MATC44H3 Chap.9 Gluing, Interlacing and Unlabelled Partitions
Gluing, Interlacing and Unlabelled Partitions
Gluing for people who stay together
To count arrangements in which certain people must be adjacent, glue them into one block, arrange the block with everyone else, and multiply by the number of orders inside the block. The lecture glues three students among seven professors and obtains 8! × 3! arrangements.
The practice set nests the idea, gluing a pair inside a larger group of four, so that each layer contributes its own factorial.
Interlacing for people kept apart
To keep certain people from sitting together, first arrange everyone else, then place the restricted people in different gaps between and around them.
The lecture's version requires professors at both ends, so only the six inner gaps between seven professors are available, giving 7! × 6 × 5 × 4 arrangements, or 7! × C(6, 3) × 3!.
When the restricted objects are identical, such as repeated letters, only the set of gaps matters and the gap count becomes a binomial coefficient.
Labelled and unlabelled parts
The lecture contrasts choosing 2 people from n, which gives C(n, 2), with splitting n people into parts of sizes 2 and n − 2. The two counts agree unless the parts have equal size: for n = 4 the split into two pairs gives only 3 partitions, half of C(4, 2), because the two pairs cannot be told apart.
The same correction applies whenever a split produces several parts of the same size and no part carries a name.
What this chapter covers
- 01
Gluing a block of adjacent people
- 02
Nested blocks
- 03
Interlacing in gaps between people
- 04
End gaps and inner gaps
- 05
Identical letters placed in gaps
- 06
Labelled versus unlabelled partitions
Worked example · free
Two people never side by side
- 1All arrangements: 8! = 40,320.
- 2Glue Priya and Omar: 7 units give 7! orders and the pair has 2 orders, so 2 × 5,040 = 10,080 have them together.
- 1Subtract: 40,320 − 10,080 = 30,240 arrangements keep them apart.
Key terms
- Gluing Technique
- The gluing technique treats people who must sit together as a single block when counting arrangements.
- Interlacing Technique
- The interlacing technique places restricted people in separate gaps between other people so that no two are adjacent.
- Unlabelled Partition
- An unlabelled partition splits a set into parts that carry no names, so equal-sized parts can be swapped.
- Labelled Partition
- A labelled partition splits a set into named parts, so swapping the contents of two parts gives a new partition.
- Inner Gap
- An inner gap is a space between two neighbouring people in a row, as opposed to a space at either end.
Gluing, Interlacing and Unlabelled Partitions FAQ
When should I use gluing and when interlacing?
Glue when a group must be together: the block becomes one unit and its internal orders multiply in. Interlace when a group must be kept apart: seat the others first and place the restricted people in different gaps. For exactly two people kept apart, gluing and subtracting from the total also works.
How many gaps do m people in a row create?
There are m − 1 gaps between neighbours and two at the ends, m + 1 in all. Read the question to see whether the ends are allowed. If the ends must be occupied by the unrestricted group, only the m − 1 inner gaps are available.
Why do I divide when splitting into equal groups without names?
Choosing one group of size k from 2k people counts each split twice, once for each group that could have been chosen first. Without names the two orders describe the same split, so divide by 2, or by the factorial of the number of equal-sized groups when there are more than two.
How do I check a gluing or interlacing answer?
Shrink the problem until you can list every arrangement, for example three people with two kept apart, and compare your formula with the list. If the formula and the list disagree, look for a missing internal order of a block, a wrong number of gaps, or a division that should not be there.
Exam move
Solve one gluing, one interlacing and one partition question of your own, and check each against a brute-force list for a smaller case, such as four people instead of eight. Write one line per factor so that each factorial and binomial coefficient is explained.
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