The sestina

A sestina is a poetic form consisting of six stanzas of six lines each, followed by an “envoi” of three lines. In each stanza, the six words at the end of the line are the same, but in a different order. So if the six lines in the first stanza end 123456, then the orders of the line endings for the next five stanzas are 615243, 364125, 532614, 451362, and 246531. You can read more at the wikipedia page, and see an example by Anthony Hecht here, and one by Ezra Pound here. The form is supposed to have been invented by one Artaud Daniel, a 12th century troubadour, and has been exercising the minds of poets and other verse-makers since.

My interest here is the permutations which are used for the line endings. If you look carefully, you’ll see that the same permutation is used each time. This means that the sestina can be described in terms of powers of a single permutation. In cyclic notation, the first permutation 615243 can be written as

(1,2,4,5,3,6).

That is, the ending of line 1 becomes the ending of line 2 in the next stanza, the ending of line 2 becomes the ending of line 4 in the next stanza, and so on. Here’s Sage to do all the hard work:

sage: S = SymmetricGroup(6)
sage: p = S((1,2,4,5,3,6))
sage: for i in range(6): print i+1,p^i
....:
1 ()
2 (1,2,4,5,3,6)
3 (1,4,3)(2,5,6)
4 (1,5)(2,3)(4,6)
5 (1,3,4)(2,6,5)
6 (1,6,3,5,4,2)

Look at stanza 5 for example, The permutation is 451362. This means that 1 goes to the third place, 2 to the sixth place, 3 to the fourth place and so on. But this is given by the cycles

(1,3,4)(2,6,5).

If we want to print out the permutations as given above, that is easy:

sage: q = p.inverse()
sage: for i in range(6):
....:     qi = q^i
....:     print i+1,[qi(i) for i in range(1,7)]
....:
1 [1, 2, 3, 4, 5, 6]
2 [6, 1, 5, 2, 4, 3]
3 [3, 6, 4, 1, 2, 5]
4 [5, 3, 2, 6, 1, 4]
5 [4, 5, 1, 3, 6, 2]
6 [2, 4, 6, 5, 3, 1]

The sestina can now described as follows: in stanza n, the permutations of the line endings are given by p^{n-1} where p = (1,2,4,5,3,6)in S_6. I wonder how much of the theory of permutation groups was known to Artaud Daniel?

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