BTT Mathematics / Primary Mathematics Learning Hub / Integer Partitions
An integer partition writes a positive whole number as a sum of positive whole numbers where order does not matter. For example, the partitions of4 are
4; 3+1; 2+2; 2+1+1; 1+1+1+1.
Thus the partition number p(4)=5.
This is different from a composition, where order matters. For partitions, 3+1 and1+3 are the same partition and are counted once.
Ferrers diagrams represent a partition using rows of dots or boxes. They make hidden symmetries visible, especially the relationship between number of parts and largest part.
This guide is Primary Mathematics enrichment. It connects Systematic Listing and Logical Counting, Visual Sequences, Combination Counting and structural representation.
Generating functions and asymptotic formulas are outside this guide. Practice focuses on exact small-n enumeration.
Partition meaning · Systematic listing · Ferrers diagrams · Conjugate partitions · Restricted partitions · 24 questions · Worked answers
1. Order does not create a new partition
A partition is normally written in non-increasing order so duplicates are obvious.
Worked example A: Partitions of3
3; 2+1; 1+1+1. Therefore p(3)=3.
Worked example B: Partitions of4
There are5, listed above.
Worked example C: Partitions of5
5
4+1
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1
So p(5)=7.
Worked example D: Partition versus composition
For5, 3+2 and2+3 are one partition but two ordered compositions.
2. Systematic listing prevents omission and duplication
List partitions by largest part. For6, begin with6, then all beginning with5, then4, then3, and so on.
Worked example E: Partitions of6
6
5+1
4+2
4+1+1
3+3
3+2+1
3+1+1+1
2+2+2
2+2+1+1
2+1+1+1+1
1+1+1+1+1+1
Therefore p(6)=11.
Worked example F: p(1) through p(7)
p(1)=1, p(2)=2, p(3)=3, p(4)=5, p(5)=7, p(6)=11, p(7)=15.
Worked example G: Largest-part grouping
Partitions of6 with largest part3 are 3+3, 3+2+1 and3+1+1+1: exactly 3.
Worked example H: Exactly two parts
Partitions of7 into exactly two positive parts are6+1,5+2,4+3: 3 partitions.
3. Ferrers diagrams turn sums into shapes
For partition 5=3+1+1, draw three boxes in the first row, then one box in each of the next two rows:
■■■
■
■
The total number of boxes is5.
Worked example I: 5=2+2+1
Diagram:
■■
■■
■
It has3 rows and largest row length2.
Worked example J: Number of parts
The number of rows equals the number of parts. For4+2+1 there are 3 parts.
Worked example K: Largest part
The length of the top row equals the largest part. For4+2+1 it is 4.
Worked example L: Total
Counting all diagram cells recovers the partitioned integer. Ferrers diagrams change representation, not quantity.
4. Reflecting a Ferrers diagram swaps rows and columns
The conjugate partition is obtained by reading column lengths instead of row lengths.
Worked example M: Conjugate of 3+1+1
The row lengths are3,1,1. Column lengths are3,1,1, so this partition is self-conjugate.
Worked example N: Conjugate of4+2+1
Columns have lengths3,2,1,1. Therefore the conjugate is 3+2+1+1.
Worked example O: Conjugate twice
Reflecting rows and columns twice returns the original Ferrers diagram. Conjugation is its own inverse.
Worked example P: Parts and largest part swap
If a partition has k parts and largest part m, its conjugate has largest part k and exactly m parts.
5. Restricted partitions ask for only part of the full list
We may count partitions with exactly k parts, with largest part at most m, into distinct parts, or into odd parts.
Worked example Q: Distinct parts of6
6, 5+1, 4+2, 3+2+1. Therefore there are 4.
Worked example R: Odd parts of6
5+1, 3+3, 3+1+1+1, 1+1+1+1+1+1. There are also 4.
Worked example S: Small identity
For6, the number of partitions into distinct parts equals the number into odd parts. This is an instance of a general theorem, but the guide only verifies small cases.
Worked example T: At most two parts of6
6,5+1,4+2,3+3: 4 partitions.
Worked example U: Exactly three parts of6
4+1+1,3+2+1,2+2+2: 3 partitions.
6. Practice: 24 original questions
Questions 1–8: Basic partitions
1. List the partitions of3.
2. How many partitions does4 have?
3. How many partitions does5 have?
4. How many partitions does6 have?
5. Is3+2 a different partition from2+3?
6. How many partitions of7 have exactly two parts?
7. List the partitions of6 whose largest part is3.
8. State p(7).
Questions 9–16: Ferrers diagrams and conjugation
9. How many rows are in the Ferrers diagram of4+2+1?
10. What is its longest row?
11. Find the conjugate of4+2+1.
12. Find the conjugate of3+1+1.
13. What happens if conjugation is applied twice?
14. A partition has5 parts and largest part3. What are the corresponding largest-part and part-count values of its conjugate?
15. What does the total number of Ferrers cells equal?
16. For2+2+1, how many rows and columns are non-empty?
Questions 17–24: Restricted partitions
17. List the partitions of6 into distinct parts.
18. How many are there?
19. List the partitions of6 into odd parts.
20. How many are there?
21. How many partitions of6 have at most two parts?
22. How many partitions of6 have exactly three parts?
23. Does equal count of distinct-part and odd-part partitions for6 prove the general theorem for all n?
24. Explain why listing by non-increasing parts avoids duplicate orderings.
7. Worked answers
Answers 1–8
1. 3;2+1;1+1+1.
2. 5.
3. 7.
4. 11.
5. No.
6. 3. 6+1,5+2,4+3.
7. 3+3,3+2+1,3+1+1+1.
8. 15.
Answers 9–16
9. 3 rows.
10. 4 cells.
11. 3+2+1+1.
12. 3+1+1.
13. The original partition returns.
14. Conjugate has largest part5 and exactly3 parts.
15. The integer being partitioned.
16. 3 rows and2 non-empty columns.
Answers 17–24
17. 6;5+1;4+2;3+2+1.
18. 4.
19. 5+1;3+3;3+1+1+1;1+1+1+1+1+1.
20. 4.
21. 4.
22. 3.
23. No. It verifies only the n=6 case.
24. Each unordered sum has one standard non-increasing form, so arrangements such as3+2 and2+3 collapse to one entry.
8. Teaching and transfer
If a learner lists the same sum repeatedly in different orders, require each partition to be written from largest part to smallest before counting it.
When enumeration feels unstructured
Group by largest part. Exhaust one group before moving to the next smaller maximum.
When Ferrers diagrams look decorative
Use them to answer structural questions: number of rows, longest row, column lengths and conjugation.
When a numerical pattern is mistaken for proof
Checking n=6 can reveal a conjecture but cannot establish a theorem for every n. Separate evidence from proof.
When conjugation feels mysterious
Rotate the viewpoint from rows to columns. Nothing is added or removed; the same cells are regrouped.
Connect to systematic counting
Integer partitions are a controlled laboratory for completeness and non-duplication, extending the habits in Systematic Listing and Logical Counting.
Continue through this enrichment collection
For stochastic state movement, use Random Walks, Transition Probabilities and Markov Chains. For compatible pairings, use Graph Matchings, Bipartite Assignment and Pairing Problems. For topological diagrams, use Knot Theory, Links, Crossings and Simple Invariants.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Partition order is non-increasing and order of summands does not create a new partition.