in c[1:]:
....: L.append((a,b))
sage: L
[(1, 4), (2, 8), (2, 7), (2, 6), (5, 9)]
```

You can check:

```
sage: prod(Permutation(t) for t in L)
[4, 8, 3, 1, 9, 2, 6, 7, 5]
sage: prod(Permutation(t) for t in L) == P
True
```

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