ai-content-maker/.venv/Lib/site-packages/sympy/combinatorics/partitions.py

from sympy.core import Basic, Dict, sympify, Tuple
from sympy.core.numbers import Integer
from sympy.core.sorting import default_sort_key
from sympy.core.sympify import _sympify
from sympy.functions.combinatorial.numbers import bell
from sympy.matrices import zeros
from sympy.sets.sets import FiniteSet, Union
from sympy.utilities.iterables import flatten, group
from sympy.utilities.misc import as_int


from collections import defaultdict


class Partition(FiniteSet):
    """
    This class represents an abstract partition.

    A partition is a set of disjoint sets whose union equals a given set.

    See Also
    ========

    sympy.utilities.iterables.partitions,
    sympy.utilities.iterables.multiset_partitions
    """

    _rank = None
    _partition = None

    def __new__(cls, *partition):
        """
        Generates a new partition object.

        This method also verifies if the arguments passed are
        valid and raises a ValueError if they are not.

        Examples
        ========

        Creating Partition from Python lists:

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3])
        >>> a
        Partition({3}, {1, 2})
        >>> a.partition
        [[1, 2], [3]]
        >>> len(a)
        2
        >>> a.members
        (1, 2, 3)

        Creating Partition from Python sets:

        >>> Partition({1, 2, 3}, {4, 5})
        Partition({4, 5}, {1, 2, 3})

        Creating Partition from SymPy finite sets:

        >>> from sympy import FiniteSet
        >>> a = FiniteSet(1, 2, 3)
        >>> b = FiniteSet(4, 5)
        >>> Partition(a, b)
        Partition({4, 5}, {1, 2, 3})
        """
        args = []
        dups = False
        for arg in partition:
            if isinstance(arg, list):
                as_set = set(arg)
                if len(as_set) < len(arg):
                    dups = True
                    break  # error below
                arg = as_set
            args.append(_sympify(arg))

        if not all(isinstance(part, FiniteSet) for part in args):
            raise ValueError(
                "Each argument to Partition should be " \
                "a list, set, or a FiniteSet")

        # sort so we have a canonical reference for RGS
        U = Union(*args)
        if dups or len(U) < sum(len(arg) for arg in args):
            raise ValueError("Partition contained duplicate elements.")

        obj = FiniteSet.__new__(cls, *args)
        obj.members = tuple(U)
        obj.size = len(U)
        return obj

    def sort_key(self, order=None):
        """Return a canonical key that can be used for sorting.

        Ordering is based on the size and sorted elements of the partition
        and ties are broken with the rank.

        Examples
        ========

        >>> from sympy import default_sort_key
        >>> from sympy.combinatorics import Partition
        >>> from sympy.abc import x
        >>> a = Partition([1, 2])
        >>> b = Partition([3, 4])
        >>> c = Partition([1, x])
        >>> d = Partition(list(range(4)))
        >>> l = [d, b, a + 1, a, c]
        >>> l.sort(key=default_sort_key); l
        [Partition({1, 2}), Partition({1}, {2}), Partition({1, x}), Partition({3, 4}), Partition({0, 1, 2, 3})]
        """
        if order is None:
            members = self.members
        else:
            members = tuple(sorted(self.members,
                             key=lambda w: default_sort_key(w, order)))
        return tuple(map(default_sort_key, (self.size, members, self.rank)))

    @property
    def partition(self):
        """Return partition as a sorted list of lists.

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> Partition([1], [2, 3]).partition
        [[1], [2, 3]]
        """
        if self._partition is None:
            self._partition = sorted([sorted(p, key=default_sort_key)
                                      for p in self.args])
        return self._partition

    def __add__(self, other):
        """
        Return permutation whose rank is ``other`` greater than current rank,
        (mod the maximum rank for the set).

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3])
        >>> a.rank
        1
        >>> (a + 1).rank
        2
        >>> (a + 100).rank
        1
        """
        other = as_int(other)
        offset = self.rank + other
        result = RGS_unrank((offset) %
                            RGS_enum(self.size),
                            self.size)
        return Partition.from_rgs(result, self.members)

    def __sub__(self, other):
        """
        Return permutation whose rank is ``other`` less than current rank,
        (mod the maximum rank for the set).

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3])
        >>> a.rank
        1
        >>> (a - 1).rank
        0
        >>> (a - 100).rank
        1
        """
        return self.__add__(-other)

    def __le__(self, other):
        """
        Checks if a partition is less than or equal to
        the other based on rank.

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3, 4, 5])
        >>> b = Partition([1], [2, 3], [4], [5])
        >>> a.rank, b.rank
        (9, 34)
        >>> a <= a
        True
        >>> a <= b
        True
        """
        return self.sort_key() <= sympify(other).sort_key()

    def __lt__(self, other):
        """
        Checks if a partition is less than the other.

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3, 4, 5])
        >>> b = Partition([1], [2, 3], [4], [5])
        >>> a.rank, b.rank
        (9, 34)
        >>> a < b
        True
        """
        return self.sort_key() < sympify(other).sort_key()

    @property
    def rank(self):
        """
        Gets the rank of a partition.

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3], [4, 5])
        >>> a.rank
        13
        """
        if self._rank is not None:
            return self._rank
        self._rank = RGS_rank(self.RGS)
        return self._rank

    @property
    def RGS(self):
        """
        Returns the "restricted growth string" of the partition.

        Explanation
        ===========

        The RGS is returned as a list of indices, L, where L[i] indicates
        the block in which element i appears. For example, in a partition
        of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is
        [1, 1, 0]: "a" is in block 1, "b" is in block 1 and "c" is in block 0.

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> a = Partition([1, 2], [3], [4, 5])
        >>> a.members
        (1, 2, 3, 4, 5)
        >>> a.RGS
        (0, 0, 1, 2, 2)
        >>> a + 1
        Partition({3}, {4}, {5}, {1, 2})
        >>> _.RGS
        (0, 0, 1, 2, 3)
        """
        rgs = {}
        partition = self.partition
        for i, part in enumerate(partition):
            for j in part:
                rgs[j] = i
        return tuple([rgs[i] for i in sorted(
            [i for p in partition for i in p], key=default_sort_key)])

    @classmethod
    def from_rgs(self, rgs, elements):
        """
        Creates a set partition from a restricted growth string.

        Explanation
        ===========

        The indices given in rgs are assumed to be the index
        of the element as given in elements *as provided* (the
        elements are not sorted by this routine). Block numbering
        starts from 0. If any block was not referenced in ``rgs``
        an error will be raised.

        Examples
        ========

        >>> from sympy.combinatorics import Partition
        >>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde'))
        Partition({c}, {a, d}, {b, e})
        >>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead'))
        Partition({e}, {a, c}, {b, d})
        >>> a = Partition([1, 4], [2], [3, 5])
        >>> Partition.from_rgs(a.RGS, a.members)
        Partition({2}, {1, 4}, {3, 5})
        """
        if len(rgs) != len(elements):
            raise ValueError('mismatch in rgs and element lengths')
        max_elem = max(rgs) + 1
        partition = [[] for i in range(max_elem)]
        j = 0
        for i in rgs:
            partition[i].append(elements[j])
            j += 1
        if not all(p for p in partition):
            raise ValueError('some blocks of the partition were empty.')
        return Partition(*partition)


class IntegerPartition(Basic):
    """
    This class represents an integer partition.

    Explanation
    ===========

    In number theory and combinatorics, a partition of a positive integer,
    ``n``, also called an integer partition, is a way of writing ``n`` as a
    list of positive integers that sum to n. Two partitions that differ only
    in the order of summands are considered to be the same partition; if order
    matters then the partitions are referred to as compositions. For example,
    4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1];
    the compositions [1, 2, 1] and [1, 1, 2] are the same as partition
    [2, 1, 1].

    See Also
    ========

    sympy.utilities.iterables.partitions,
    sympy.utilities.iterables.multiset_partitions

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Partition_%28number_theory%29
    """

    _dict = None
    _keys = None

    def __new__(cls, partition, integer=None):
        """
        Generates a new IntegerPartition object from a list or dictionary.

        Explanation
        ===========

        The partition can be given as a list of positive integers or a
        dictionary of (integer, multiplicity) items. If the partition is
        preceded by an integer an error will be raised if the partition
        does not sum to that given integer.

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> a = IntegerPartition([5, 4, 3, 1, 1])
        >>> a
        IntegerPartition(14, (5, 4, 3, 1, 1))
        >>> print(a)
        [5, 4, 3, 1, 1]
        >>> IntegerPartition({1:3, 2:1})
        IntegerPartition(5, (2, 1, 1, 1))

        If the value that the partition should sum to is given first, a check
        will be made to see n error will be raised if there is a discrepancy:

        >>> IntegerPartition(10, [5, 4, 3, 1])
        Traceback (most recent call last):
        ...
        ValueError: The partition is not valid

        """
        if integer is not None:
            integer, partition = partition, integer
        if isinstance(partition, (dict, Dict)):
            _ = []
            for k, v in sorted(partition.items(), reverse=True):
                if not v:
                    continue
                k, v = as_int(k), as_int(v)
                _.extend([k]*v)
            partition = tuple(_)
        else:
            partition = tuple(sorted(map(as_int, partition), reverse=True))
        sum_ok = False
        if integer is None:
            integer = sum(partition)
            sum_ok = True
        else:
            integer = as_int(integer)

        if not sum_ok and sum(partition) != integer:
            raise ValueError("Partition did not add to %s" % integer)
        if any(i < 1 for i in partition):
            raise ValueError("All integer summands must be greater than one")

        obj = Basic.__new__(cls, Integer(integer), Tuple(*partition))
        obj.partition = list(partition)
        obj.integer = integer
        return obj

    def prev_lex(self):
        """Return the previous partition of the integer, n, in lexical order,
        wrapping around to [1, ..., 1] if the partition is [n].

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> p = IntegerPartition([4])
        >>> print(p.prev_lex())
        [3, 1]
        >>> p.partition > p.prev_lex().partition
        True
        """
        d = defaultdict(int)
        d.update(self.as_dict())
        keys = self._keys
        if keys == [1]:
            return IntegerPartition({self.integer: 1})
        if keys[-1] != 1:
            d[keys[-1]] -= 1
            if keys[-1] == 2:
                d[1] = 2
            else:
                d[keys[-1] - 1] = d[1] = 1
        else:
            d[keys[-2]] -= 1
            left = d[1] + keys[-2]
            new = keys[-2]
            d[1] = 0
            while left:
                new -= 1
                if left - new >= 0:
                    d[new] += left//new
                    left -= d[new]*new
        return IntegerPartition(self.integer, d)

    def next_lex(self):
        """Return the next partition of the integer, n, in lexical order,
        wrapping around to [n] if the partition is [1, ..., 1].

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> p = IntegerPartition([3, 1])
        >>> print(p.next_lex())
        [4]
        >>> p.partition < p.next_lex().partition
        True
        """
        d = defaultdict(int)
        d.update(self.as_dict())
        key = self._keys
        a = key[-1]
        if a == self.integer:
            d.clear()
            d[1] = self.integer
        elif a == 1:
            if d[a] > 1:
                d[a + 1] += 1
                d[a] -= 2
            else:
                b = key[-2]
                d[b + 1] += 1
                d[1] = (d[b] - 1)*b
                d[b] = 0
        else:
            if d[a] > 1:
                if len(key) == 1:
                    d.clear()
                    d[a + 1] = 1
                    d[1] = self.integer - a - 1
                else:
                    a1 = a + 1
                    d[a1] += 1
                    d[1] = d[a]*a - a1
                    d[a] = 0
            else:
                b = key[-2]
                b1 = b + 1
                d[b1] += 1
                need = d[b]*b + d[a]*a - b1
                d[a] = d[b] = 0
                d[1] = need
        return IntegerPartition(self.integer, d)

    def as_dict(self):
        """Return the partition as a dictionary whose keys are the
        partition integers and the values are the multiplicity of that
        integer.

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict()
        {1: 3, 2: 1, 3: 4}
        """
        if self._dict is None:
            groups = group(self.partition, multiple=False)
            self._keys = [g[0] for g in groups]
            self._dict = dict(groups)
        return self._dict

    @property
    def conjugate(self):
        """
        Computes the conjugate partition of itself.

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> a = IntegerPartition([6, 3, 3, 2, 1])
        >>> a.conjugate
        [5, 4, 3, 1, 1, 1]
        """
        j = 1
        temp_arr = list(self.partition) + [0]
        k = temp_arr[0]
        b = [0]*k
        while k > 0:
            while k > temp_arr[j]:
                b[k - 1] = j
                k -= 1
            j += 1
        return b

    def __lt__(self, other):
        """Return True if self is less than other when the partition
        is listed from smallest to biggest.

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> a = IntegerPartition([3, 1])
        >>> a < a
        False
        >>> b = a.next_lex()
        >>> a < b
        True
        >>> a == b
        False
        """
        return list(reversed(self.partition)) < list(reversed(other.partition))

    def __le__(self, other):
        """Return True if self is less than other when the partition
        is listed from smallest to biggest.

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> a = IntegerPartition([4])
        >>> a <= a
        True
        """
        return list(reversed(self.partition)) <= list(reversed(other.partition))

    def as_ferrers(self, char='#'):
        """
        Prints the ferrer diagram of a partition.

        Examples
        ========

        >>> from sympy.combinatorics.partitions import IntegerPartition
        >>> print(IntegerPartition([1, 1, 5]).as_ferrers())
        #####
        #
        #
        """
        return "\n".join([char*i for i in self.partition])

    def __str__(self):
        return str(list(self.partition))


def random_integer_partition(n, seed=None):
    """
    Generates a random integer partition summing to ``n`` as a list
    of reverse-sorted integers.

    Examples
    ========

    >>> from sympy.combinatorics.partitions import random_integer_partition

    For the following, a seed is given so a known value can be shown; in
    practice, the seed would not be given.

    >>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1])
    [85, 12, 2, 1]
    >>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1])
    [5, 3, 1, 1]
    >>> random_integer_partition(1)
    [1]
    """
    from sympy.core.random import _randint

    n = as_int(n)
    if n < 1:
        raise ValueError('n must be a positive integer')

    randint = _randint(seed)

    partition = []
    while (n > 0):
        k = randint(1, n)
        mult = randint(1, n//k)
        partition.append((k, mult))
        n -= k*mult
    partition.sort(reverse=True)
    partition = flatten([[k]*m for k, m in partition])
    return partition


def RGS_generalized(m):
    """
    Computes the m + 1 generalized unrestricted growth strings
    and returns them as rows in matrix.

    Examples
    ========

    >>> from sympy.combinatorics.partitions import RGS_generalized
    >>> RGS_generalized(6)
    Matrix([
    [  1,   1,   1,  1,  1, 1, 1],
    [  1,   2,   3,  4,  5, 6, 0],
    [  2,   5,  10, 17, 26, 0, 0],
    [  5,  15,  37, 77,  0, 0, 0],
    [ 15,  52, 151,  0,  0, 0, 0],
    [ 52, 203,   0,  0,  0, 0, 0],
    [203,   0,   0,  0,  0, 0, 0]])
    """
    d = zeros(m + 1)
    for i in range(m + 1):
        d[0, i] = 1

    for i in range(1, m + 1):
        for j in range(m):
            if j <= m - i:
                d[i, j] = j * d[i - 1, j] + d[i - 1, j + 1]
            else:
                d[i, j] = 0
    return d


def RGS_enum(m):
    """
    RGS_enum computes the total number of restricted growth strings
    possible for a superset of size m.

    Examples
    ========

    >>> from sympy.combinatorics.partitions import RGS_enum
    >>> from sympy.combinatorics import Partition
    >>> RGS_enum(4)
    15
    >>> RGS_enum(5)
    52
    >>> RGS_enum(6)
    203

    We can check that the enumeration is correct by actually generating
    the partitions. Here, the 15 partitions of 4 items are generated:

    >>> a = Partition(list(range(4)))
    >>> s = set()
    >>> for i in range(20):
    ...     s.add(a)
    ...     a += 1
    ...
    >>> assert len(s) == 15

    """
    if (m < 1):
        return 0
    elif (m == 1):
        return 1
    else:
        return bell(m)


def RGS_unrank(rank, m):
    """
    Gives the unranked restricted growth string for a given
    superset size.

    Examples
    ========

    >>> from sympy.combinatorics.partitions import RGS_unrank
    >>> RGS_unrank(14, 4)
    [0, 1, 2, 3]
    >>> RGS_unrank(0, 4)
    [0, 0, 0, 0]
    """
    if m < 1:
        raise ValueError("The superset size must be >= 1")
    if rank < 0 or RGS_enum(m) <= rank:
        raise ValueError("Invalid arguments")

    L = [1] * (m + 1)
    j = 1
    D = RGS_generalized(m)
    for i in range(2, m + 1):
        v = D[m - i, j]
        cr = j*v
        if cr <= rank:
            L[i] = j + 1
            rank -= cr
            j += 1
        else:
            L[i] = int(rank / v + 1)
            rank %= v
    return [x - 1 for x in L[1:]]


def RGS_rank(rgs):
    """
    Computes the rank of a restricted growth string.

    Examples
    ========

    >>> from sympy.combinatorics.partitions import RGS_rank, RGS_unrank
    >>> RGS_rank([0, 1, 2, 1, 3])
    42
    >>> RGS_rank(RGS_unrank(4, 7))
    4
    """
    rgs_size = len(rgs)
    rank = 0
    D = RGS_generalized(rgs_size)
    for i in range(1, rgs_size):
        n = len(rgs[(i + 1):])
        m = max(rgs[0:i])
        rank += D[n, m + 1] * rgs[i]
    return rank