permuta 1.5.0

A comprehensive high performance permutation library.

Permuta is a Python library for working with perms (short for permutations), patterns, and mesh patterns.

If you need support, you can join us in our Discord support server.

Installing

To install Permuta on your system, run:

pip install permuta

It is also possible to install Permuta in development mode to work on the source code, in which case you run the following after cloning the repository:

./setup.py develop

To run the unit tests:

pip install -r test_requirements.txt
./setup.py test

Using Permuta

Once you’ve installed Permuta, it can be imported by a Python script or an interactive Python session, just like any other Python library:

>>> from permuta import *

Importing * from it supplies you with the ‘Perm’ and ‘PermSet’ classes along with the ‘AvoidanceClass’ class (with alias ‘Av’) for generating perms avoiding a set of patterns. It also gives you the ‘MeshPatt’ class and some other submodules which we will not discuss in this readme.

Creating a single perm

Permutations are zero-based in Permuta and can be created using any iterable.

>>> Perm()  # Empty perm
Perm(())
>>> Perm([])  # Another empty perm
Perm(())
>>> Perm((0, 1, 2, 3)) # The zero-based version of 1234
Perm((0, 1, 2, 3))
>>> Perm((2, 1, 3)) # Warning: it will initialise with any iterable
Perm((2, 1, 3))
>>> Perm((2, 1, 3), check=True) # If you are unsure, you can check
Traceback (most recent call last):
    ...
ValueError: Element out of range: 3
>>> Perm((4, 2, 3, 0, 0), check=True)
Traceback (most recent call last):
    ...
ValueError: Duplicate element: 0
>>> Perm("123", check=True)
Traceback (most recent call last):
    ...
TypeError: ''1'' object is not an integer

Permutations can also be created using some specific class methods.

>>> Perm.from_string("201")  # strings
Perm((2, 0, 1))
>>> Perm.one_based((1, 3, 2, 4)) # one-based iterable of integers
Perm((0, 2, 1, 3))
>>> Perm.to_standard("a2gsv3") # standardising any iterable using '<'
Perm((2, 0, 3, 4, 5, 1))
>>> Perm.from_integer(210) # an integer between 0 and 9876543210
Perm((2, 1, 0))
>>> Perm.from_integer(321) # any integer given is standardised
Perm((2, 1, 0))
>>> Perm.from_integer(201)
Perm((2, 0, 1))

Printing perms gives zero-based strings.

>>> print(Perm(()))
ε
>>> print(Perm((2, 1, 0)))
210
>>> print(Perm((6, 2, 10, 9, 3, 8, 0, 1, 5, 11, 4, 7)))
(6)(2)(10)(9)(3)(8)(0)(1)(5)(11)(4)(7)

The avoids, contains, and occurrence methods enable working with patterns:

>>> p = Perm((0,2,1,3))
>>> p.contains(Perm((2, 1, 0)))
False
>>> p.avoids(Perm((0, 1)))
False
>>> list(p.occurrences_of(Perm((1, 0))))
[(1, 2)]
>>> list(Perm((0, 1)).occurrences_in(p))
[(0, 1), (0, 2), (0, 3), (1, 3), (2, 3)]

The basic symmetries are implemented:

>>> [p.reverse(), p.complement(), p.inverse()]
[Perm((3, 1, 2, 0)), Perm((3, 1, 2, 0)), Perm((0, 2, 1, 3))]

To take direct sums and skew sums we use + and -:

>>> q = Perm((0, 1, 2, 3, 4))
>>> p + q
Perm((0, 2, 1, 3, 4, 5, 6, 7, 8))
>>> p - q
Perm((5, 7, 6, 8, 0, 1, 2, 3, 4))

There are numerous practical methods available:

>>> list(p.fixed_points())
[0, 3]
>>> list(p.ascents())
[0, 2]
>>> list(p.descents())
[1]
>>> list(p.inversions())
[(1, 2)]
>>> p.major_index()
2

Creating a perm class

You might want the set of all perms:

>>> all_perms = PermSet()
>>> print(all_perms)
<The set of all perms>

Perm classes can be specified with a basis:

>>> basis = [Perm((1, 0, 2)), Perm((1, 2, 0))]
>>> basis
[Perm((1, 0, 2)), Perm((1, 2, 0))]
>>> perm_class = Av(basis)
>>> perm_class
Av((Perm((1, 0, 2)), Perm((1, 2, 0))))

When a basis consists of a single element you can pass it directly to Av:

>>> q = Perm((1,0))
>>> len(Av(q).of_length(100))
1

You can ask whether a perm belongs to the perm class:

>>> Perm((3, 2, 1, 0)) in perm_class
True
>>> Perm((0, 2, 1, 3)) in perm_class
False

You can get the n-th perm of the class or iterate:

>>> sorted([perm_class[n] for n in range(8)])
[Perm(()), Perm((0,)), Perm((0, 1)), Perm((1, 0)), Perm((0, 1, 2)), Perm((0, 2, 1)), Perm((2, 0, 1)), Perm((2, 1, 0))]
>>> perm_class_iter = iter(perm_class)
>>> sorted([next(perm_class_iter) for _ in range(8)])
[Perm(()), Perm((0,)), Perm((0, 1)), Perm((1, 0)), Perm((0, 1, 2)), Perm((0, 2, 1)), Perm((2, 0, 1)), Perm((2, 1, 0))]

(BEWARE: Lexicographic order is not guaranteed at the moment!)

The subset of a perm class where the perms are a specific length

You can define a subset of perms of a specific length in the perm class:

>>> perm_class_14 = perm_class.of_length(14)
>>> perm_class_14
Av((Perm((1, 0, 2)), Perm((1, 2, 0)))).of_length(14)

You can ask for the size of the subset because it is guaranteed to be finite:

>>> len(perm_class_14)
8192

The iterating and containment functionality is the same as with perm_class, but indexing has yet to be implemented:

>>> Perm((2, 1, 0)) in perm_class_14
False
>>> Perm((0, 13, 1, 12, 2, 3, 4, 11, 5, 10, 6, 7, 8, 9)) in perm_class_14
True
>>> Perm(range(10)) - Perm(range(4)) in perm_class_14
False
>>> next(iter(perm_class_14)) in perm_class_14
True

The BiSC algorithm

The BiSC algorithm can tell you what mesh patterns are avoided by a set of permutations. Although the output of the algorithm is only guaranteed to describe the finite inputted set of permutations, the user usually hopes that the patterns found by the algorithm describe an infinite set of permutatations. To use the algorithm we first need to import it.

>>> from permuta.bisc import *

A classic example of a set of permutations described by pattern avoidance are the permutations sortable in one pass through a stack. We start by loading a function stack_sortable which returns True for permutations that satisfy this property. The user now has two choices: Run auto_bisc(stack_sortable) and let the algorithm run without any more user input. It will try to use sensible values, starting by learning small patterns from small permutations, and only considering longer patterns when that fails. If the user wants to have more control over what happens that is also possible and we now walk through that: We input the property into bisc and ask it to search for patterns of length 3.

>>> from permuta.bisc.permsets.perm_properties import stack_sortable
>>> bisc(stack_sortable, 3)
I will use permutations up to length 7
{3: {Perm((1, 2, 0)): [set()]}}

When this command is run without specifying what length of permutations you want to consider, bisc will create permutations up to length 7 that satisfy the property of being stack-sortable. The output means: There is a single length 3 pattern found, and its underlying classical pattern is the permutation Perm((1, 2, 0)). Ignore the [set()] in the output for now. We can use show_me to get a better visualization of the patterns found. In this call to the algorithm we also specify that only permutations up to length 5 should be considered.

>>> SG = bisc(stack_sortable, 3, 5)
>>> show_me(SG)
There are 1 underlying classical patterns of length 3
There are 1 different shadings on 120
The number of sets to monitor at the start of the clean-up phase is 1
<BLANKLINE>
Now displaying the patterns
<BLANKLINE>
 | | |
-+-●-+-
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-●-+-+-
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-+-+-●-
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<BLANKLINE>

We should ignore the The number of sets to monitor at the start of the clean-up phase is 1 message for now.

We do not really need this algorithm for sets of permutations described by the avoidance of classical patterns. Its main purpose is to describe sets with mesh patterns, such as the West-2-stack-sortable permutations

>>> from permuta.bisc.permsets.perm_properties import West_2_stack_sortable
>>> SG = bisc(West_2_stack_sortable, 5, 7)
>>> show_me(SG)
There are 2 underlying classical patterns of length 4
There are 1 different shadings on 1230
There are 1 different shadings on 2130
The number of sets to monitor at the start of the clean-up phase is 1
There are 1 underlying classical patterns of length 5
There are 1 different shadings on 42130
<BLANKLINE>
Now displaying the patterns
<BLANKLINE>
 | | | |
-+-+-●-+-
 | | | |
-+-●-+-+-
 | | | |
-●-+-+-+-
 | | | |
-+-+-+-●-
 | | | |
<BLANKLINE>
 |▒| | |
-+-+-●-+-
 | | | |
-●-+-+-+-
 | | | |
-+-●-+-+-
 | | | |
-+-+-+-●-
 | | | |
<BLANKLINE>
 |▒| | | |
-●-+-+-+-+-
 | |▒| | |
-+-+-+-●-+-
 | | | | |
-+-●-+-+-+-
 | | | | |
-+-+-●-+-+-
 | | | | |
-+-+-+-+-●-
 | | | | |
<BLANKLINE>

This is good news and bad news. Good because we quickly got a description of the set we were looking at, that would have taken a long time to find by hand. The bad news is that there is actually some redundancy in the output. To understand better what is going on we will start by putting the permutations under investigation in a dictionary, which keeps them separated by length.

>>> A, B = create_bisc_input(7, West_2_stack_sortable)

This creates two dictionaries with keys 1, 2, …, 7 such that A[i] points to the list of permutations of length i that are West-2-stack-sortable, and B[i] points to the complement. We can pass the A dictionary directly into BiSC since only the permutations satisfying the property are used to find the patterns. We can use the second dictionary to check whether every permutation in the complement contains at least one of the patterns we found.

>>> SG = bisc(A, 5, 7)
>>> patterns_suffice_for_bad(SG, 7, B)
Starting sanity check with bad perms
Now checking permutations of length 0
Now checking permutations of length 1
Now checking permutations of length 2
Now checking permutations of length 3
Now checking permutations of length 4
Now checking permutations of length 5
Now checking permutations of length 6
Now checking permutations of length 7
Sanity check passes for the bad perms
(True, [])

In this case it is true that every permutation in B, up to length 7, contains at least one of the patterns found. Had that not been the case a list of permutations would have been outputted (instead of just the empty list).

Now, we claim that there is actually redundancy in the patterns we found, and the length 4 mesh patterns should be enough to describe the set. This can occur and it can be tricky to theoretically prove that one mesh pattern is implied by another pattern (or a set of others, as is the case here). We use the dictionary B again and run

>>> bases, dict_numbs_to_patts = run_clean_up(SG, B)
<BLANKLINE>
The bases found have lengths
[2]

There is one basis of mesh patterns found, with 2 patterns

>>> show_me_basis(bases[0], dict_numbs_to_patts)
<BLANKLINE>
Displaying the patterns in the basis
<BLANKLINE>
 | | | |
-+-+-●-+-
 | | | |
-+-●-+-+-
 | | | |
-●-+-+-+-
 | | | |
-+-+-+-●-
 | | | |
<BLANKLINE>
 |▒| | |
-+-+-●-+-
 | | | |
-●-+-+-+-
 | | | |
-+-●-+-+-
 | | | |
-+-+-+-●-
 | | | |
<BLANKLINE>

This is the output we were expecting. There are several other properties of permutations that can be imported from permuta.bisc.permsets.perm_properties, such as smooth, forest-like, Baxter, Simsun, quick_sortable, etc.

Both bisc and auto_bisc can accept input in the form of a property, or a list of permutations (satisfying some property).

License

BSD-3: see the LICENSE file.