# Sestinas and Sage

According to [1], a sestina is a highly structured poem consisting of six six-line stanzas followed by a tercet (called its envoy or tornada), for a total of thirty-nine lines. The same set of six words ends the lines of each of the six-line stanzas, but in a shuffled order each time. The shuffle used is very similar to the Mongean shuffle.

Define $f_n(k) = 2k$, if k <= n/2 and $f_n(k) = 2n+1-2k$, if $k > n/2.$ Let $p = (p_1,...,p_n) \in S_n$, where $p_j = f_n(p_{j-1})$ and $S_n$ is the symmetric group of order $n$. From [2], we have the following result.

Theorem: If p is an n-cycle then 2n+1 is a prime.

Call such a prime a “sestina prime”. Which primes are sestina primes?

Here is Python/Sage code for this permutation:


def sestina(n):
"""
Computes the element of the symmetric group S_n associated to the shuffle above.

EXAMPLES:
sage: sestina(4)
(1,2,4)
sage: sestina(6)
(1,2,4,5,3,6)
sage: sestina(8)
(1,2,4,8)(3,6,5,7)
sage: sestina(10)
(1,2,4,8,5,10)(3,6,9)
sage: sestina(12)
(1,2,4,8,9,7,11,3,6,12)(5,10)
sage: sestina(14)
(1,2,4,8,13,3,6,12,5,10,9,11,7,14)
sage: sestina(16)
(1,2,4,8,16)(3,6,12,9,15)(5,10,13,7,14)
sage: sestina(18)
(1,2,4,8,16,5,10,17,3,6,12,13,11,15,7,14,9,18)
sage: sestina(20) (1,2,4,8,16,9,18,5,10,20)(3,6,12,17,7,14,13,15,11,19)
sage: sestina(22) (1,2,4,8,16,13,19,7,14,17,11,22)(3,6,12,21)(5,10,20)(9,18)

"""
def fcn(k, n):
if k<=int(n/2):
return 2*k
else:
return 2*n+1-2*k
L = [fcn(k,n) for k in range(1,n+1)]
G = SymmetricGroup(n)
return G(L)



And here is an example due to Ezra Pound [3]:

                                  I

Damn it all! all this our South stinks peace.
You whoreson dog, Papiols, come! Let’s to music!
I have no life save when the swords clash.
But ah! when I see the standards gold, vair, purple, opposing
And the broad fields beneath them turn crimson,
Then howl I my heart nigh mad with rejoicing.

II

In hot summer have I great rejoicing
When the tempests kill the earth’s foul peace,
And the light’nings from black heav’n flash crimson,
And the fierce thunders roar me their music
And the winds shriek through the clouds mad, opposing,
And through all the riven skies God’s swords clash.

III

Hell grant soon we hear again the swords clash!
And the shrill neighs of destriers in battle rejoicing,
Spiked breast to spiked breast opposing!
Better one hour’s stour than a year’s peace
With fat boards, bawds, wine and frail music!
Bah! there’s no wine like the blood’s crimson!

IV

And I love to see the sun rise blood-crimson.
And I watch his spears through the dark clash
And it fills all my heart with rejoicing
And prys wide my mouth with fast music
When I see him so scorn and defy peace,
His lone might ’gainst all darkness opposing.

V

The man who fears war and squats opposing
My words for stour, hath no blood of crimson
But is fit only to rot in womanish peace
Far from where worth’s won and the swords clash
For the death of such sluts I go rejoicing;
Yea, I fill all the air with my music.

VI

Papiols, Papiols, to the music!
There’s no sound like to swords swords opposing,
No cry like the battle’s rejoicing
When our elbows and swords drip the crimson
And our charges ’gainst “The Leopard’s” rush clash.
May God damn for ever all who cry “Peace!”

VII

And let the music of the swords make them crimson
Hell grant soon we hear again the swords clash!
Hell blot black for always the thought “Peace”!



References:

[1] http://en.wikipedia.org/wiki/Sestina

[2] Richard Dore and Anton Geraschenko,”Sestinas and Primes” posted to http://stacky.net/wiki/index.php?title=Course_notes, and http://math.berkeley.edu/~anton/written/sestina.pdf

[3] Ezra Pound, “Sestina: Altaforte” (1909), (originally published int the English Review, 1909)

[4] John Bullitt, N. J. A. Sloane and J. H. Conway , http://oeis.org/A019567