
from collections import defaultdict

def binomial_coefficients(n):
    """Return a dictionary containing pairs :math:`{(k1,k2) : C_kn}` where
    :math:`C_kn` are binomial coefficients and :math:`n=k1+k2`.
    Examples
    ========

    >>> from sympy.ntheory import binomial_coefficients
    >>> binomial_coefficients(9)
    {(0, 9): 1, (1, 8): 9, (2, 7): 36, (3, 6): 84, (4, 5): 126, (5, 4): 126, (6, 3): 84, (7, 2): 36, (8, 1): 9, (9, 0): 1}

    See Also
    ========

    binomial_coefficients_list, multinomial_coefficients
    """
    d = {(0, n):1, (n, 0):1}
    a = 1
    for k in range(1, n//2+1):
        a = (a * (n-k+1))//k
        d[k, n-k] = d[n-k, k] = a
    return d

def binomial_coefficients_list(n):
    """ Return a list of binomial coefficients as rows of the Pascal's
    triangle.

    Examples
    ========

    >>> from sympy.ntheory import binomial_coefficients_list
    >>> binomial_coefficients_list(9)
    [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]

    See Also
    ========

    binomial_coefficients, multinomial_coefficients
    """
    d = [1] * (n+1)
    a = 1
    for k in range(1, n//2+1):
        a = (a * (n-k+1))//k
        d[k] = d[n-k] = a
    return d

def multinomial_coefficients0(m, n, _tuple=tuple, _zip=zip):
    """Return a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}``
    where ``C_kn`` are multinomial coefficients such that
    ``n=k1+k2+..+km``.

    For example:

    >>> from sympy import multinomial_coefficients
    >>> multinomial_coefficients(2, 5) # indirect doctest
    {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1}

    The algorithm is based on the following result:

       Consider a polynomial and its ``n``-th exponent::

         P(x) = sum_{i=0}^m p_i x^i
         P(x)^n = sum_{k=0}^{m n} a(n,k) x^k

       The coefficients ``a(n,k)`` can be computed using the
       J.C.P. Miller Pure Recurrence [see D.E.Knuth, Seminumerical
       Algorithms, The art of Computer Programming v.2, Addison
       Wesley, Reading, 1981;]::

         a(n,k) = 1/(k p_0) sum_{i=1}^m p_i ((n+1)i-k) a(n,k-i),

       where ``a(n,0) = p_0^n``.
    """

    if not m:
        if n:
            return {}
        else:
            return {(): 1}
    if m == 2:
        return binomial_coefficients(n)
    symbols = [(0,)*i + (1,) + (0,)*(m-i-1) for i in range(m)]
    s0 = symbols[0]
    p0 = [_tuple(aa - bb for aa, bb in _zip(s, s0)) for s in symbols]
    r = {_tuple(aa*n for aa in s0): 1}
    l = [0] * (n*(m - 1) + 1)
    l[0] = list(r.items())
    for k in range(1, n*(m - 1) + 1):
        d = defaultdict(int)
        for i in range(1, min(m, k + 1)):
            nn = (n + 1)*i - k
            if not nn:
                continue
            t = p0[i]
            for t2, c2 in l[k - i]:
                tt = _tuple([aa + bb for aa, bb in _zip(t2, t)])
                d[tt] += nn*c2
                if not d[tt]:
                    del d[tt]
        r1 = [(t, c//k) for (t, c) in d.items()]
        l[k] = r1
        r.update(r1)
    return r

def multinomial_coefficients(m, n):
    r"""Return a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}``
    where ``C_kn`` are multinomial coefficients such that
    ``n=k1+k2+..+km``.

    For example:

    >>> from sympy.ntheory import multinomial_coefficients
    >>> multinomial_coefficients(2, 5) # indirect doctest
    {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1}

    The algorithm is based on the following result:

    .. math::
        \binom{n}{k_1, \ldots, k_m} =
        \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots}

    Code contributed to Sage by Yann Laigle-Chapuy, copied with permission
    of the author.

    See Also
    ========

    binomial_coefficients_list, binomial_coefficients
    """
    if not m:
        if n:
            return {}
        else:
            return {(): 1}
    if m == 2:
        return binomial_coefficients(n)
    if m >= 2*n and n > 1:
        return dict(multinomial_coefficients_iterator(m, n))
    t = [n] + [0] * (m - 1)
    r = {tuple(t): 1}
    if n:
        j = 0 # j will be the leftmost nonzero position
    else:
        j = m
    # enumerate tuples in co-lex order
    while j < m - 1:
        # compute next tuple
        tj = t[j]
        if j:
            t[j] = 0
            t[0] = tj
        if tj > 1:
            t[j + 1] += 1
            j = 0
            start = 1
            v = 0
        else:
            j += 1
            start = j + 1
            v = r[tuple(t)]
            t[j] += 1
        # compute the value
        # NB: the initialization of v was done above
        for k in range(start, m):
            if t[k]:
                t[k] -= 1
                v += r[tuple(t)]
                t[k] += 1
        t[0] -= 1
        r[tuple(t)] = (v * tj) // (n - t[0])
    return r

def multinomial_coefficients_iterator(m, n, _tuple=tuple):
    """multinomial coefficient iterator

    This routine has been optimized for `m` large with respect to `n` by taking
    advantage of the fact that when the monomial tuples `t` are stripped of
    zeros, their coefficient is the same as that of the monomial tuples from
    ``multinomial_coefficients(n, n)``. Therefore, the latter coefficients are
    precomputed to save memory and time.

    >>> from sympy.ntheory.multinomial import multinomial_coefficients
    >>> m53, m33 = multinomial_coefficients(5,3), multinomial_coefficients(3,3)
    >>> m53[(0,0,0,1,2)] == m53[(0,0,1,0,2)] == m53[(1,0,2,0,0)] == m33[(0,1,2)]
    True

    Examples
    ========

    >>> from sympy.ntheory.multinomial import multinomial_coefficients_iterator
    >>> it = multinomial_coefficients_iterator(20,3)
    >>> next(it)
    ((3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 1)
    """
    if m < 2*n or n == 1:
        mc = multinomial_coefficients(m, n)
        for k, v in mc.items():
            yield(k, v)
    else:
        mc = multinomial_coefficients(n, n)
        mc1 = {}
        for k, v in mc.items():
            mc1[_tuple(filter(None, k))] = v
        mc = mc1

        t = [n] + [0] * (m - 1)
        t1 = _tuple(t)
        b = _tuple(filter(None, t1))
        yield (t1, mc[b])
        if n:
            j = 0 # j will be the leftmost nonzero position
        else:
            j = m
        # enumerate tuples in co-lex order
        while j < m - 1:
            # compute next tuple
            tj = t[j]
            if j:
                t[j] = 0
                t[0] = tj
            if tj > 1:
                t[j + 1] += 1
                j = 0
            else:
                j += 1
                t[j] += 1

            t[0] -= 1
            t1 = _tuple(t)
            b = _tuple(filter(None, t1))
            yield (t1, mc[b])
