Quickstart ========== Symmetria offers three classes for representing permutations: ``Permutation``, ``Cycle``, and ``CycleDecomposition``. Each of these classes has its own set of advantages and disadvantages depending on the specific goal you want to achieve. Below is a brief example showcasing the capabilities of every of the aforementioned classes. Let's start by defining a permutation and exploring how we can represent it in various formats. .. tab-set:: .. tab-item:: Permutation :sync: permutation .. code-block:: python from symmetria import Permutation permutation = Permutation(1, 3, 4, 5, 2, 6) permutation # Permutation(1, 3, 4, 5, 2, 6) str(permutation) # (1, 3, 4, 5, 2, 6) permutation.cycle_notation() # (1)(2 3 4 5)(6) permutation.one_line_notation() # 134526 .. tab-item:: Cycle :sync: cycle .. code-block:: python from symmetria import Cycle cycle = Cycle(1, 3, 4, 5, 2, 6) cycle # Cycle(1, 3, 4, 5, 2, 6) cycle.cycle_notation() # (1 3 4 5 2 6) str(cycle) # (1 3 4 5 2 6) int(cycle) # 134526 .. tab-item:: CycleDecomposition :sync: cycledecomposition .. code-block:: python from symmetria import Cycle, CycleDecomposition cd = CycleDecomposition(Cycle(1, 2), Cycle(3, 4)) cd # CycleDecomposition(Cycle(1, 2), Cycle(3, 4)) cd.cycle_notation() # (1 2)(3 4) str(cd) # (1 2)(3 4) cd[0], cd[1] # Cycle(1, 2), Cycle(3, 4) Permutation objects are easy to manipulate. They implement nearly every standard functionality of basic Python objects. As a rule of thumb, if something seems intuitively possible, you can probably do it. .. tab-set:: .. tab-item:: Permutation :sync: permutation .. code-block:: python from symmetria import Permutation idx = Permutation(1, 2, 3) permutation = Permutation(1, 3, 2) if permutation: print(f"The permutation {permutation} is not the identity.") if idx == Permutation(1, 2, 3): print(f"The permutation {idx} is the identity permutation.") if permutation != idx: print(f"The permutations {permutation} and {idx} are different.") # The permutation (1, 3, 2) is not the identity. # The permutation (1, 2, 3) is the identity permutation. # The permutations (1, 3, 2) and (1, 2, 3) are different. .. tab-item:: Cycle :sync: cycle .. code-block:: python from symmetria import Cycle idx = Cycle(1) cycle = Cycle(1, 3, 2) if cycle: print(f"The cycle {cycle} is not the identity.") if idx == Cycle(1): print(f"The cycle {idx} is the identity cycle.") if cycle != idx: print(f"The cycles {cycle} and {idx} are different.") # The cycle (1 3 2) is not the identity. # The cycle (1 2 3) is the identity permutation. # The cycles (1 3 2) and (1 2 3) are different. .. tab-item:: CycleDecomposition :sync: cycledecomposition .. code-block:: python from symmetria import Cycle, CycleDecomposition idx = CycleDecomposition(Cycle(1), Cycle(2), Cycle(3)) cd = CycleDecomposition(Cycle(1, 2), Cycle(3)) if cd: print(f"The cycle decomposition {cd} is not the identity.") if idx == CycleDecomposition(Cycle(1), Cycle(2), Cycle(3)): print(f"The cycle decomposition {cd} is the identity.") if cd != idx: print(f"The cycle decompositions {cd} and {idx} are different.") # The cycle decomposition (1 2)(3) is not the identity. # The cycle decomposition (1)(2)(3) is the identity. # The cycle decompositions (1 2)(3) and (1)(2)(3) are different. Basic arithmetic operations are implemented. .. tab-set:: .. tab-item:: Permutation :sync: permutation .. code-block:: python from symmetria import Permutation permutation = Permutation(3, 1, 4, 2) multiplication = permutation * permutation # Permutation(4, 3, 2, 1) power = permutation ** 2 # Permutation(4, 3, 2, 1) inverse = permutation ** -1 # Permutation(2, 4, 1, 3) identity = permutation * inverse # Permutation(1, 2, 3, 4) .. tab-item:: CycleDecomposition :sync: cycledecomposition .. code-block:: python from symmetria import Cycle, CycleDecomposition cd = CycleDecomposition(Cycle(1, 4, 2), Cycle(3)) multiplication = cd * cd # CycleDecomposition(Cycle(1, 2, 4), Cycle(3)) power = cd ** 2 # CycleDecomposition(Cycle(1, 2, 4), Cycle(3)) inverse = cd ** -1 # CycleDecomposition(Cycle(2, 4, 1), Cycle(3)) identity = cd * inverse # CycleDecomposition(Cycle(1), Cycle(2), Cycle(3), Cycle(4)) Actions on different objects are also implemented. .. tab-set:: .. tab-item:: Permutation :sync: permutation .. code-block:: python from symmetria import Permutation permutation = Permutation(3, 2, 4, 1) permutation(3) # 4 permutation("abcd") # 'dbac' permutation(["I", "love", "Python", "!"]) # ['!', 'love', 'I', 'Python'] .. tab-item:: Cycle :sync: cycle .. code-block:: python from symmetria import Cycle cycle = Cycle(1, 3, 4, 2) cycle(3) # 4 cycle("abcd") # 'bdac' cycle(["I", "love", "Python", "!"]) # ['love', '!', 'I', 'Python'] .. tab-item:: CycleDecomposition :sync: cycledecomposition .. code-block:: python from symmetria import Cycle, CycleDecomposition cd = CycleDecomposition(Cycle(1, 3), Cycle(2, 4)) cd(3) # 1 cd("abcd") # 'cdab' cd(["I", "love", "Python", "!"]) # ['Python', '!', 'I', 'love'] Moreover, many methods are already implemented. If what you are looking for is not available, let us know as soon as possible. .. tab-set:: .. tab-item:: Permutation :sync: permutation .. code-block:: python from symmetria import Permutation permutation = Permutation(3, 2, 4, 1) permutation.order() # 3 permutation.support() # {1, 3, 4} permutation.sgn() # 1 permutation.cycle_decomposition() # CycleDecomposition(Cycle(1, 3, 4), Cycle(2)) permutation.cycle_type() # (1, 3) permutation.is_derangement() # False permutation.is_regular() # False permutation.inversions() # [(1, 2), (1, 4), (2, 4), (3, 4)] permutation.ascents() # [2] permutation.descents() # [1, 3] .. tab-item:: Cycle :sync: cycle .. code-block:: python from symmetria import Cycle cycle = Cycle(2, 3, 5, 7, 6) cycle.order() # 5 cycle.support() # {2, 3, 5, 7, 6} cycle.sgn() # 1 cycle.cycle_decomposition() # CycleDecomposition(Cycle(1), Cycle(2, 3, 5, 7, 6), Cycle(4)) cycle.is_derangement() # True cycle.inversions() # [(4, 5)] .. tab-item:: CycleDecomposition :sync: cycledecomposition .. code-block:: python from symmetria import Cycle, CycleDecomposition cd = CycleDecomposition(Cycle(1, 3), Cycle(2, 4)) cd.order() # 2 cd.support() # {1, 2, 3, 4} cd.sgn() # 1 cd.is_even() # True cd.cycle_type() # (2, 2) cd.is_derangement() # True cd.is_regular() # True cd.inversions() # [(1, 3), (1, 4), (2, 3), (2, 4)] cd.ascents() # [1, 3] cd.descents() # [2] Click [here](https://symmetria.readthedocs.io/en/latest/pages/API_reference/elements/index.html) for an overview of all the functionalities implemented in `symmetria`.