Combinatorics Through Guided Discovery, 2004a

3.3. Partitions of Integers 69
The Twentyfold Way: A Table of Distribution Problems
k objects and conditions
n recipients and mathematical
on how they are received
model for distribution
Distinct
Identical
1. Distinct
no conditions
2. Distinct
Each gets at most one
3. Distinct
Each gets at least one
4. Distinct
Each gets exactly one
5. Distinct,
order matters
6. Distinct,
order matters
Each gets at least one
7. Identical
no conditions
8. Identical
Each gets at most one
9. Identical
Each gets at least one
10. Identical
Each gets exactly one
n k
functions
n k
k-element
permutations
S(k, n)n!
onto functions
k! =n!
permutations
(k + n − 1) k
ordered functions
(k) n (k − 1) k−n
ordered
onto functions
( n+k−1 )
k
multisets
( n k )
subsets
( k−1
n−1 )
compositions
(n parts)
1ifk = n;
0 otherwise
∑ k
i=1
S(n, i)
set partitions (≤ n parts)
1ifk ≤ n;
0 otherwise
S(k, n)
set partitions (n parts)
1ifk = n;
0 otherwise
∑ n
i=1
L(k, i)
broken permutations
(≤ n parts)
L(k, n) =( k )(k
n
− 1)k−n
broken permutations
(n parts)
∑ n
i=1
P(k, i)
number partitions
(≤ n parts)
1ifk ≤ n;
0 otherwise
P(k, n)
number partitions
(n parts)
1ifk = n;
0 otherwise
Table 3.3.4: The number of ways to distribute k objects to n recipients, with restrictions
on how the objects are received
3.3.4 Partitions into distinct parts
Often Q(k, n) is used to denote the number of partitions of k into distinct parts,
that is, parts that are different from each other.
Problem 172. Show that
Q(k, n) ≤ 1 n!
( ) k − 1
.
n − 1
(h)