def solve(C, N):
average = 0
for m in range(1, 200):
foo = 0
for i in range(1, C + 1):
foo += (-1)**(i + 1) * gmpy.comb(C, i) * A(N, C - i)**m
p = A(N, C)**m - foo
bar = 0
for i in range(1, C + 1):
bar += (-1)**(i + 1) * gmpy.comb(C, i) * A(N, C - i)**(m - 1)
p = p - (A(N, C)**(m - 1) - bar) * A(N, C)
average += (p / A(N, C)**m) * m
return average
def get_number_combinations(tile_length, row_length=ROW_LENGTH):
s = 0
for num_tiles in range(1, row_length / tile_length + 1):
total_tiles = row_length - (num_tiles * tile_length) + num_tiles
comb = gmpy.comb(total_tiles, num_tiles)
s += comb
return s
def p053():
sum = 0
for n in tqdm(range(1, 101)):
for r in range(1, n + 1):
if (comb(n, r)) > 1000000:
sum += 1
print(sum)
def probs(post_state):
""" Get all possible sub-states from post_state and their probabilities """
# print 'not cached'
s_prime= set()
sub_states = [0]*len(post_state)
#if post_state == (0,0,0):
# return {(0,0,0) : 0}
'''
For each element in the post state, figure out all ways it can be spread out.
Example for (5,4,3):
So figure out all the ways (5,0,0) can be spread out (4,1,0),(4,0,1),(3,2,0)...
along with the probability of each new sub-state happening.
Then recombine all the sub-states together to have every possible resulting
state, and calculate the resulting probability of each distinct new state.
'''
for pos in range(len(post_state)):
sub_states[pos] = set()
el = post_state[pos]
if el == 0:
sub_states[pos].add(((0,0,0), 1))
continue
'''
Figure out all attainable states and their likelihood.
I'm iterating through every possible state, and only saving those with
the right total number of cows.
A bit of base 13 trickery going on here, replace 13 by 10 and it'll make
sense
'''
for s in states():
if(sum(s) == el):
p = t_probs[pos]
#Calculate the probability of this state happening
prob = (p[0]**s[0])*(p[1]**s[1])*(p[2]**s[2])
#Correct the probability for states that can occur multiple ways
prob *= comb(el, max(s)) #Uses combinations
#If it's possible, save it
if(prob > 0):
sub_states[pos].add((s, prob))
#Now for combine all the sub_states together in every possible way
new_states = [join_states(j_states) for j_states in cartesian_product(sub_states)]
prebirth = concat_multiples(new_states)
#Now calculate birth probabilities:
postbirth = list()
for state in new_states:
#This returns all the different amounts of cows that can be produced, along with probability.
offspring = breed(state[0][1])
for off in offspring:
if sum(state[0]) + off > H:
calves = H - sum(state[0])
num_young = state[0][0] + calves
else:
num_young = state[0][0] + off
postbirth.append(((num_young, state[0][1], state[0][2]),
state[1] * offspring[off]))
ret_states = concat_multiples(postbirth)
return ret_states
def scherbak(a):
#a = reduce(a)
s=int(sum(a)/2)+2
end=len(a)
subt=[]; t=[]
for substring_len in range(1,end):
it=list(range(substring_len))
while it[0]<=end-substring_len-2:
subt.append(0); subset=[a[i] for i in it]
if sum(subset)+substring_len-s>=end-3:
subt[-1]=int(comb(sum(subset)+substring_len-s, end-3))
it=nextsubset(list(it), end)
subt.append(0); subset=[a[i] for i in it]
if sum(subset)+substring_len-s>=end-3:
subt[-1]=int(comb(sum(subset)+substring_len-s, end-3))
t.append(sum(subt)*(-1)**(end+1-substring_len)); subt=[]
return(sum(t))
def binomial(k, n):
"""Return the number of combinations of k elements among a collection of
size n."""
if k < 0:
return 0
elif k > n:
return 0
else:
return comb(int(n), int(k))
def scherbak(a):
s=int(sum(a)/2)+2
end=len(a)
subt=[]; t=[]
x=1 #Lengths of substrings being iterated
while x<end:
it=list(range(x))
while it[0]<=end-x-2:
subt.append(0); subset=[a[i] for i in it]
if sum(subset)+x-s>=end-3:
subt[-1]=int(comb(sum(subset)+x-s, end-3))
it=nextsubset(list(it), end)
subt.append(0); subset=[a[i] for i in it]
if sum(subset)+x-s>=end-3:
subt[-1]=int(comb(sum(subset)+x-s, end-3))
t.append(sum(subt)*(-1)**(end+1-x)); subt=[]
x+=1
return(sum(t))
def main():
"""Main Program"""
counter = 0
for n in range(1, 101):
for r in range(1, n):
c = gmpy.comb(n, r)
if c > 1000000:
counter += 1
print counter
def binomial(k, n):
"""Return the number of combinations of k elements among a collection of
size n."""
if k < 0:
return 0
elif k > n:
return 0
else:
return comb(int(n), int(k))
def get_number_combinations(num_tiles=0, accum_tile_length=0, row_length=ROW_LENGTH):
if accum_tile_length > row_length:
return 0
total_tiles = num_tiles + row_length - accum_tile_length
s = gmpy.comb(total_tiles, num_tiles)
for color_tile_length in ALL_COLORS:
s += get_number_combinations(num_tiles + 1, accum_tile_length +
color_tile_length)
return s
def score(internal, degree, number_of_nodes, total_nodes):
"""
Returns the score for a single community
"""
community_score = 0.
external = degree - (2 * internal)
if number_of_nodes == 0:
return 0
elif number_of_nodes == 1:
return 0
else:
internal_density = internal / float(comb(number_of_nodes, 2))
return internal_density
def pure(n, rank):
answer = 0
if rank == 1:
return 1
if (n, rank) in mem:
return mem[(n, rank)]
for i in xrange(1, rank):
answer += pure(rank, i) * int(comb(n - rank - 1, rank - i - 1))
mem[(n, rank)] = answer
return answer
def score(internal, degree, number_of_nodes, total_nodes) :
"""
Returns the score for a single community
"""
community_score = 0.
external = degree - (2*internal)
if number_of_nodes == 0 :
pass
elif number_of_nodes == 1 :
pass
else :
separability = internal / float(internal + external)
internal_density = internal / float(comb(number_of_nodes,2))
community_score = separability * internal_density
actual_score = (number_of_nodes / float(total_nodes)) * community_score
return actual_score
def score(internal, degree, number_of_nodes, total_nodes) :
"""
Returns the score for a single community
"""
community_score = 0.
external = degree - (2*internal)
if number_of_nodes == 0 :
pass
elif number_of_nodes == 1 :
pass
else :
separability = internal / float(internal + external)
internal_density = internal / float(comb(number_of_nodes,2))
community_score = separability * internal_density
actual_score = (number_of_nodes / float(total_nodes)) * community_score
return actual_score
def membership_function(X_test,y_test,class_value,alpha,prior_probability,total_X_val):
X_test = np.matrix(X_test)
#print prior_probability
#gjx = np.zeros(X_test.shape[0])
gjx =[]
for i in range(X_test.shape[0]):
for j in range(X_test.shape[1]):
sum_gj =0.0
combination = gm.comb(int(total_X_val[i]),int(X_test[i,j]))
if(combination!=0 and alpha[j]!=0 and (1-alpha[j])!=0):
sum_gj+= (mt.log(combination)) + (X_test[i,j] * mt.log(alpha[j])) + ((total_X_val[i]-X_test[i,j]) * mt.log(1-alpha[j]))
else:
sum_gj+= 0
sum_gj+= mt.log(abs(prior_probability))
"""j=1
val1 = [(mt.log(gm.comb(int(total_X_val[j]),int(X_test[j,i])))) for i in range(X_test.shape[1])]
val2 = sum([(X_test[j,i] * mt.log(alpha[i])) for i in range(X_test.shape[1])])
val3 = sum([((total_X_val[j] - X_test[j,i]) * mt.log(1-alpha[i])) for i in range(X_test.shape[1])])
total = val1 + val2 + val3 + mt.log(prior_probability)
print total_X_val[j]+1
print X_test[j,18]
print mt.log(gm.comb(1183,347))
print "Val 1 :",val1
print "Val 2 :",val2
print "Val 3 :",val3
print "Total :",total
exit(0)"""
gjx.append(sum_gj)
return np.array(gjx)
def nbcomb(n, k):
if k == 1 or k == n - 1:
return 1
if (n, k) in results.results:
return results.results[(n, k)]
answer = 0
x = 1
while x < k:
val = nbcomb(k, x)
answer += gmpy.comb(n - k - 1, k - x - 1) * val
x += 1
answer = int(answer) % 100003
results.results[(n, k)] = answer
return answer
def f(n, k):
if n < 0 or k < 0:
return 0
if n <= k:
return 0
if n == k + 1:
return 1
# print "%d %d" % (n, k)
if d[n][k] != -1:
return d[n][k]
res = 0
t = n - k - 1
for i in range(t + 1):
res += f(k, k - i - 1) * comb(t, i)
res %= MOD
d[n][k] = res
return res
#!/usr/bin/env python
import gmpy
count = 0
for n in range(1, 101):
for r in range(1, n):
c = gmpy.comb(n,r)
if c > 1000000:
count += 1
print count
def memoized_comb(x, n):
if memoize_comb[x][n] == -1:
memoize_comb[x][n] = comb(x, n) % MOD
return memoize_comb[x][n]
def binomial(p, n, k):
return gmpy.comb(n, k) * (pow(p, k)) * pow(1 - p, n - k)
def K1(p, q, tau, T):
#K1 function
#
# USAGE: [k1 k1d k1dd] = K1(p,q,tau,T)
#
# PARAMETERS:
# p,q ~ Degree of the interpolation polynomial of the first and second
# signals.
# tau ~ Point (modulo T) in which the cross-correlation function is
# evaluated.
# T ~ Sampling period of the discrete signals.
#
# RETURN VALUE:
# k1,k1d,k1dd ~ Value of the function, the first and the second derivative
# at tau respectively.
#
# DESCRIPTION:
# The K1 function corresponds to the following formula
# k1 = sum_{p=0}^q choose(q,k) * (T-tau)^(q-k) * tau^(p+k+1) / (p+k+1)
# and its derivatives (1st and 2nd) wigh respect to tau. T and tau may
# be two vectors of the same size, or vector and number.
#
# REFERENCES:
# X. Alameda-Pineda and R. Horaud. Geometrically-constrained time delay
# estimation-based sound source localisation (gTDESSL). Research Report
# RR-7988, INRIA, June 2012.
#
# see also K2, CCInterpolation
# Copyright 2012, Xavier Alameda-Pineda
# INRIA Grenoble Rhone-Alpes
# E-mail: [email protected]
# This is part of the gtde program.
#
# gtde is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
### Compute the function's value
k1 = np.zeros((1,tau.size))
for r in np.arange(0., q+1):
k1 = k1 + gp.comb(q, int(r)) * (T-tau)**(q-r) * tau**(p+r+1.) / float(p+r+1.)
### Compute its first derivative
k1d = np.zeros((1,tau.size))
for r in np.arange(0., q):
k1d = k1d + gp.comb(q, int(r)) * (T-tau)**(q-r-1.) * tau**(r+p) * ( T-(1.+p+q)*tau / float(r+p+1.) )
# r = q may give some numerical problems, special formula
k1d = k1d + tau**(q+p)
### Compute its second derivative
k1dd = np.zeros((1,tau.size))
# -----
# r = 0
# -----
if p >= 1.:
if q >= 2.:
# Same formula
r = 0.
k1dd = k1dd + gp.comb(q, int(r)) * (T-tau)**(q-r-2.) * tau**(r+p-1.) * ( (r+p)*T**2. - 2.*(q+p)*T*tau + (q+p)*(q+p+1)*tau**2. / float(r+p+1.) )
elif q == 1.:
k1dd = k1dd - tau**(p-1) * ( p*(tau-T) + 2*tau )
else:
k1dd = k1dd + p*tau**(p-1)
else:
# Special cases
if q >= 2.:
k1dd = k1dd + q*(T-tau)**(q-2) * ( (q+1)*tau - 2*T )
elif q == 1.:
k1dd = k1dd-2.
# -------
# r = q-1, exists if q >= 1, but for q = 1, the r = q will take
# care of it
# -------
if q >= 2.:
k1dd = k1dd + q*tau**(q+p-2) * ( (q+p)*(T-tau) - (T+tau) )
# -----
# r = q, it exists for q >= 1, since for q = 0, r = q will take
# care of it
# -----
if q >= 1.:
k1dd = k1dd + (q+p)*tau**(q+p-1)
# Intermediate terms
for r in np.arange(1., (q-2.)+1):
k1dd = k1dd + gp.comb(q, int(r)) * (T-tau)**(q-r-2.) * tau**(r+p-1.) * ( float((r+p)*T**2.) - 2*(q+p)*T*tau + (q+p)*(q+p+1.)*tau**2/float(r+p+1) )
return [k1, k1d, k1dd]
with timer("scipy"):
for i in range(0, 10000,100):
for j in range(0,i,10):
scipy_choose(i,j)
with timer("sympy"):
for i in range(0, 10000,100):
for j in range(0,i,10):
sympy_choose(i,j)
with timer("gmpy"):
for i in range(0, 10000,100):
for j in range(0,i,10):
comb(i,j)
with timer("choose1"):
for i in range(0, 10000,100):
for j in range(0,i,10):
choose1(i,j)
# OverflowError: integer division result too large for a float
# with timer("choose2"):
# for i in range(0, 10000,100):
# for j in range(0,i,10):
# choose2(i,j)
# tooooo slow
# with timer("choose3"):
def K2(p, q, tau, T):
# K2 K2 function
#
# USAGE: [k2 k2d k2dd] = K2(p,q,tau,T)
#
# PARAMETERS:
# p,q ~ Degree of the interpolation polynomial of the first and second
# signals.
# tau ~ Point (modulo T) in which the cross-correlation function is
# evaluated.
# T ~ Sampling period of the discrete signals.
#
# RETURN VALUE:
# k2,k2d,k2dd ~ Value of the function, the first and the second derivative
# at tau respectively.
#
# DESCRIPTION:
# The K2 function corresponds to the following formula
# K2 = sum_{k=0}^q choose(q,k) * (tau)^(q-k) * ( T^(k+p+1) - tau^(k+p+1)) / (k+p+1)
# and its derivatives (1st and 2nd) wigh respect to tau. T and tau may
# be two vectors of the same size, or vector and number.
#
# REFERENCES:
# X. Alameda-Pineda and R. Horaud. Geometrically-constrained time delay
# estimation-based sound source localisation (gTDESSL). Research Report
# RR-7988, INRIA, June 2012.
#
# see also K1, CCInterpolation
# Copyright 2012, Xavier Alameda-Pineda
# INRIA Grenoble Rhone-Alpes
# E-mail: [email protected]
#
# This is part of the gtde program.
#
# gtde is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
# Compute the function's value
k2 = np.zeros((1, tau.size))
for r in np.arange(0, q + 1):
k2 = k2 + gp.comb(q, int(r)) * (-tau) ** (q - r) * (T ** (p + r + 1.0) - tau ** (p + r + 1.0)) / float(
p + r + 1.0
)
#% If asked, compute its derivative
k2d = np.zeros((1, tau.size))
#% The general formula works for r < q
for r in np.arange(0, q):
k2d = k2d + gp.comb(q, int(r)) * (-tau) ** (q - r - 1.0) * (
(r - q) * (T ** (r + p + 1.0) - tau ** (r + p + 1.0)) / float(r + p + 1.0) + tau ** (r + p + 1.0)
)
k2d = k2d - tau ** (q + p)
#% If asked, compute its second derivative
k2dd = np.zeros((1, tau.size))
#% The general formula works for r < q-1
for r in np.arange(0, q - 1):
k2dd = k2dd + gp.comb(q, int(r)) * (-1.0) ** (q - r - 2.0) * (
(1.0 + r - q)
* (r - q)
* (T ** float(p) * tau ** (q - 1.0) * (T / tau) ** (r + 1.0) - tau ** (q + p - 1.0))
/ float(r + p + 1.0)
+ (r - p - 2.0 * q) * tau ** (q + p - 1.0)
)
#% If both q and p are 0, do not add anything
if q + p > 0.0:
#% Term for r = q-1
k2dd = k2dd + q * (q + p + 1.0) * tau ** (q + p - 1.0)
#% Term for r = q
k2dd = k2dd - (q + p) * tau ** (q + p - 1.0)
return [k2, k2d, k2dd]
#!/usr/bin/env python from gmpy import comb print comb(2 * 20,20)
def binomial_exact(k, p, n):
return gmpy.comb(n, k) * p**k * (1-p)**(n-k)
def A(k, n):
return gmpy.comb(n, k) * gmpy.fac(k)
"""
What I did was stopped calculating when i first encountered > 1Million, and calculated manually from that point.
The different formula for even and odd "n", i found by observing nCr results from 23,24,25,26,27.
Using GMPY library with Python made this extremely fast. < 1ms
"""
from gmpy import comb
num = 0
for n in xrange(20,101):
for r in xrange(1,n):
if comb(n,r) > 1000000:
# print n,
if n%2: more = 2*(n/2 - r +1)
else: more = 2*(n/2 - r) + 1
num += more
break
print num
# one-liner
from gmpy import comb
sum(2*(n/2 -r +1) if n%2 else 2*(n/2 -r)+1 for n in xrange(20,101) for r in xrange(1,n) if comb(n,r) > 1000000)
def bin_coef_sum(n):
return sum(gmpy.comb(n, i) ** 2 for i in xrange(n+1))
#!/usr/local/bin/python
from __future__ import division
import gmpy
num = 0
for i in xrange(569):
num += gmpy.comb(1000,i)
dem = 2**1000
print num/dem
#!/usr/bin/env python from gmpy import comb print comb(2 * 20, 20)
