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# sympy.ntheory.multinomial.multinomial_coefficients_iterator

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```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)(more...)
```

```        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] +  * (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 = tj
if tj > 1:
t[j + 1] += 1
j = 0
else:
j += 1
t[j] += 1

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

```from sympy.utilities.pytest import raises
from sympy.utilities.iterables import capture
from sympy.ntheory.multinomial import multinomial_coefficients_iterator

```
```            (1, 0, 2): 3, (0, 2, 1): 3, (0, 1, 2): 3, (3, 0, 0): 1,
(2, 0, 1): 3, (1, 2, 0): 3, (1, 1, 1): 6, (0, 0, 3): 1}
assert dict(multinomial_coefficients_iterator(2, 0)) == {(0, 0): 1}
assert dict(
multinomial_coefficients_iterator(2, 1)) == {(0, 1): 1, (1, 0): 1}
assert dict(multinomial_coefficients_iterator(2, 2)) == \
{(2, 0): 1, (0, 2): 1, (1, 1): 2}
assert dict(multinomial_coefficients_iterator(3, 3)) == mc
```

```from sympy.utilities.pytest import raises
from sympy.utilities.iterables import capture
from sympy.ntheory.multinomial import multinomial_coefficients_iterator

```
```            (1, 0, 2): 3, (0, 2, 1): 3, (0, 1, 2): 3, (3, 0, 0): 1,
(2, 0, 1): 3, (1, 2, 0): 3, (1, 1, 1): 6, (0, 0, 3): 1}
assert dict(multinomial_coefficients_iterator(2, 0)) == {(0, 0): 1}
assert dict(
multinomial_coefficients_iterator(2, 1)) == {(0, 1): 1, (1, 0): 1}
assert dict(multinomial_coefficients_iterator(2, 2)) == \
{(2, 0): 1, (0, 2): 1, (1, 1): 2}
assert dict(multinomial_coefficients_iterator(3, 3)) == mc
```