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sympy__sympy-12481

Based on my knowledge of the sympy codebase and the issue description, I can create the patch. The issue is in the sympy/combinatorics/permutations.py file where the Permutation class constructor (__new__ method) validates cycles and…

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Content

Based on my knowledge of the sympy codebase and the issue description, I can create the patch. The issue is in the `sympy/combinatorics/permutations.py` file where the `Permutation` class constructor (`__new__` method) validates cycles and raises a `ValueError` when cycles are non-disjoint.

The fix needs to:
1. Remove the check that raises `ValueError` for non-disjoint cycles
2. Instead, apply the cycles in left-to-right order to compute the resulting permutation

Here is the patch:

--- a/sympy/combinatorics/permutations.py
+++ b/sympy/combinatorics/permutations.py
@@ -895,12 +895,8 @@ class

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