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Here is a sample session

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

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