Source code for sympy.simplify.combsimp

from __future__ import print_function, division

from sympy.core import Mul
from sympy.core.basic import preorder_traversal
from sympy.core.function import count_ops
from sympy.functions.combinatorial.factorials import binomial, factorial
from sympy.functions import gamma
from sympy.simplify.gammasimp import gammasimp, _gammasimp

from sympy.utilities.timeutils import timethis


@timethis('combsimp')
def combsimp(expr):
    r"""
    Simplify combinatorial expressions.

    This function takes as input an expression containing factorials,
    binomials, Pochhammer symbol and other "combinatorial" functions,
    and tries to minimize the number of those functions and reduce
    the size of their arguments.

    The algorithm works by rewriting all combinatorial functions as
    gamma functions and applying gammasimp() except simplification
    steps that may make an integer argument non-integer. See docstring
    of gammasimp for more information.

    Then it rewrites expression in terms of factorials and binomials by
    rewriting gammas as factorials and converting (a+b)!/a!b! into
    binomials.

    If expression has gamma functions or combinatorial functions
    with non-integer argument, it is automatically passed to gammasimp.

    Examples
    ========

    >>> from sympy.simplify import combsimp
    >>> from sympy import factorial, binomial, symbols
    >>> n, k = symbols('n k', integer = True)

    >>> combsimp(factorial(n)/factorial(n - 3))
    n*(n - 2)*(n - 1)
    >>> combsimp(binomial(n+1, k+1)/binomial(n, k))
    (n + 1)/(k + 1)

    """

    expr = expr.rewrite(gamma)
    if any(isinstance(node, gamma) and not node.args[0].is_integer
        for node in preorder_traversal(expr)):
        return gammasimp(expr);

    expr = _gammasimp(expr, as_comb = True)
    expr = _gamma_as_comb(expr)
    return expr


def _gamma_as_comb(expr):
    """
    Helper function for combsimp.

    Rewrites expression in terms of factorials and binomials
    """

    expr = expr.rewrite(factorial)

    from .simplify import bottom_up

    def f(rv):
        if not rv.is_Mul:
            return rv
        rvd = rv.as_powers_dict()
        nd_fact_args = [[], []] # numerator, denominator

        for k in rvd:
            if isinstance(k, factorial) and rvd[k].is_Integer:
                if rvd[k].is_positive:
                    nd_fact_args[0].extend([k.args[0]]*rvd[k])
                else:
                    nd_fact_args[1].extend([k.args[0]]*-rvd[k])
                rvd[k] = 0
        if not nd_fact_args[0] or not nd_fact_args[1]:
            return rv

        hit = False
        for m in range(2):
            i = 0
            while i < len(nd_fact_args[m]):
                ai = nd_fact_args[m][i]
                for j in range(i + 1, len(nd_fact_args[m])):
                    aj = nd_fact_args[m][j]

                    sum = ai + aj
                    if sum in nd_fact_args[1 - m]:
                        hit = True

                        nd_fact_args[1 - m].remove(sum)
                        del nd_fact_args[m][j]
                        del nd_fact_args[m][i]

                        rvd[binomial(sum, ai if count_ops(ai) <
                                count_ops(aj) else aj)] += (
                                -1 if m == 0 else 1)
                        break
                else:
                    i += 1

        if hit:
            return Mul(*([k**rvd[k] for k in rvd] + [factorial(k)
                    for k in nd_fact_args[0]]))/Mul(*[factorial(k)
                    for k in nd_fact_args[1]])
        return rv

    return bottom_up(expr, f)