def test_binomial_negative_values():
assert_equal(binomial(3, 2), 3)
assert_equal(binomial(3, -2), 0)
assert_equal(binomial(-3, 2), 0)
assert_equal(binomial(-3, -2), 0)
assert_equal(binomial(2, 3), 0)
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 number_of_k_resolutions_upper_constraints_wrong(n, t_up):
"""
k resolutions with uphill constraint, wrong version, examined at the end of section 3.
:param n: number of element extracted
:param t_up: constraints uphill vector
:return:
"""
return binomial(n - sum(t_up) + len(t_up) - 1, len(t_up) - 1)
def number_of_k_resolutions_n(n, k):
"""
R_{k}^{n}
k resolution of n with no constraints, starting from 0
:param n,k: integer indexes
:return: number of k-resolutions of n whose elements can include zero
"""
return binomial(n + k - 1, k - 1)
def number_of_k_resolutions_lower_constraints(n, low_t):
"""
$R_{(a_1,a_2, \dots , a_k )^{\downarrow} }^{n}$
k resolutions with lower constraint
:param n:
:param low_t: vector of non-negative integers $(a_1,a_2, \dots , a_k)$ representing lower constraints
:return: number of k-resolution with lower constraints.
"""
return binomial(n - sum(low_t) + len(low_t) - 1, len(low_t) - 1)
def test_binomial_with_tartaglia_triangle():
def tartaglia_matrix(d):
m = np.zeros([d, d])
m[:, 0] = np.ones(d)
for r in range(1, d):
for c in range(1, d):
m[r, c] = m[r-1, c-1] + m[r-1, c]
return m
l = 10
m1 = np.ones([l, l])
m2 = tartaglia_matrix(l)
for rr in range(l):
for cc in range(l):
m1[rr, cc] = binomial(rr, cc)
assert_array_equal(m1, m2)
