# Mathematics of Haiku

2/15/17

A traditional haiku has a certain structure and various "rules". Of course, both of these things can be changed, altered, modified, by any degree, or strictly adhered to. One of the main things, at least for so-called "American haiku" is a syllable structure of 575, or

5
7
5

which means 5 syllables for the first line, 7 syllables for the second line, and 5 syllables for the third line.

What is interesting is that for this syllable structure alone, there are 25-1*27-1*25-1 (I wrote it like that for a reason) possible syllable structures, or 24*26*24 = 16*64*16 = 16,384.

This happens because 5 syllables (for example) can be expressed 16 ways like:

5
41 (ie. 4 syllables and 1 syllable, so 5 total syllables)
32
311
23
221
212
2111
14
131
122
1211
113
1121
1112
11111

This breaking down of an integer is called "integer composition". Curiously, if we didn't care about order, for example 113 is the same as 131, 311, etc., then it is called "integer partition". There is tons of research in mathematics on these topics, and they are related with number theory and binary numbers.

Similarly 7 (again, from the 575 structure) can be broken down 64 ways. In general, n can be broken down 2n-1 ways. Here is the breakdown for 7

7
61
52
511
43
421
412
4111
34
331
322 (ie. 3 syllable, 2 syllable, and 2 syllable, so 7 total syllables)
3211
313
3121
3112
31111
25
241
232
2311
223
2221
2212
22111
214
2131
2122
21211
2113
21121
21112
211111
16
151
142
1411
133
1321
1312
13111
124
1231
1222
12211
1213
12121
12112
121111
115
1141
1132
11311
1123
11221
11212
112111
1114
11131
11122
111211
11113
111121
111112
1111111

Of those 5 syllable possibilities, say the 23 one, there are so many 2 syllable words and so many 3 syllable words, so the 23 possibility has millions of possibilities itself. The 1112 probably has trillions.

Therefore, even using very small word lists for 1, 2, 3, 4, 5, 6, and 7 syllable words, I calculate total possible random haikus of over 1066. What I am saying is, even if your vocabulary is somewhat limited, which it probably is not, you have the creative capacity inside you for making more haiku than the number of atoms in the universe.