What are the 15 possible rhyme schemes for a Quatrain?

Question

Wikipedia says that for a Quatrain there are fifteen possible rhyme schemes.

What are these? I cannot find a full list anywhere.

Answer 1

Let's build the verse line by line.

• The first one is A (by definition).
• The second line either rhymes with the first (A) or not (B). So that's two possibilities.
• If we have AA, the next line is either A or B (two possibilities); if we have AB, the next line can be A, B or C (three possibilities).
• For the final line:
  • AAA can lead to AAAA or AAAB
  • AAB can lead to AABA, AABB or AABC
  • ABA can lead to ABAA, ABAB or ABAC
  • ABB can lead to ABBA, ABBB or ABBC
  • ABC can lead to ABCA, ABCB, ABCC or ABCD.

In general, the number of possible rhyme schemes for a poem with n lines is given by the so-called Bell number. In fact, Wikipedia explicitly lists this purpose and the list you're looking for:

Rhyme schemes

The Bell numbers also count the rhyme schemes of an n-line poem or stanza. A rhyme scheme describes which lines rhyme with each other, and so may be interpreted as a partition of the set of lines into rhyming subsets. Rhyme schemes are usually written as a sequence of Roman letters, one per line, with rhyming lines given the same letter as each other, and with the first lines in each rhyming set labeled in alphabetical order. Thus, the 15 possible four-line rhyme schemes are AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, and ABCD.