def number_of_k_resolutions_upper_constraints(n, up_t, verbose=False):
"""
Fatti non foste a computare come bruti.
$R_{(a_1,a_2, \dots , a_k )^{\uparrow} }^{n}$
:param n: positive integer
:param up_t: vector of non-negative integers $(a_1,a_2, \dots , a_k)$ representing upper constraints
:return: $\binom{n+k-1}{k-1} - \sum_{m=1}^{k} (-1)^{m+1}
\sum_{1\leq i_1 < i_2 < \dots <i_m \leq k}
\binom{n-\sum_{l=1}^{m} a_{i_{l}} - m + k - 1}{k-1}$
"""
k = len(up_t)
p = power_lists(range(1, k + 1))
s = 0
for l in p:
s += ((-1)**(len(l))) * binomial(n + k - 1 - len(l) - sum([up_t[i - 1] for i in l]), k - 1)
if verbose:
print p
print 'emmenems of each color in the pocket : ' + str(up_t)
print 'emmenems in the hand : ' + str(n)
print 'numbers of handfuls : ' + str(s)
return s
def test_power_set_simple(verbose=True):
s1 = ['a', 'b', 'c']
ans_ground = [[], ['a'], ['b'], ['c'], ['a', 'b'], ['a', 'c'], ['b', 'c'], ['a', 'b', 'c']]
ans = power_lists(s1)
if verbose:
print ''
print 'Power set of ' + str(s1) + ':'
print ans
assert_equal(len(ans), len(ans_ground))
for i in range(len(ans)):
assert_true(ans[i] in ans_ground)
