Mathematical forms in poetry 4 – Permutations

In the latter part of the 12th century Arnaut Daniel, a troubadour from Ribérac in what is now the Dordogne, entertained the courts of southern Europe with poems on themes of chivalry and courtly love. Daniel’s poetry, written in his native Occitan, is characterised by technical virtuosity, with complex rhyme and metrical schemes and intricate structures. Although only a few of his poems are still extant, his gifts have impressed successive generations of poets: Dante, Petrarch and Ezra Pound all held him in the highest regard.

Daniel is generally credited with inventing the sestina and his poem ‘Lo ferm voler qu’el cor m’intra’, dating from around 1200, is the earliest known example of this poetic form. A sestina consists of six stanzas of six lines each, with each stanza featuring the same end-words in a set sequence of permutations.

In addition, a traditional sestina concludes with a three-line envoi that contains some or all of these end-words. If we denote the end-words by the letters A, B, C, D, E, F, then they rotate through the stanzas in the following sequence:

You will see from this scheme that the order of the end-words is unique to each stanza. (The Wikipedia article on the sestina includes interesting alternative presentations of the pattern of end-words.)

The tight constraints make sestinas challenging but also fun to write. Dante and Petrarch both composed a number of sestinas. Over the centuries numerous poets have taken on this technically demanding form, including writers as diverse as Sir Philip Sidney, Rudyard Kipling, Seamus Heaney and Elizabeth Bishop. Ruth Holzer’s ‘For Dylan Thomas on His Hundredth Birthday’is a fine example by a gifted contemporary poet, with end-words wild, sky, end, hills, wave, love. Here are the opening stanza and the envoi:

In old Carmarthenshire a boy ran wild
beneath the lamb-white clouds and larking sky,
not knowing that his paradise would end,
that he would lose those sheltering green hills,
the bay and every green transparent wave,
that innocence would vanish, and all love.


A hundred autumns fill the sky. Still coursing wild,
his words are foxes in the hills. They never end,
his psalms of love, his praise of star and wave.  

The form has given rise to many adaptations, such as Swinburne’s rhyming double sestina ‘The Complaint of Lisa’ or Marvin Thompson’s ‘Triptych’, in which each of the nine stanzas has seven lines. ‘Six Words’ by Lloyd Schwartz is a delightfully witty sestina consisting entirely of permutations of the words yes, no, maybe, sometimes, always and never, made meaningful by adroit use of font variation and punctuation. 

The American poet Marie Ponsot invented a condensed form called the tritina, which she described (somewhat inaccurately from a mathematical point of view) as ‘the square root of the sestina’. A tritina consists of three tercets with three end-words in the sequence 123, 312, 213, and a single final line containing all three end-words. The result can be charming, lighter and more playful than a sestina: Tamar Yosellof has an interesting discussion on this form which includes examples by Ponsot, Susan Watson and Yosellof herself.

This leads to the question: for what numbers m is an m-tina possible? The question was first investigated in the 1960s by the French poet Raymond Queneau, cofounder of the Oulipo movement, and his mathematician colleague Jacques Roubaud, and has continued to be a fruitful topic of mathematical research. (For the interested reader, links to two papers on this subject are given in the references, both of which include tables of suitable m-tina numbers. In practice however writing, say, a 194-tina is definitely not for the faint-hearted!)

Queneau was fascinated by the creative possibilities of using mathematics in his writing. In 1961 he published ‘Cent mille milliards de poémes’ (‘A Hundred Thousand Billion Poems’), a sequence of ten sonnets constructed in such a way that any given line from one sonnet can be interchanged at random with the corresponding line from any of the other sonnets. There are therefore 10 possibilities for the first line, 10 for the second line and so on, generating a total of 1014 distinct sonnets. No one has ever read all the permutations in their entirety: by Queneau’s own estimate it would take 190,258,751 years, reading night and day without a break, to do so. 

Francois Le Lionnais, co-founder with Queneau of the Oulipo movement, devised the concept of a multi-choice narrative, in which the reader can choose a route through a story or poem. JoAnne Growney shares a fine example of this approach on her blog ‘Intersections – Poetry with Mathematics’: ‘Multiple Choice’ by Harry Mathews is accompanied by a decision tree illustrating the possible paths.

Phil Vernon’s ‘Metro-man’, from his collection Poetry After Auschwitz, also offers a choice of routes through a poem. There are five stanzas which can be read in any order, giving 5×4×3×2×1=120 different versions. The non-linear layout invites the reader to explore the multiple options open to them.

The lipogram is another form of constrained writing that was taken up by the Oulipo movement, although its roots go back to ancient Greece. The word lipogram is derived from the Greek λειπογράμματος (leipogrammatos), meaning ‘to leave out a letter’ and describes a piece of writing in which certain letters of the alphabet are omitted. My poem ‘Scylla and Charybdis’ is a lipogram in which each word is a permutation of some or all of the letters in the title: a, b, c, d, h, i, l, n, r, s and y. Experimental poet Luke Bradford’s Zoolalia is a collection of lipograms constrained by the letters that make up the Latin names of various animal species. The result is lyrical and evocative, as in his ‘Arctic Hare’, Lepus arcticus:

a restless, alert spirit,
it traces a trail,
races past a pastel icescape as pale as erasure.

An anagram is a special case of a lipogram, in which all the letters of a word or phrase are rearranged to form a different word or phrase. In 1936 the American lexicographer and cryptographer David Shulman wrote a sonnet called ‘Washington Crossing the Delaware’, with each of the fourteen lines an exact rearrangement of the letters that constitute the title. 

The Oulipian poet Michelle Grangaud specialises in anagrams: her book Stations is a collection of poems that are anagrams of the names of Paris metro stations.  She has invented a poetic form known as a sestanagrammatina, which is a combination of the sestina and the anagram. Each line in her sestina ‘Le grand incendie de Londres’is an anagram of the poem’s title.

Anthony Etherin has written a number of anagram poems, including sonnets, a villanelle and a sestanagrammatina for Pablo Picasso. From his collection Stray Arts (and Other Inventions), here’s his aptly named ‘Permutations’:

Atoms erupt in
mutant prose. I
turn a poem – its
matter is upon 
me, to trap us in
utopian terms….

At resumption,
I must open art,
or input a stem
torn up as time – 
use important

In his most recent publication Etherin has pushed the poetic possibilities of the anagram even further. The Utu Sonnets is a sequence of seven sonnets, each of which – in addition to other constraints – is an exact anagram of all the others. You can read my review of this remarkable work here.

In the 1950s the innovative artist, writer and sound poet Brion Gysin experimented with what is known as the cut-up technique, cutting up and rearranging pieces of text. This led him to write a series of permutation poems, each consisting of permutations of the words in the title. Together with the mathematician Ian Sommerville he pioneered the use of computer programming to generate permutated text: an early example of the application of technology in poetry. Gysin’s most famous permutation poem is entitled ‘I AM THAT I AM’: it has several versions, which you can read, along with his other permutation poems, here.

Many of Gysin’s books were published by the London-based Writer’s Forum, established by the publisher and experimental poet Bob Cobbing. Cobbing’s concrete ‘Square Poem’ (which also features in my blog on square poems) has 12 lines, each of which is a permutation of the words ‘This is a square poem’.

Howard Bergerson, editor of Word Ways – a journal devoted to recreational linguistics – invented the vocabularyclept poem, which is formed by rearranging the words of an existing poem. In 1969 he challenged the journal’s readers to construct a new poem using the vocabulary of his own 24 lines long ‘Winter Retrospect’. A solution was submitted by the English poet, puzzle enthusiast and gifted recreational mathematician J. A. Lindon, who gave his version the same title even though he had not referred to Bergerson’s original.  Lindon’s account of how he went about finding his solution makes fascinating reading. 

More recently, mathematical educator and poet Susan Gerofsky has drawn inspiration from ancient traditions of bellringing, with their complex combinatorial patterns, to create poems based on word permutations. In a 2018 paper co-authored with three others, she considers how the Plain Hunt on 4 can be used in a variety of creative applications, including the writing of poetry. Her ‘Desert Poem’ follows the Plain Hunt on 4 structure as shown in the accompanying diagram. (In a fine example of multidisciplinary collaboration, Gerofsky and ceramic artist Nevena Tadic jointly created a beautiful ceramic mobile of this poem.)

Wings over dry land
Over wings, land dry
Over land, wings dry
Land over dry wings
Land dry over wings
Dry land wings over
Dry wings land over
Wings dry over land --
Wings over dry land.

Stephanie Strickland’s innovative collection Ringing the Changes also draws on bell-ringing traditions, taking as its starting point permutations of ringing a set of seven bells in sequence (a process known as change ringing). In place of bells, however, Strickland has used shareable code to generate text. As she notes in her Afterword, ‘Sounding from a bell tower changes are samples of sound, but in this book they are samples of language.’ The effect is an intricate weaving of juxtaposed texts from a range of sources, inviting the reader to reflect on change, shifting perspectives and interconnectivity in a dynamic context. Strickland gives us a sense of this approach in Liberty Ring! which she describes as ‘a toy interactive companion to Ringing the Changes’. With each ‘sounding’ of the bell (by clicking on it with the mouse cursor) a set of seven statements appears on the computer screen for us to read and contemplate, in an experience that is both playful and profound.

There has been no space here to mention the use of permutations in rhyme and metrical patterns, in German and Russian combinatorial literature or in other standard poetic forms such as the villanelle and pantoum (links on some of these topics are given in the list of references). From ancient traditions and mediaeval troubadour songs to contemporary code-generated text, permutations and poetry interweave in creative works of dazzling virtuosity.  

With thanks to Ruth Holzer, Luke Bradford, Anthony Etherin and Susan Gerofsky for giving me permission to quote their poetry.  


Birken, Marcia and Coon, Anne C. (2008) Discovering Patterns in Mathematics and Poetry. Amsterdam, Rodopi.

Bonch-Osmolovskaya, Tatiana (2010) ‘Some Possibilities of Russian Combinatorial Literature’. Bridges 2010: Mathematics, Music, Art, Architecture, Culture. Available at

Bradford, Luke (2021) Zoolalia. Penteract Press

Champneys, A. R., Hjorth, P. G., & Man, H. (2018). ‘The numbers lead a dance: Mathematics of the Sestina’. In 

Non-linear partial differential equations, mathematical physics, and stochastic analysis: the Helge Holden anniversary volume (pp. 55-71). European Mathematical Society Publishing House. Available online at

Cramer, Florian (2000) ‘Combinatory Poetry and Literature in the Internet’. Available at 

Dumas, Jean-Guillaume. (2008). ‘Caractérisation des Quenines et leur représentation spirale. Mathématiques et sciences humaines’. 184. 10.4000/msh.10946. Available at:

Etherin, Anthony (2019) Stray Arts (and Other Inventions). Penteract Press

Etherin, Anthony (2021) The Utu Sonnets. Penteract Press

Gerofsky, Susan (2020) ‘Two New Combinatoric Poetry Forms:
Braided Bellringing PH4 Poems & Anagrammatic, Anglo Saxon- inspired Poems’.Bridges 2020 Conference Proceedings. Available at

Gerofsky, Susan (2021) Seven Strands of Alphabetical Braided Crows. (Film) Available at

Gerofsky, S., Knoll, E., Taylor, T. and Campbell-Cousins, A. (2018) ‘Experiencing Group Structure: Observing, Creating and Performing the Plain Hunt on 4 via Music, Poetry, Visual and Culinary Arts’. Bridges 2018 Conference Proceedings. Available at 

Growney, JoAnne: Intersections – Poetry with Mathematics. Available at

May D. (2020) ‘Poems Structured by Mathematics’ in Sriraman B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham.

Strickland, Stephanie (2020) Ringing the Changes. Denver, Counterpath

Terry, Philip (ed) (2019) The Penguin Book of Oulipo. Penguin Random House UK

Vernon, Phil (2020) Poetry After Auschwitz. London, SPM Publications

Revised version posted on 28th April 2021.

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