Juuggo
4/25/2017 - 3:51 AM

## The number of ways of splitting n items in k non-empty sets.

The number of ways of splitting n items in k non-empty sets.

``````# One Gold Star
# Question 1-star: Stirling and Bell Numbers

# The number of ways of splitting n items in k non-empty sets is called
# the Stirling number, S(n,k), of the second kind. For example, the group
# of people Dave, Sarah, Peter and Andy could be split into two groups in
# the following ways.

# 1.   Dave, Sarah, Peter         Andy
# 2.   Dave, Sarah, Andy          Peter
# 3.   Dave, Andy, Peter          Sarah
# 4.   Sarah, Andy, Peter         Dave
# 5.   Dave, Sarah                Andy, Peter
# 6.   Dave, Andy                 Sarah, Peter
# 7.   Dave, Peter                Andy, Sarah

# so S(4,2) = 7

# If instead we split the group into one group, we have just one way to
# do it.

# 1. Dave, Sarah, Peter, Andy

# so S(4,1) = 1

# or into four groups, there is just one way to do it as well

# 1. Dave        Sarah          Peter         Andy

# so S(4,4) = 1

# If we try to split into more groups than we have people, there are no
# ways to do it.

# The formula for calculating the Stirling numbers is

# =====================================
#  S(n, k) = k*S(n-1, k) + S(n-1, k-1)
# =====================================

# Furthermore, the Bell number B(n) is the number of ways of splitting n
# into any number of parts, that is,

# B(n) is the sum of S(n,k) for k =1,2, ... , n.

# Write two procedures, stirling and bell. The first procedure, stirling
# takes as its inputs two positive integers of which the first is the
# number of items and the second is the number of sets into which those
# items will be split. The second procedure, bell, takes as input a
# positive integer n and returns the Bell number B(n).

def stirling(n, k):
# n == k; k = 1; n < k
if n < k:
return 0
if n == k or k == 1:
return 1
return k * stirling(n - 1, k) + stirling(n - 1, k-1)

def bell(n):
result = 0
for i in range(1, n+1):
result += stirling(n, i)
return result

#print stirling(1,1)
#>>> 1
#print stirling(2,1)
#>>> 1
#print stirling(2,2)
#>>> 1
#print stirling(2,3)
#>>>0

#print stirling(3,1)
#>>> 1
#print stirling(3,2)
#>>> 3
#print stirling(3,3)
#>>> 1

#print stirling(4,1)
#>>> 1
#print stirling(4,2)
#>>> 7
#print stirling(4,3)
#>>> 6
#print stirling(4,4)
#>>> 1

#print stirling(5,1)
#>>> 1
#print stirling(5,2)
#>>> 15
#print stirling(5,3)
#>>> 25
#print stirling(5,4)
#>>> 10
#print stirling(5,5)
#>>> 1

#print stirling(20,15)
#>>> 452329200

print bell(1)
#>>> 1
print bell(2)
#>>> 2
print bell(3)
#>>> 5
print bell(4)
#>>> 15
print bell(5)
#>>> 52
print bell(15)
#>>> 1382958545
``````