Convert between different ways of expressing permutations

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I am not sure about your exact question, but you might start with:

sage: x = Permutation([2,1,4,3])
sage: y = Permutation([3,2,1,4])
sage: x.cycle_tuples()
[(1, 2), (3, 4)]
sage: y.cycle_tuples()
[(1, 3), (2,), (4,)]

Regarding Kazhdan-Lusztig polynomial for Weyl groups, you can have a look at :

• https://doc.sagemath.org/html/en/refe...
• https://doc.sagemath.org/html/en/them...

Here is a sample session

sage: S4 = SymmetricGroup(4)
sage: R.<q> = LaurentPolynomialRing(QQ)
sage: KL = KazhdanLusztigPolynomial(S4,q)
sage: x = S4([1,2,3,4])
sage: y = S4([2,3,4,1])
sage: KL.P(x, y)
1
sage: y = S4([4,2,3,1])
sage: KL.P(x, y)
1 + q