COMPUTER PROGRAMMING
VOLUME 4 PRE-FASCICLE 4B
A DRAFT OF SECTION 7.2.1.7:
HISTORY OF COMBINATORIAL
GENERATION
DONALD E.KNUTH StanfordUniversity
ADDISON{WESLEY
6
77
Seealsohttp://www-s-faulty. sta nf or d. ed u/~ kn ut h/ sg b.h tm lforinformation
aboutTheStanfordGraphBase,inludingdownloadablesoftwarefordealingwiththe
graphsusedinmanyoftheexamplesinChapter7.
Anotherpage,http://www-s-faulty .s ta nf or d.e du /~ kn ut h/ pro gr am s. ht ml, on-
tainsauxiliaryprogramsbytheauthor.
Seealsohttp://www-s-faulty. sta nf or d. ed u/~ kn ut h/ mm ixw ar e. ht ml fordown-
loadablesoftwaretosimulatetheMMIXomputer.
Copyright
2004byAddison{Wesley
Allrightsreserved.Nopartofthispubliationmaybereprodued,storedinaretrieval
system,ortransmitted,inanyform, orby anymeans,eletroni, mehanial,photo-
opying, reording, or otherwise, without the prior onsent of the publisher, exept
that the oÆial eletroni le may be used to print single opies for personal (not
ommerial)use.
Zerothprinting(revision7),28Otober2005
I like to work in a variety of elds
inorder to spreadmy mistakes morethinly.
| VICTOR KLEE (1999)
This booklet ontains draft material that I'm irulating to experts in the
eld, in hopes that they an help remove its most egregious errors before too
many other people see it. I am also, however, posting it on the Internet for
ourageous and/or random readers who don't mind the risk of reading a few
pagesthathavenotyetreahedaverymaturestate. Beware: Thismaterialhas
notyetbeenproofreadasthoroughlyasthemanusriptsofVolumes1,2,and3
were at the time of theirrst printings. And those arefully-hekedvolumes,
alas,were subsequentlyfoundtoontainthousandsofmistakes.
Giventhisaveat, Ihopethatmyerrorsthistime willnotbesonumerous
and/orobtrusivethatyouwillbedisouragedfromreadingthematerialarefully.
Ididtrytomakeitbothinterestingandauthoritative,asfarasitgoes. Butthe
eldissovast,Iannothopetohavesurroundeditenoughtoorralitompletely.
ThereforeIbegyoutoletmeknowaboutanydeieniesyoudisover.
Toputthematerialinontext,thisisSetion7.2.1.7ofalong,longhapter
onombinatorialalgorithms. Chapter7willeventuallyllthreevolumes(namely
Volumes 4A, 4B, and 4C), assuming that I'm able to remain healthy. It will
begin with a short review of graph theory, with emphasis on some highlights
of signiant graphsin theStanford GraphBase, from whih I will bedrawing
manyexamples. Then omesSetion 7.1, whih deals withthetopi of bitwise
manipulations. (I drafted about 60 pages about that subjet in 1977, but
those pagesneedextensiverevision; meanwhile I'vedeidedto workfor awhile
on the material that follows it, so that I an get a better feel for how muh
to ut.) Setion 7.2 is about generating all possibilities, and it begins with
Setion7.2.1: GeneratingBasiCombinatorialPatterns|whih,inturn,begins
withSetion 7.2.1.1, \Generatingalln-tuples,"Setion7.2.1.2, \Generatingall
permutations," :::, Setion 7.2.1.6, \Generating all trees." (Readers of the
presentbookletshouldhavealreadylookedatthosesetions,draftsofwhihare
availableasPre-Fasiles2A,2B,3A,3B,and4A.)Thestageisnowset forthe
mainontentsofthisbooklet,Setion7.2.1.7: \Historyandfurther referenes."
Setion 7.2.2will dealwithbaktrakingingeneral. Andso it will ontinue,if
allgoeswell; anoutlineofthe entireChapter 7 asurrentlyenvisaged appears
onthetaop webpagethatisitedonpageii.
WritingabouthistoryisextraordinarilydiÆult,notonlybeausethesoure
materialsarewidelysatteredbutalsobeauseImustoperateatthelimitofmy
ability to understand languages other than English. Furthermore, fats about
reallifearemuhmoreompliatedthanfatsaboutmathematis. Nosummary
anadequatelyonveythetruefeelingsofaneraorthetruespiritofa ulture,
yet the story that I'm trying to tell in this setion overs many enturies of
developmentinmanydierentpartsoftheworld. Thestoryisfasinating,and
many partsof itdo notseemtohavebeentoldbefore, at leastnotinEnglish.
ThereforeI'mkeentohaveprofessionalhistoriansofmathematistakealookat
whatI'vebeenabletopiee together,hopingthattheywill notbetoo shoked
byblundersthathaveresultedfrommypresentignoraneand/orinompetene.
I hopealso to get advie from people of many dierent ultures whoknow of
relevanttraditionsthathavenotyetbeenwellstudiedbyprofessionalhistorians.
Theanswertoexerise 6posestwo historialproblemsthat Ihaven'tbeen
abletoresolve. Iurgentlyneedyourhelpalsowithrespettosomeexerisesthat
ImadeupasIwaspreparingthismaterial. Iertainlydon'tliketoreeiveredit
forthingsthathavealreadybeenpublishedbyothers,andmostoftheseresults
are quite natural \fruits" that were just waiting to be \pluked." Therefore
pleasetellmeifyouknowwhodeservestoberedited,withrespettotheideas
foundinexerises2,8,10,17,20, 26,and/or27.
Ishallhappilypayander'sfeeof$2.56foreaherrorinthisdraftwhenitis
rstreportedtome,whetherthaterrorbetypographial,tehnial,orhistorial.
ThesamerewardholdsforitemsthatIforgottoputintheindex. Andvaluable
suggestions forimprovementsto the text areworth 32/eah. (Furthermore, if
younda bettersolutiontoanexerise,I'llatuallyrewardyouwithimmortal
gloryinsteadofmeremoney,bypublishingyournameintheeventualbook: )
Cross referenes to yet-unwritten material sometimes appear as `00'; this
impossiblevalueisaplaeholderfortheatualnumberstobesuppliedlater.
Happyreading!
Stanford,California D.E.K.
12Otober 2004
[This subjet℄ has a relation
to almosteveryspeies of useful knowledge
that themindof man an beemployed upon.
| JAMES BERNOULLI,Ars Conjetandi (1713)
7.2.1.7. History and further referenes. Earlyworkon thegeneration of
ombinatorialpatterns beganas ivilizationitself wastaking shape. The story
isquitefasinating,andwewillseethatitspansmanyulturesinmanypartsof
theworld,withtiestopoetry,musi,andreligion.Thereisspaeheretodisuss
onlysome oftheprinipalhighlights;but perhaps a fewglimpsesinto thepast
willstimulatethereadertodigdeeperintotherootsofthesubjet,astheworld
getseversmallerandasglobalsholarshipontinuestoadvane.
Lists of binary n-tuples an be traed bak thousands of years to anient
China,India,and Greee. Themostnotablesoure|beauseitstillis a best-
sellingbook inmodern translations|is the Chinese I Ching or Yijing, whose
namemeans\theBibleofChanges." Thisbook,whihisoneofthevelassis
ofConfuianwisdom,onsistsessentiallyof2 6
=64hapters;andeahhapter
is symbolizedby a hexagramformed from sixlines, eah of whih is either
(\yin")or (\yang"). Forexample,hexagram1ispureyang, ; hexagram2
ispureyin, ;andhexagram64intermixesyinandyang,withyangontop: .
Hereistheompletelist:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
(1)
Thisarrangementof the64possibilitiesisalledKingWen's ordering,beause
thebasitextoftheIChinghastraditionallybeenasribedtoKingWen(.1100
B.C.),thelegendaryprogenitoroftheChoudynasty. Anienttextsare,however,
notoriouslydiÆultto datereliably,and modernhistorianshavefoundnosolid
evidenethatanyoneatuallyompiledsuhalistofhexagramsbeforethethird
enturyB.C.
Notiethatthehexagramsof(1)ourinpairs: Thosewithoddnumbersare
immediatelyfollowedbytheirtop-to-bottomreetions,exeptwhen reetion
would make no hange; and the eight symmetrial diagrams are paired with
their omplements (1 = 2, 27 = 28, 29 = 30 , 61 = 62). Hexagrams that are
omposedfromtwotrigramsthatrepresentthefourbasielementsheaven( ),
earth( ),re( ),andwater( )havealsobeenplaedjudiiously. Otherwise
thearrangement appears to beessentially random, as if a person untrainedin
mathematis kept listing dierent possibilities until being unable to ome up
with any more. A few intriguingpatterns do exist between the pairs, but no
morethanarepresentbyoinideneinthedigitsof(see3.3{(1)).
Yinand yang represent omplementaryaspetsof theelementaryfores of
nature,alwaysintension,alwayshanging. TheIChing issomewhatanalogous
toathesaurusinwhihthehexagramsserveasanindextoaumulatedwisdom
aboutfundamentaloneptslikegiving( ), reeiving( ), modesty ( ),joy
( ), fellowship ( ), withdrawal ( ), peae ( ), onit ( ), organization
( ), orruption ( ), immaturity ( ), elegane ( ), et. One an hoose
a pair of hexagrams at random, obtaining the seond from the rst by, say,
independently hanging eah yin to yang (or vie versa) with probability1/4;
this tehnique yields 4096 ways to ponder existential mysteries, as well as a
Markovproessbywhihhangeitself mightperhapsgivemeaningtolife.
A stritly logialwayto arrange thehexagrams was eventuallyintrodued
aboutA.D.1060byShaoYung. Hisordering,whihproeededlexiographially
from to to to to toto to (readingeah hexagramfrom
bottomtotop),wasmuhmoreuser-friendlythantheKingWenorder,beause
a random pattern ould now be found quikly. When G. W. Leibniz learned
aboutthissequeneofhexagramsin1702,hejumpedtotheerroneousonlusion
that Chinese mathematiians had one been familiar with binary arithmeti.
[SeeFrankSwetz, Mathematis Magazine 76 (2003), 276{291. Further details
abouttheIChing anbefound,forexample,inJosephNeedham'sSieneand
CivilisationinChina2(CambridgeUniversityPress,1956),304{345;R.J.Lynn,
TheClassiofChanges(NewYork:ColumbiaUniversityPress,1994).℄
AnotheranientChinesephilosopher,YangHsiung,proposedasystembased
on81ternarytetragramsinsteadof64binaryhexagrams. HisCanonofSupreme
Mystery,written.2B.C.,hasreentlybeentranslatedintoEnglishbyMihael
Nylan (Albany, NewYork: 1993). Yang desribed a omplete, hierarhial ter-
narytreestrutureinwhihthereare3regions,with3provinesineahregion,
3departmentsineahprovine,3familiesineahdepartment,and9shortpoems
alled\appraisals"foreah family,hene729appraisalsinall|makingalmost
exatly2appraisalsforeveryday intheyear. Histetragramswerearrangedin
stritlexiographi order whenread top-to-bottom: , , , , , , ,
:::, . Infat,asexplainedonpage28ofNylan'sbook,Yangpresentedasimple
waytoomputetherankofeahtetragram,asifusingaradix-3numbersystem.
ThushewouldnothavebeensurprisedorimpressedbyShaoYung'ssystemati
orderingofbinaryhexagrams,although Shaolivedmorethan1000yearslater.
Indianprosody. Binaryn-tupleswerestudiedinaompletelydierentontext
bypunditsinanientIndia,whoinvestigatedthepoeti metersofsaredVedi
hants. Syllables in Sanskrit are either short (.) or long (_), and the study
of syllable patterns is alled \prosody." Modern writers use the symbols ^
and instead of. and _. A typial Vedi verse onsists of four lines with
nsyllablesperline,forsomen8;prosodiststhereforesoughtawaytolassify
all2 n
possibilities. ThelassiworkChandah
.
sastra byPingala,_ writtenbefore
A.D.400andprobablymuhearlier(theexatdateisquiteunertain),desribed
proedures by whih one ouldreadilyndtheindex k ofanygivenpattern of
^sand s,aswellastondthekthpattern,givenk. Inotherwords,Pingala_
explainedhow torank anygiven patternas wellas to unrank anygivenindex;
thushewentbeyondtheworkofYangHsiung,whohadonsideredrankingbut
not unranking. Pingala's_ methods were also related to exponentiation, as we
havenotedearlierinonnetionwithAlgorithm4.6.3A.
The next important step was taken by a prosodist named Kedara in his
workVr
.
ttaratnakara,thoughtto havebeenwritteninthe8thentury. Kedara
gave a step-by-step proedure for listing all then-tuples from ::: to
^ ::: to ^ ::: to ^^ ::: to ^::: to ^ ^:::
toto^^^:::^,essentiallyAlgorithm7.2.1.1Mintheaseofradix2. His
method may well have beenthe rst-ever expliit algorithm for ombinatorial
sequenegeneration. [SeeB.vanNooten,J.IndianPhilos.21(1993),31{50.℄
Poeti meters an also be regardedas rhythms,with one beat foreah ^
andtwo beatsforeah . Ann-syllablepatternan involvebetweennand2n
beats,butmusialrhythmssuitableformarhingordaninggenerallyarebased
ona xed numberof beats. Therefore itwas natural to onsider theset of all
sequenesof^sand sthathaveexatlymbeats,forxedm. Suh patterns
are now alledMorse ode sequenes of length m, and we know from exerise
4.5.3{32 that there are exatly F
m+1
of them. For example, the 21 sequenes
whenm=7are
^ , ^ ,^^^ , ^ , ^^ ^ ,
^ ^^ , ^^ ^ ,^^^^^ , ^,
^^ ^,^ ^ ^, ^^ ^, ^^^^ ^,
^ ^^, ^ ^^,^^^ ^^, ^^^,
^^ ^^^,^ ^^^^, ^^^^^,^^^^^^^.
(2)
InthiswayIndianprosodistswereledtodisovertheFibonaisequene,aswe
haveobservedinSetion1.2.8.
Moreover, the anonymous author of Prakr
.
ta Paingala (.1320) disovered
elegantalgorithmsfor rankingandunranking withrespet tom-beatrhythms.
To nd thekth pattern, one startsby writingdown m^s, then expressesthe
dierened=F
m+1
kasa sumofFibonainumbersF
j
1
++F
j
t
;hereF
j
1
isthelargestFibonainumberthatisdandF
j
2
isthelargestd F
j
1 ,et.,
ontinuinguntiltheremainderiszero. Thenbeatsj 1andjaretobehanged
from^^ to , forj =j
1 , :::, j
t
. Forexample, togetthe 5thelementof (2)
weompute21 5=16=13+3=F
7 +F
4
;theansweris^^ ^ .
A few yearslater, Narayan
. a Pan
. d
.
itatreated the more generalproblem of
ndingall ompositions of m whose parts are q, where q is any givenposi-
tiveinteger. As a onsequene he disovered the qth-order Fibonai sequene
5.4.2{(4), whih was destined to be used 600 years later in polyphase sorting;
he also developed the orresponding ranking and unranking algorithms. [See
ParmanandSingh,HistoriaMathematia12 (1985),229{244, andexerise16.℄
Pingala_ gavespeialode-namestoallthethree-syllablemeters,
= m (m), ^ = t (t),
^ = y (y), ^ ^ = j(j),
^ = r (r), ^^ = B (bh),
^^ = s(s), ^^^ = n (n),
(3)
and students of Sanskrit have been expeted to memorize them ever sine.
Somebody long ago devised a lever way to reall these odes, by inventing
thenonsense wordyamatarajabhanasalagam (ymAtArAjBAnslgAm);thepoint
isthatthetensyllablesofthiswordan bewritten
ya
^
ma ta ra ja
^ bha na
^ sa
^ la
^ gam
(4)
and eah three-syllable pattern ours just after its ode name. Theorigin of
yama:::lagam is obsure, but Subhash Kak [Indian J. History of Siene 35
(2000),123{127℄ hastraedit bakat leastto C. P.Brown'sSanskrit Prosody
(1869),page28;thusitqualiesastheearliestknownappearaneofa\deBruijn
yle"that enodesbinaryn-tuples.
Meanwhile, in Europe. Inasimilarway, lassiGreekpoetry wasbasedon
groupsofshortand/orlongsyllablesalled\metrialfeet,"analogoustobarsof
musi. Eah basi typeof foot aquired a Greekname; forexample, two short
syllables`^^'werealledapyrrhi,andtwolongsyllables` 'werealleda
spondee,beausethoserhythmswereusedrespetivelyinasongofwar(purrqh)
orasongofpeae(sponda). Greeknamesformetrifeetweresoonassimilated
intoLatinandeventuallyintomodernlanguages, inludingEnglish:
^ arsis
thesis
^^ pyrrhi
^ iambus
^ trohee
spondee
^^^ tribrah
^^ anapest
^ ^ amphibrah
^ bahius
^^ datyl
^ amphimaer
^ palimbahius
molossus
^^^^ proeleusmati
^^^ fourth pon
^^ ^ third pon
^^ minor ioni
^ ^^ seond pon
^ ^ diiambus
^ ^ antispast
^ rstepitrite
^^^ rstpon
^^ horiambus
^ ^ ditrohee
^ seond epitrite
^^ majorioni
^ third epitrite
^ fourth epitrite
dispondee
(5)
Alternative names, like \horee" instead of \trohee," or \reti" instead of
\amphimaer,"werealsoinommonuse. Moreover,bythetimeDiomedeswrote
his Latin grammar (approximately A.D. 375), eah of the 32 ve-syllable feet
hadaquiredatleastonename. Diomedesalsopointedouttherelationbetween
omplementarypatterns; hestated forexamplethat tribrahand molossusare
\ontrarius," as areamphibrah and amphimaer. Buthe alsoregardeddatyl
as the ontrary of anapest,and bahius as the ontrary of palimbahius, al-
thoughtheliteralmeaningofpalimbahiusisatually\reversebahius." Greek
prosodistshadnostandardorderinwhihtolisttheindividualpossibilities,and
theform ofthenamesmakesitlearthatnoonnetiontoaradix-twonumber
systemwas ontemplated. [See H. Keil,Grammatii Latini 1 (1857),474{482;
W.vonChrist,MetrikderGriehenundRomer (1879),78{79.℄
Surviving fragmentsof a work by Aristoxenus alled Elementsof Rhythm
(.325 B.C.) showthat thesame terminologywas applied also to musi. And
indeed,thesametraditionslivedonaftertheRenaissane;forexample,wend
onpage 32of AthanasiusKirher's Musurgia Universalis 2(Rome: 1650),and
Kirherwentontodesribeallofthethree-noteandfour-noterhythmsof(5).
Earlylistsofpermutations. We'vetraedthehistoryofformulasforounting
permutations in Setion 5.1.2; but nontrivial lists of permutations were not
publisheduntilhundreds ofyearsafter theformulan!wasdisovered. Therst
suhtabulationurrentlyknownwasompiledbytheItalianphysiianShabbetai
DonnoloinhisommentaryonthekabbalistiSeferYetzirah,writteninA.D.946.
Table1showshislistforn=5asitwassubsequentlyprintedinWarsaw(1884).
(TheHebrewletters inthis table are typeset ina rabbinial font traditionally
usedfor ommentaries; notie that theletter hangesits shapeto when it
appears at the left end of a word.) Donnolowent on to list 120 permutations
of the six-letter word , all beginning with shin ( ); then he noted that
120more ouldbeobtainedwitheahofthe othervelettersinfront, making
720inall. His listsinvolvedgroupingsofsixpermutations,but inahaphazard
fashion that led him into error (see exerise 4). Although he knew how many
permutationsthereweresupposedtobe,andhowmanyshouldstartwithagiven
letter,heevidentlydidn'thaveanalgorithmforgeneratingthem.
Table1
AMEDIEVAL LIST OF PERMUTATIONS
Aompletelistofall720 permutationsoffa;b;;d;e;fgappearedonpages
668{671ofJeremiasDrexel'sOrbisPhaethon(Munih:1629;alsoonpages526{
531of theCologneeditionin1631). Heoereditas proofthat a manwith six
guests ouldseat them dierentlyat lunh and dinner every day for a year|
altogether360 days, beause therewerevedays offasting duringHolyWeek.
Shortly afterwards, Marin Mersenne exhibited all 720 permutations of the six
tones fut;re;mi;fa;sol;lag, on pages 111{115 of his Traitez de la Voix et des
Chants (Volume 2 of Harmonie Universelle, 1636); then on pages 117{128 he
presentedthesamedatainmusialnotation:
Drexel's table was organized lexiographially by olumns; Mersenne's tables
werelexiographiwithrespettotheorderut<re<mi<fa<sol<la,begin-
ningwith\ut,re,mi,fa,sol,la"andendingwith\la,sol,fa,mi,re,ut." Mersennealso
prepared a \grandetimmense" manusript that listedall 40,320permutations
ofeight noteson672foliopages,followedbyrankingandunrankingalgorithms
[BibliothequenationaledeFrane,FondsFranais, no.24256℄.
We sawin Setion 7.2.1.2 that the important ideaof plain hanges, Algo-
rithm7.2.1.2P,was inventedinEnglandafewyearslater.
Methods for listing all permutations of a multiset with repeated elements
wereoftenmisunderstoodbyearlyauthors. Forexample,whenBhaskaraexhib-
itedthepermutationsoff4;5;5;5;8ginsetion271ofhis L
l avat
(.1150),he
gavetheminthefollowingorder:
48555 84555 54855 58455 55485
55845 55548 55584 45855 45585
45558 85455 85545 85554 54585
58545 55458 55854 54558 58554
(6)
Mersenneused a slightlymore sensiblebut notompletely systematiorderon
page 131 of his book when he listed sixty anagrams of the Latin name IESVS.
When Athanasius Kirher wanted to illustrate the 30 permutations of a ve-
notemelodyonpages10and11ofMusurgiaUniversalis2(1650),thislakofa
systemgothimintotrouble(seeexerise5):
(7)
But John Wallis knew better. On page 117 of his Disourse of Combinations
(1685)heorretlylistedthe60anagramsof\messes" inlexiographiorder,if
weletm<e<s;andonpage126hereommendedrespetingalphabetiorder
\thatwemaybethemore sure,nottomiss any."
Wewill seelaterthat theIndian mathematiianNarayan
. a Pan
. d
.
itahadal-
readydevelopedatheoryofpermutationgenerationinthe14thentury,although
hisworkremainedalmosttotallyunknown.
Seki'slist. TakakazuSeki(1642{1708)wasaharismatiteaherandresearher
whorevolutionizedthe studyof mathematis in17th-enturyJapan. Whilehe
wasstudyingtheeliminationofvariablesfromsimultaneoushomogeneousequa-
tions, he was led to expressions suh as a
1 b
2 a
2 b
1 and a
1 b
2
3 a
1 b
3
2 +
a
2 b
3
1 a
2 b
1
3 +a
3 b
1
2 a
3 b
2
1
, whih we now reognize as determinants.
In1683 he published a booklet about this disovery, introduing an ingenious
shemeforlistingallpermutationsinsuhawaythathalfofthemwere\alive"
(even)andtheotherhalfwere\dead"(odd).Startingwiththeasen=2,when
`12'wasaliveand`21'was dead,heformulatedthefollowingrulesforn>2:
1) Takeeverylivepermutationforn 1,inreaseallitselementsby1,andinsert
1infront. Thisruleprodues(n 1)!=2\basipermutations"off1;:::;ng.
2) Fromeahbasipermutation,form2n othersbyrotation andreetion:
a
1 a
2 :::a
n 1 a
n
; a
2 :::a
n 1 a
n a
1
; :::; a
n a
1 a
2 :::a
n 1
; (8)
a
n a
n 1 :::a
2 a
1
; a
1 a
n a
n 1 :::a
2
; :::; a
n 1 :::a
2 a
1 a
n
: (9)
Ifnisodd,thoseintherstrowarealiveandthoseintheseondaredead;
ifniseven,thoseineahrowarealternativelyalive,dead,:::,alive,dead.
For example, when n = 3 the only basi permutation is 123. Thus 123, 231,
312arealivewhile321,132,213aredead,andwe'vesuessfullygeneratedthe
sixtermsof a 33 determinant. Thebasi permutations forn=4 are 1234,
1342,1423;andfrom,say,1342wegeta setofeight,namely
+1342 3421+4213 2134+2431 1243+3124 4321; (10)
alternatelyalive(+) anddead( ). A44 determinantthereforeinludes the
termsa
1 b
3
4 d
2 a
3 b
4
2 d
1
+ a
4 b
3
2 d
1
andsixteenothers.
Seki's rulefor permutation generation isquite pretty, but unfortunately it
hasa serious problem: It doesn't work when n > 4. His error seems to have
goneunreognizedforhundreds ofyears. [SeeY. Mikami,TheDevelopment of
Mathematis in China and Japan (1913), 191{199; Takakazu Seki's Colleted
Works(Osaka:1974),18{20, :: ;andexerises 7{8.℄
Listsofombinations. Theearliestexhaustivelistofombinations knownto
havesurvivedtheravagesoftimeappearsinthelastbookofSusruta'swell-known
Sanskrittreatise onmediine,Chapter63, writtenbeforeA.D.600andperhaps
muh earlier. Noting that mediine an be sweet, sour, salty, peppery, bitter,
and/or astringent, Susruta's book diligently listed the (15;20;15;6;1;6) ases
thatarisewhenthosequalitiesourtwo,three,four,ve,six,andoneatatime.
Bhaskararepeatedthisexampleinsetions110{114ofL
lavat
,andobserved
thatthesamereasoningappliestosix-syllablepoetimeterswithagivennumber
oflongsyllables. Buthesimplymentionedthetotals,(6;15;20;15;6;1),without
listingtheombinations themselves. Insetions274 and 275,heobservedthat
thenumbers(n)(n 1):::(n k+1)=(k(k 1):::(1))enumerateompositions
(thatis,orderedpartitions)aswellasombinations;againhegavenolist.
To avoidprolixity thisis treatedina brief manner;
for thesieneof alulationis an oeanwithout bounds.
| Bhaskara (.1150)
Anisolatedbut interestinglistofombinationsappearedintheremarkable
algebra text Al-Bahir 'l-h
.
isab (TheShining Book of Calulation), writtenby
al-Samaw'alof Baghdad when hewas only19 yearsold (1144). Inthe losing
partofthatworkhepresentedalistof 10
6
=210simultaneouslinearequations
in10unknowns:
Al-Samaw'al'sArabioriginal Equivalentmodernnotation
65 654321 m (1) x1+x2+x3+x4+x5+x6=65
70 754321 o (2) x1+x2+x3+x4+x5+x7=70
75 854321 ~ (3) x1+x2+x3+x4+x5+x8=75
.
.
.
.
.
.
91 1098764 ¢ (209)x
4 +x
6 +x
7 +x
8 +x
9 +x
10
=91
100 1098765 Ý (210)x
5 +x
6 +x
7 +x
8 +x
9 +x
10
=100 (11)
Eah ombination of ten things taken six at a time yielded one of his equa-
tions. Hispurposewasevidentlytodemonstratethatover-determinedequations
an still have a unique solution|whih in this ase was (x
1
;x
2
;:::;x
10 ) =
(1;4;9;16;25;10;15;20;25;5). [Salah Ahmad and Roshdi Rashed,Al-Bahir en
Algebred'As-Samaw'al(Damasus:1972),77{82, 248{231.℄
Rolling die. Some glimmerings of elementary ombinatoris arose also in
medievalEurope,espeiallyinonnetionwiththequestionoflistingallpossible
outomeswhen three dieare thrown. There are, of ourse, 8
3
= 56 ways to
hoose3thingsfrom6whenrepetitionsareallowed. GamblingwasoÆiallypro-
hibited;yetthese56waysbeameratherwellknown. InaboutA.D.965,Bishop
Wibold ofCambrai innorthernFrane deviseda gamealledLudus Clerialis,
so that members of the lergy ould enjoy rolling die while remaining pious.
Hisideawasto assoiateeah possiblerollwithone of56virtues,aording to
thefollowingtable:
q q q
love
q
q q
q
q q q
q q q
perseverane q q
q q
q
q q
q q
q hospitality q q
q
q q q
q q q q
q q q
mortiation
q q
q q
faith
q
q q q
q q q q
q kindness q q
q q
q
q q q
q q q
eonomy q q
q
q q
q q
q q q
q q
q innoene
q q
q q
q
hope
q
q q q
q q q
q q
q modesty q q
q q q
q q q q
q patiene q q
q
q q
q q
q q q q
q q q
ontrition
q q
q q q
q justie
q
q q q
q q q q
q q q
resignation q q
q q q
q q q
q q
q zeal q
q q
q q q
q q q
q q q
q q q
onfession
q q
q q
q q
q prudene q
q q
q q
q q q
q q
q gentleness q q
q q q
q q q q
q q q
poverty q q q
q q q q
q q q q
q maturity
q q
q q q
q q q
temperane q
q q
q q
q q q q
q q q
generosity q q
q q
q q
q q q
q q
q softness q q q
q q q q
q q q
q q
q soliitude
q
q q
q q
ourage q
q q q
q q q
q q q
q q q
wisdom q
q
q q
q q
q q q q
q q q
virginity q q q
q q q q
q q q q
q q q
onstany
q
q q
q q
q
peae q
q
q q
q q
remorse q q
q q q
q q q
q q q
q q q
respet q q q
q q q
q q
q q q
q q
q intelligene
q
q q
q q q
q hastity q q
q q
q q
q
joy q
q q
q q
q
q q
q
piety q
q q
q q q
q q
q q q q
q q q
sighing
q
q q
q q
q q
q mery q
q
q q
q q q
q sobriety q q
q
q q
q
q q q
q indulgene q q q
q q q q
q q q
q q q
q q q
weeping
q
q q
q q q
q q q
obediene q q
q q
q q
q q
q satisfation q q
q
q q
q
q q
q q
q prayer q q
q q
q q q
q q
q q q
q q
q heerfulness
q
q q
q
q q
q
fear q
q
q q
q q q
q q q
sweetness q q
q
q q
q
q q q
q q q
aetion q q
q q
q q q
q q
q q q q
q q q
ompassion
q
q q
q
q q q
q foresight q q
q q
q
q q
q
leverness q q
q
q q q
q q q q
q judgment q q
q q
q q q q
q q q
q q q
q q q
self-ontrol
q
q q
q
q q
q q
q disretion q q
q q
q
q q q
q simpliity q q
q
q q q
q q q
q q
q vigilane q q q
q q q
q q q
q q q
q q q
q q q
humility
Playerstookturns,andthersttorolleahvirtueaquiredit. Afterallpossibil-
itieshadarisen,themostvirtuousplayerwon. Wiboldnotedthatlove(aritas)
is the best virtue of all. He gavea ompliated soring system by whih two
virtuesouldbeombinedifthesumofpipsonallsixoftheirdie was 21;for
example,love+humility or hastity+intelligeneould bepairedin thisway,
andsuh ombinationsrankedaboveany individual virtue. Healsoonsidered
moreomplexvariantsofthegameinwhihvowelsappearedonthedieinstead
ofspots,so thatvirtuesouldbelaimediftheirvowelswerethrown.
Wibold's table of virtues was presented in lexiographi order, as above,
when it was rst desribed by Balderi in his Chronion Cameraense, about
150yearslater. [PatrologiaLatina134(Paris:1884), 1007{1016.℄ Butanother
medievalmanusriptpresentedthepossibledierollsinquitea dierentorder:
q q q
q q q
q q q
q q q
q q q
q q q
q q
q q
q q q
q q
q q q
q q
q q q q
q q q q
q q q q
q q q
q
q q
q
q q
q
q q
q q
q q
q q q
q q q
q q q
q q q
q q q
q q
q q
q
q q q
q q q
q q q
q q q
q q q
q q q q
q q q
q q q
q q q
q q
q
q q q
q q q
q q q
q q q
q q
q q q
q q q
q q q
q q q
q
q q
q q
q q q
q q
q q q q
q q q
q q
q q
q q q
q q
q q q q
q q q
q q
q q q
q q
q q q
q
q q
q q
q q q
q q
q q q
q q
q q
q q q
q q
q q
q q q
q q q q
q q q q
q q q
q q q
q q q q
q q q
q q
q q q q
q q q q
q q q
q
q q q
q q q q
q q q
q q q
q q q q
q q
q q
q
q q
q
q q q
q q q
q q
q
q q
q
q q
q q
q q q
q
q q
q
q q q
q q q
q
q q
q
q q
q q
q
q q
q
q
q q
q q
q q q
q q q
q q
q q
q q
q q
q
q q
q q
q q q
q q q
q q
q q
q
q q
q q
q q q
q q q
q q q
q q
q q
q q
q q q
q q q
q q q
q q
q
q q
q q
q q q
q q q
q q
q q
q q q q
q q q
q q
q q q q
q q q
q
q q q
q q q
q
q q
q q
q
q q
q
q q q
q q q
q q q
q q q
q q q
q q q
q q q
q q
q q q
q q q
q q
q
q
q q
q q
q q q
q
q
q q q
q q q
q q
q q
q q q
q
q q q
q q q
q q
q q
q q q
q q q
q q q
q q
q q
q q
q q q
q q q
q q
q
q q
q q
q q q
q
q q q
q q q
q
q q
q q
q q q q
q q q
q q
q q
q q q q
q q
q q q
q q q
q q q
q q q
q
q q q
q q q
q
q
q q q
q q q
q q
q
q q
q q
q q
q q q
q
q q
(12)
Inthisasetheauthor knewhow todealwith repeatedvalues, buthad a very
ompliated,adhowaytohandletheasesinwhihalldieweredierent. [See
D.R.Bellhouse,InternationalStatistialReview 68(2000),123{136.℄
An amusing poem entitled \Chaune of the Dyse," attributed to John
Lydgate, was writtenin the early 1400s for use at parties. Its opening verses
invite eah person to throw three die; then the remaining verses, whih are
indexedindereasinglexiographiorderfrom q q q
q q q
q q q
q q q
q q q
q q q
to q q q
q q q
q q q
q q q
q q
q q
q toto q q q
,
give56 harater skethes that light-heartedly desribe thethrower. [The full
text was published by E. P. Hammond in Englishe Studien 59 (1925), 1{16;
atranslationintomodernEnglishwouldbedesirable.℄
Iprayto god that euery wightmayaste
Vpon three dyseryght asis in hysherte
Whether he be rehelesse or stedfaste
Somootehe lawghen outher elles smerte
He that is giltyhis lyfe to onverte
Theythat introuthe hauesuredmany athrowe
Mootether haune fal asthey moote be knowe.
|The Chaune of the Dyse (.1410)
Ramon Llull. Signiant ripples of ombinatorial onepts also emanated
fromanenergeti and quixotiCatalanpoet,novelist,enylopedist,eduator,
mysti,andmissionarynamedRamonLlull (.1232{1316). Llull's approahto
knowledge was essentially to identifybasi priniplesand then to ontemplate
ombiningthem inallpossibleways.
For example, one hapter in his Ars Compendiosa Inveniendi Veritatem
(.1274)beganbyenumeratingsixteenattributesofGod: Goodness,greatness,
eternity,power,wisdom,love,virtue,truth,glory,perfetion,justie,generosity,
mery, humility, sovereignty, and patiene. Then Llull wrote 16
2
=120 short
essaysofabout80wordseah,onsideringGod'sgoodnessasrelatedtogreatness,
God'sgoodnessasrelatedtoeternity,andsoon,endingwithGod'ssovereigntyas
relatedtopatiene. Inanotherhapterheonsideredsevenvirtues(faith,hope,
harity,justie,prudene,fortitude,temperane)andsevenvies(gluttony,lust,
greed, sloth,pride, envy, anger), with 14
2
=91 subhapters todeal witheah
pairinturn. Other hapters were systematiallydividedina similar way, into
8
2
=28, 15
2
=105, 4
2
=6,and 16
2
=120subsetions. (Onewonders what
might have happened if he had beenfamiliar with Wibold's list of 56 virtues;
would hehaveproduedommentariesonall 56
2
=1540oftheirpairs?)
Fig. 44. Illustrationsinamanusriptpresentedby Ramon Llullto
thedogeofVeniein1280. [FromhisArsDemonstrativa,Bibliotea
Mariana,VI200,folio3 v
.℄
Llullillustratedhismethodologybydrawingirulardiagramslikethosein
Figure 44. Theleft-handirle inthis illustration, Deus, namessixteen divine
attributes|essentiallythesame sixteen listedearlier, exeptthat love(amor)
wasnowalledwill(voluntas),andthenalfourwerenowrespetivelysimpliity,
rank, mery, and sovereignty. Eah attribute is assigned a ode letter, and
the illustration depits their interrelations as the omplete graph K
16
on ver-
ties (B;C;D;E;F;G;H;I;K;L;M;N;O;P;Q;R). The right-hand gure, virtutes
etvitia,showsthesevenvirtues(b;;d;e;f;g;h)interleavedwiththesevenvies
(i;k;l;m;n;o;p);in theoriginalmanusript virtuesappeared inblue ink while
vies appeared in red. Notie that in this ase his illustration depited two
independentompletegraphs K
7
, one ofeah olor. (Henolonger bothered to
ompareeahindividualvirtuewitheahindividualvie,sineeveryvirtuewas
learlybetterthaneveryvie.)
Llull used the same approah to write about mediine: Instead of juxta-
posing theologial onepts, his Liber Prinipiorum Mediin (.1275) on-
sidered ombinations of symptoms and treatments. And he also wrote books
Fig.45. Llullianillustrations
fromamanusriptpresentedto
thequeenofFrane,.1325.
[BadisheLandesbibliothekKarls-
ruhe,CodexSt.Peterperg.92,
folios28 v
and39 v
.℄
onphilosophy, logi, jurisprudene, astrology, zoology, geometry, rhetori, and
hivalry|morethan200worksinall. Itmustbeadmitted,however,thatmuh
ofthismaterialwashighlyrepetitive;moderndataompressiontehniqueswould
probablyredue Llull'soutputtoasize muh lessthanthatof, say,Aristotle.
He eventually deided to simplify his system by working primarily with
groupsofninethings. See,forexample,Fig.45,whereirleAnowlistsonlythe
rstnine of God's attributes (B;C;D;E;F;G;H;I;K). The 9
2
= 36assoiated
pairs(BC;BD;:::;IK)appearinthestairstephartattherightofthatirle. By
addingtwomorevirtues,namelypatieneandompassion|aswellastwomore
vies, namelylying and inonsisteny|he ouldtreat virtuesvis-a-vis virtues
andviesvis-a-vis vies withthesame hart. Healso proposed usingthesame
hart to arry out an interesting sheme for voting, in an eletion with nine
andidates[see I.MLeanandJ.London, StudiaLulliana32(1992),21{37℄.
The enirled triangles at the lower left of Fig. 45 illustrate another key
aspetofLlull'sapproah. Triangle(B;C;D)standsfor(dierene,onordane,
ontrariness);triangle(E;F;G)standsfor(beginning,middle,ending);andtrian-
gle(H;I;K)standsfor(greater,equal,less). Thesethreeinterleavedappearanes
ofK
3
representthreekindsofthree-valuedlogi. Llullhadexperimentedearlier
with other suh triplets, notably `(true, unknown, false)'. We an get an idea
ofhowhe usedthe trianglesbyonsidering howhe dealt withombinationsof
thefour basi elements(earth, air,re, water): Allfour elementsare dierent;
earth is onordant with re, whih onords with air, whih onords with
water, whih onordswithearth; earth isontrary to air,and reis ontrary
to water; these onsiderations omplete an analysis with respet to triangle
(B;C;D). Turningtotriangle(E;F;G),henotedthatvariousproessesinnature
beginwith one element dominating another; then a transition or middle state
ours,until agoalisreahed, likeairbeoming warm. For triangle(H;I;K)he
said that in general we have re >air > water > earth with respet to their
\spheres," their \veloities," and their \nobilities"; nevertheless we also have,
for example, air >re with respet to supporting life, while airand rehave
equalvaluewhentheyareworkingtogether.
LlullprovidedthevertialtableattherightofFig.45asafurtheraid. (See
exerise11 below.) Healsointroduedmovableonentriwheels,labeledwith
the letters (B;C;D;E;F;G;H;I;K) and with other names, so that many things
ould be ontemplated simultaneously. In this way a faithful pratitioner of
the Llullian art ould be sure to have all the bases overed. [Llull may have
seen similar wheels that were used innearby Jewish ommunities;see M.Idel,
J.WarburgandCourtauldInstitutes51(1988),170{174andplates 16{17.℄
Severalenturieslater,AthanasiusKirherpublishedanextensionofLlull's
system as part of a large tome entitled Ars Magna Siendisive Combinatoria
(Amsterdam: 1669), with ve movable wheels aompanying page 173 of that
book. KirheralsoextendedLlull'srepertoireofompletegraphsK
n
byprovid-
ingillustrationsofompletebipartitegraphsK
m;n
;forexample,Fig.46istaken
from page 171 of Kirher's book, and his page 170 ontains a glorious piture
ofK
18;18 .
Fig.46. K9;9 aspre-
sentedbyAthanasius
Kirherin1669.
It is an investigative andinventive art.
When ideas areombined in allpossibleways,
thenewombinations start themindthinkingalongnovelhannels
and oneisled to disoverfresh truths andarguments.
|MARTIN GARDNER,Logi MahinesandDiagrams (1958)
The most extensive modern development of Llull-like methods is perhaps
TheShillingerSystemofMusialCompositionbyJosephShillinger(NewYork:
Carl Fisher, 1946), a remarkable two-volume work that presents theories of
rhythm, melody, harmony, ounterpoint,omposition,orhestration, et., from
a ombinatorial perspetive. On page 56, for example, Shillinger lists the 24
permutations offa;b;;dgin theGray-ode orderof plain hanges (Algorithm
7.2.1.2P);thenonpage57heappliesthemnottopithesbutrathertorhythms,
tothedurationsofnotes. Onpage364heexhibitsthesymmetrialyle
(2;0;3;4;2;5;6;4;0;1;6;2;3;1;4;5;3;6;0;5;1); (13)
a universal yle of 2-ombinations for the seven objets f0;1;2;3;4;5;6g; in
otherwords,(13)is anEulerian trailinK
7 : All
7
2
=21 pairsof digitsour
exatly one. Suh patterns are grist for a omposer's mill. But we an be
gratefulthatShillinger'sbetterstudents(likeGeorgeGershwin)didnotommit
themselvesentirelyto astritlymathematialsenseof aesthetis.
Taquet, van Shooten, and Izquierdo. Threeadditionalbooksrelatedto
ourstorywerepublishedduringthe1650s. AndreTaquetwroteapopulartext,
ArithmetiTheoriaetPraxis (Louvain:1656),that wasreprintedandrevised
oftenduringthenextftyyears. Neartheend,onpages376and377,hegavea
proedureforlistingombinationstwo ata time,thenthreeata time,et.
FransvanShooten'sExeritationesMathemati(Leiden:1657)wasmore
advaned. Onpage373 helistedallombinationsinanappealinglayout
a
b:ab
:a:b:ab
d:ad:bd:abd:d: ad:bd:abd
(14)
andheproeededonthenextfewpagesto extendthis patternto theletterse,
f,g,h,i,k,\etsiininnitum." Onpage376heobservedthatoneanreplae
(a;b;;d)by(2;3;5;7)in(14)togetthedivisorsof210 thatexeedunity:
2
3 6
5 10 15 30
7 14 21 42 35 70 105 210
(15)
Andonthefollowingpageheextendedtheideato
a
a:aa
b:ab: aab
:a:aa:b: ab:aab
(16)
therebyallowingtwoa's. Hedidn'treallyunderstandthisextension,though;his
nextexample
a
a:aa
a:aaa
b:ab:aab: aaab
b:bb:abb:aabb:aaabb
(17)
wasbothed,indiatingthelimitsofhisknowledgeatthetime. (Seeexerise13.)
Onpage411vanShootenobservedthattheweights(a;b;;d)=(1;2;4;8)
ouldbeassignedin(14),leadingto
1
2 3
4 5 6 7
8 9 10 11 12 13 14 15
(18)
afteraddition. Buthedidn'tseetheonnetionwithradix-2arithmeti.
Sebastian Izquierdo's two-volume work Pharus Sientiarum (Lyon: 1659),
\TheLighthouseofSiene,"inludedanielyorganizeddisussionofombina-
torisentitledDisputatio29,DeCombinatione. Hegaveadetaileddisussionof
four key parts of Stanley'sTwelvefold Way, namelythe n-tuples, n-variations,
n-multiombinations,and n-ombinationsof mobjetsthat appear intherst
tworowsandthersttwo olumnsofTable7.2.1.4{1.
InSetions81{84ofDeCombinationehelistedallombinationsofmletters
takennata time,for2n5 andnm9,alwaysinlexiographi order;
healsotabulated themfor m=10and20intheases n=2and 3. Butwhen
he listed the m n
variations of m things taken n at a time, he hose a more
ompliatedordering(seeexerise14).
Izquierdowas rstto disovertheformula
m+n 1
n
for ombinationsof m
thingstakennatatimewithunlimitedrepetition;thisruleappearedinx48{x51
ofhiswork. Butinx105,whenheattemptedtolistallsuhombinationsinthe
asen = 3,he didn't know that there was a simpleway to doit. In fat, his
listingofthe56asesform=6wasratherliketheold,awkwardorderingof(12).
Combinations with repetition were not well understood until James Ber-
noulli'sArsConjetandi, \TheArt ofGuessing," ame outin1713. InPart2,
Chapter 5, Bernoulli simply listed the possibilitiesin lexiographi order, and
showed thattheformula
m+n 1
n
followsbyindutionasaneasyonsequene.
[NioloTartagliahad,inidentally,omelosetodisoveringthisformulainhis
Generaltrattatodinumeri,etmisure 2(Venie:1556),17 r
and 69 v
;sohadthe
MaghrebimathematiianIbnMun`iminhis 13th-enturyFiqhal-H
. isab.℄
Thenullase. Beforeweonludeourdisussionofearlyworkonombinations,
we should notforget a small yetnoble steptakenby John Wallison page 110
of his Disourse of Combinations (1685), where he speially onsidered the
ombination ofm things taken0 at a time: \It ismanifest, That,if wewould
takeNone,thatis,ifwewouldleaveAll;thereanbebutoneasethereof,what
everbetheNumberofthingsexposed." Furthermore,onpage113,heknewthat
0
0
=1: \(for,here,totakeall,orto leaveall,isbutoneandthesame ase.)"
However,when hegavea table ofn! forn24, hedidnotgo so faras to
pointoutthat0!=1,orthatthereisexatlyonepermutationoftheemptyset.
The workofNarayan
.
a. AremarkablemonographentitledGan
.
itaKaumud
(\TreatiseonCalulation"),writtenbyNarayan
. aPan
. d
.
itain1356,hasreently
beome known in detail to sholars outside of India for the rst time, thanks
to anEnglishtranslationbyParmanandSingh[Gan
.
itaBharat 20(1998), 25{
82; 21 (1999), 10{73; 22 (2000), 19{85; 23 (2001), 18{82; 24 (2002), 35{98℄.
Chapter13ofhiswork,subtitledAnka_ Pasa(\ConatenationofNumbers"),was
devoted to ombinatorial generation. Indeed, although the97 \sutras" of this
hapterwereratherrypti,theypresentedaomprehensivetheoryofthesubjet
thatantiipateddevelopmentsintherestoftheworldbyseveralhundredyears.
Forexample,Narayan
.
adealtwithpermutationgenerationinsutras49{55a,
wherehegavealgorithmstolistallpermutationsofasetindereasingolexor-
der,togetherwithalgorithmstorankagivenpermutationandtounrankagiven
serialnumber. Inthis wayheessentiallydisoveredthefatorialrepresentation
ofpositiveintegers. Theninsutras57{60heextendedthealgorithmstohandle
generalmultisets;forexample,helistedthepermutationsoff1;1;2;4gas
1124;1214;2114;1142;1412;4112;1241;2141;1421;4121;2411;4211;
againindereasingolexorder.
Narayan
.
a's sutras88{92 dealt withsystematigenerationofombinations.
Besidesillustratingtheombinationsoff1;:::;8gtaken3ata time,namely
(678;578;478;:::; 134;124;123);
healsoonsideredabit-stringrepresentationoftheseombinationsinthereverse
order(inreasing olex):
(11100000;11010000;10110000;:::;00010011;00001011;00000111):
Healmost,butnotquite,disovered Theorem7.2.1.3L.
Thus we an legitimately regard Narayan
. a Pan
. d
.
ita as the founder of the
sieneofombinatorialgeneration|eventhough,likemanyotherpioneerswho
were signiantly\aheadof theirtime," hiswork onthesubjet neverbeame
wellknowneveninhisownountry.
Permutable poetry. Let's turn now to a urious question that attrated
theattentionof several prominentmathematiians in theseventeenth entury,
beause it sheds onsiderable lighton the stateof ombinatorial knowledge in
Europeat that time. A Jesuitpriestnamed BernardBauhuishad omposed a
famousone-linetributetotheVirginMary,inLatin hexameter:
Tot tibisuntdotes,Virgo,quotsideralo. (19)
[\Thou hast as many virtues, O Virgin, as there are stars in heaven"; see
his Epigrammatum Libri V (Cologne: 1615), 49.℄ His verse inspired Eryius
Puteanus, a professor at the University of Louvain, to write a book entitled
PietatisThaumata(Antwerp:1617),presenting1022permutationsofBauhuis's
words. Forexample,Puteanuswrote
107 Totdotestibi,quotlosuntsidera,Virgo.
270 Dotestot, losuntsideraquot,tibiVirgo.
329 Dotes, losuntquotsidera,Virgo tibitot.
384 Sideraquotlo, totsuntVirgotibidotes.
725 Quotlosuntsidera, totVirgotibidotes.
949 SuntdotesVirgo,quotsidera, tottibilo.
1022 SuntlototVirgotibi,quotsidera,dotes.
(20)
Hestoppedat 1022,beause1022wasthenumberofvisible starsinPtolemy's
well-knownatalogoftheheavens.
Theideaofpermutingwords inthis waywaswellknownat thetime;suh
wordplay was what JuliusSaliger had alled\Proteus verses" inhis Poeties
LibriSeptem(Lyon:1561),Book2,Chapter30. TheLatinlanguagelendsitself
topermutationslike(20),beauseLatinwordendingstendtodenethefuntion
ofeahnoun,makingtherelativewordordermuhlessimportanttothemeaning
of a sentene than it is in English. Puteanus did state, however, that he had
speially avoidedunsuitablepermutations suhas
Sideratotlo,Virgo,quotsunttibidotes, (21)
beausetheywould plaeanupper boundontheVirgin'svirtuesratherthan a
lowerbound. [Seepages12and103 ofhisbook.℄
Of ourse there are 8!= 40;320 ways to permute the words of (19). But
thatwasn'tthepoint;mostofthosewaysdon't\san." EahofPuteanus's1022
verses obeyed the strit rules of lassial hexameter, the rules that had been
followedbyGreekandLatinpoetssinethedaysofHomerandVergil,namely:
i) Eahwordonsistsofsyllablesthatareeitherlong( )or short(^);
ii) Thesyllablesofeahlinebelongtoone of32patterns,
n
^^
on
^^
on
^^
on
^^
o
^^
n
^ o
: (22)
Inotherwords therearesixmetrialfeet,whereeahoftherstfouriseithera
datylor aspondeeintheterminologyof(5); thefthfootshould bea datyl,
andthelastiseithertroheeorspondee.
TherulesforlongversusshortsyllablesinLatinpoetryaresomewhattriky
ingeneral, but the eight words of Bauhuis's verse an be haraterized by the
followingpatterns:
tot= ; tibi= n
^^
^ o
; sunt= ; dotes= ;
Virgo= n
^ o
; quot= ; sidera= ^^; lo= : (23)
Notie that poets had two hoies when they used the words `tibi' or `Virgo'.
Thus,forexample,(19)tsthehexameterpattern
Tot
^
ti-
^
bi suntdo- tes, Vir- go,quot si-
^
de-
^
ra -lo.
(24)
(Datyl, spondee, spondee, spondee, datyl, spondee; \dum-diddy dum-dum
dum-dumdum-dumdum-diddydum-dum."Theommasrepresentslightpauses,
alled\suras,"whenthewordsareread;theydon'tonernushere,although
Puteanusinsertedthemarefullyintoeah ofhis1022permutations.)
A natural questionnow arises: IfwepermuteBauhuis's words at random,
whataretheoddsthattheysan? Inotherwords,howmanyofthepermutations
obeyrules(i)and(ii),giventhesyllablepatternsin(23)? G.W.Leibnizraised
thisquestion, amongothers, inhis Dissertatiode ArteCombinatoria (1666),a
workpublishedwhenhewasapplyingforapositionattheUniversityofLeipzig.
At this time Leibnizwas just 19 years old, largely self-taught, and his under-
standingofombinatoriswasquitelimited;forexample,hebelievedthatthere
are 600 permutations of fut;ut;re;mi;fa;solg and 480 of fut;ut;re;re;mi;fag,
andheevenstatedthat(22)represents76possibilitiesinsteadof32. [Seex5and
x8inhisProblem6.℄
But Leibniz did realize that it would be worthwhile to develop general
methods for ounting all permutations that are \useful," in situations when
many permutations are \useless." He onsidered several examples of Proteus
verses,enumeratingsome ofthesimpleronesorretlybut makingmanyerrors
whenthewords wereompliated. AlthoughhementionedPuteanus'swork, he
didn'tattempttoenumeratethesannable permutationsof(19).
AmuhmoresuessfulapproahwasintroduedafewyearslaterbyJean
Prestetinhis
ElemensdesMathematiques (Paris:1675),342{438. Prestetgave
a lear exposition leadingto theonlusion that exatly2196 permutations of
Bauhuis'sversewouldyieldaproperhexameter. However,hesoonrealizedthat
he had forgotten to ount quite a few ases|inluding those numbered 270,
384,and725in(20). Soheompletelyrewrotethismaterialwhenhepublished
Nouveaux
ElemensdesMathematiquesin1689. Pages127{133ofPrestet'snew
book weredevotedto showingthatthetruenumberofsannablepermutations
was3276,almost 50%largerthanhis previoustotal.
MeanwhileJohnWallishadtreatedtheprobleminhisDisourseofCombi-
nations (London: 1685), 118{119, published as a supplement to his Treatise of
Algebra. Afterexplainingwhyhebelievedtheorretnumbertobe3096,Wallis
admittedthat he may haveoverlooked some possibilities and/orountedsome
asesmorethanone;\butIdonot,atpresent,diserneithertheoneandother."
AnanonymousreviewerofWallis'sworkremarkedthatthetruenumberof
metriallyorretpermutationswasatually2580|buthegavenoproof[Ata
Eruditorum5 (1686), 289℄. The reviewerwas almost ertainly G. W. Leibniz
himself, although no lue to the reasoning behind the number 2580 has been
foundamong Leibniz'svoluminousunpublishednotes.
Finally James Bernoulli entered the piture. In his inaugural leture as
Dean of Philosophy at the University of Basel, 1692, he mentioned the tot-
tibi enumeration problem and stated that a areful analysis is neessary to
obtain the orret answer|whih, he said, was 3312(!). His proof appeared
posthumouslyintherst editionof hisArs Conjetandi(1713), 79{81. [Those
pageswere, inidentally, omitted from later editionsof that famous book, and
fromhisolletedworks,beausehedidn'tatuallyintendthemforpubliation;
aproofreaderhadinsertedthembymistake. SeeDieWerkevonJakobBernoulli
3(Basel:Birkhauser,1975),78,98{106,108,154{155.℄
Sowhowasright? Arethere2196sannablepermutations,or3276,or3096,
or2580,or3312? W.A.Whitworthand W.E.Hartleyonsideredthequestion
anewinTheMathematialGazette2(1902),227{228,wheretheyeahpresented
elegant arguments and onluded that the true total was in fat none of the
above. Theirjoint answer,2880,representedthersttime that anytwo math-
ematiianshadindependentlyometo thesame onlusionaboutthisproblem.
But exerises 21 and 22, below, reveal the truth: Bernoulli is vindiated,
and everybody else was wrong. Moreover, a study of Bernoulli's systemati
andarefullyindented3-page derivationindiatesthathewas suessfulhiey
beause he adhered faithfully to a disipline that we now all the baktrak
method. WeshallstudythebaktrakmethodthoroughlyinSetion7.2.2,where
we will also see that the tot-tibiquestion is readilysolved as a speial ase of
theexatoverproblem.
Even thewisest andmost prudent peopleoftensuerfrom
whatLogiiansall insuÆient enumerationof ases.
| JAMES BERNOULLI (1692)
Setpartitions. ThepartitionsofasetseemtohavebeenstudiedrstinJapan,
where a parlor game alled genji-ko (\Genji inense") beame popular among
upperlasspeopleaboutA.D.1500. Thehostofagatheringwouldseretlyselet
ve pakets of inense, some of whih might be idential, and he would burn
them one at a time. Theguests would tryto disernwhih of thesentswere
thesameandwhihweredierent;inotherwords,theywouldtrytoguesswhih
ofthe$
5
=52partitionsoff1;2;3;4;5ghadbeenhosenbytheirhost.
Fig.47. Diagramsusedto represent setpartitions
in16thenturyJapan. [Fromaopyintheolle-
tionofTamakiYanoatSaitamaUniversity.℄
Soonitbeameustomarytorepresentthe52possibleoutomesbydiagrams
likethoseinFig.47. For example,theuppermostdiagram ofthat illustration,
whenreadfromrighttoleft,wouldindiatethatthersttwosentsareidential
andsoarethelastthree;thusthepartitionis12j345. Theothertwodiagrams,
similarly, are pitorial ways to representthe respetive partitions 124j35 and
1j24j35. As an aid to memory, eah of the 52 patterns was named after a
hapterofLadyMurasaki'sfamous11th-enturyTaleofGenji,aordingtothe
followingsequene[EnylopediaJaponi (Tokyo:Sanseido,1910),1299℄:
(25)
(One again, as we've seen in many other examples, thepossibilities were not
arrangedinanypartiularlylogialorder.)
Theappealingnature ofthesegenji-kopatternsled manyfamiliestoadopt
themasheraldirests. Forexample,thefollowingstylizedvariantsof(25)were
foundinstandardatalogs ofkimonopatternsearlyinthe20thentury:
[SeeFumieAdahi,JapaneseDesignMotifs(NewYork:Dover,1972),150{153.℄
Earlyinthe1700s,TakakazuSekiandhisstudentsbegantoinvestigatethe
number ofset partitions $
n
forarbitrary n,inspiredby theknown resultthat
$
5
=52. YoshisukeMatsunagafoundformulasforthenumberofsetpartitions
whenthere are k
j
subsets ofsize n
j
for 1 j t, withk
1 n
1
++k
t n
t
=n
(see the answer to exerise 1.2.5{21). He also disovered the basi reurrene
relation7.2.1.5{(14),namely
$
n+1
=
n
0
$
n +
n
1
$
n 1 +
n
2
$
n 2
++
n
n
$
0
; (26)
bywhihthevaluesof$
n
anreadilybeomputed.
Matsunaga'sdisoveriesremainedunpublisheduntilYoriyukiArima'sbook
Shuki Sanpo ame outin 1769. Problem 56 of that book asked thereader to
solvethe equation \$
n
= 678570" for n; and Arima's answer, worked out in
detail(withreditdulygiventoMatsunaga),was n=11.
Shortly afterwards, Masanobu Saka studied thenumber
n
k
of ways that
ann-setanbepartitionedintoksubsets,inhisworkSanpo-Gakkai(1782). He
disovered thereurrene formula
n
n+1
k o
= k n
n
k o
+ n
n
k 1 o
; (27)
andtabulatedtheresultsforn11. JamesStirling,inhis Methodus Dieren-
tialis(1730),haddisoveredthenumbers
n
k
inapurelyalgebraiontext;thus
Saka was therstpersontorealizetheirombinatorialsigniane.
Aninterestingalgorithmforlistingset partitionswassubsequentlydevised
byToshiakiHonda(seeexerise24). Furtherdetailsaboutgenji-koanditsrela-
tiontothehistoryofmathematisanbefoundinJapanese artilesbyTamaki
Yano,SugakuSeminar 34,11(Nov.1995),58{61;34,12(De.1995),56{60.
Setpartitions remainedvirtuallyunknown inEuropeuntil muhlater, ex-
ept for three isolated inidents. First, George and/or Rihard Puttenham
published The Arte of English Poesie in 1589, and pages 70{72 of that book
ontaindiagramssimilartothoseofgenji-ko. For example,thesevendiagrams
(28)
were usedtoillustratepossiblerhymeshemesfor5-linepoems, \whereofsome
ofthembeharsherandunpleasauntertotheearethenothersomebe." Butthis
visuallyappealinglistwasinomplete(seeexerise25).
Seond, an unpublished manusript of G. W. Leibnizfrom the late 1600s
shows that he had tried to ount the number of ways to partition f1;:::;ng
into three or foursubsets, but with almost nosuess. Heenumerated
n
2 by
a very umbersome method, whih would not haveled himto see readily that
n
2
= 2 n 1
1. He attempted to ompute
n
3 and
n
4
onlyfor n 5, and
made several numerial slips leading to inorret answers. [See E. Knobloh,
StudiaLeibnitianaSupplementa 11(1973),229{233; 16(1976),316{321.℄
ThethirdEuropeanappearaneofsetpartitionshadaompletelydierent
harater. JohnWallisdevotedthe third hapter ofhis Disourse ofCombina-
tions(1685)toquestionsabout\aliquotparts,"theproperdivisorsofnumbers,
andinpartiularhestudiedthesetofallwaystofatorizeagiveninteger. This
questionisequivalenttothestudyofmultiset partitions;forexample,thefator-
izationsofp 3
q 2
rareessentiallythesameasthepartitionsoffp;p;p;q;q;rg,when
p,q,andr areprimenumbers. Wallisdevisedanexellentalgorithmforlisting
allfatorizationsofagivenintegern,essentiallyantiipatingAlgorithm7.2.1.5M
(seeexerise28). Buthedidn'tinvestigatetheimportantspeialasesthatarise
when nis the powerof a prime(equivalent to integer partitions)or when n is
squarefree(equivalenttosetpartitions). Thus,althoughWalliswasabletosolve
themoregeneralproblem,itsomplexitiesparadoxiallydeetedhimfromdis-
overingpartitionnumbers,Bellnumbers,orStirlingsubsetnumbers,orfromde-
visingsimplealgorithmsthatwouldgenerateintegerpartitionsorsetpartitions.
Integer partitions. Partitions of integers arrived on the sene even more
slowly. Bishop Wibold (. 965) knew the partitions of n into exatly three
parts 6. So did Galileo, who wrote a memo about them (. 1627) and also
studiedtheirfrequenyofourreneasrollsofthreedie. [\Sopralesopertede
idadi,"inGalileo'sOpere,Volume8,591{594;helistedpartitionsindereasing
lexiographiorder.℄
Mersennelistedthepartitionsof9intoanynumberofparts,onpage130of
hisTraitezdelaVoixetdesChants(1636). Witheahpartition9=a
1
++a
k
healsoomputedthemultinomialoeÆient9!=(a
1
!:::a
k
!);aswe'veseenearlier,
he was interestedinountingvarious melodies, and he knewfor example that
there are 9!=(3!3!3!) = 1680 melodies on the nine notes fa;a;a;b;b;b;;;g.
But he failed to mentionthe ases 8+1 and 3+2+1+1+1+1, probably
beausehehadn'tlistedthepossibilitiesinanysystematiway.
Leibniz onsidered two-part partitions in Problem 3 of his Dissertatio de
ArteCombinatoria(1666),andhisunpublishednotesshowthathesubsequently
spent onsiderable time trying to enumerate the partitions that have three or
more summands. He alled them \diserptions," or (less frequently) \divul-
sions"|inLatin of ourse|or sometimes\setions"or \dispersions"or even
\partitions." Hewas interested inthem primarily beause of their onnetion
withthe monomialsymmetrifuntions P
x a
1
i
1 x
a
2
i
2
:::. Buthis manyattempts
ledtoalmosttotalfailure,exeptintheaseofthreesummands,whenhealmost
(butnotquite)disoveredtheformulafor
n
3
inexerise7.2.1.4{31. Forexample,
hearelesslyountedonly21partitionsof8,forgettingthease2+2+2+1+1;
and he got only 26 for p(9), after missing 3 +2+2+2, 3+2+2 +1+1,
2+2+2+1+1+1,and2+2+1+1+1+1+1|inspiteof thefatthat
hewastryingtolistpartitionssystematiallyindereasinglexiographiorder.
[SeeE.Knobloh,StudiaLeibnitianaSupplementa11(1973),91{258;16(1976),
255{337;HistoriaMathematia1(1974),409{430.℄
Abrahamde Moivrehadthe rstrealsuess withpartitions, inhis paper
\AMethodofRaisinganinniteMultinomialtoanygivenPower,orExtrating
anygivenRootofthesame"[PhilosophialTransations19(1697),619{625and
Fig.5℄. He proved that theoeÆient of z m+n
in(az+bz 2
+z 3
+) m
has
onetermforeahpartitionof n;forexample,theoeÆientofz m+6
is
m
6
a m 6
b 6
+5 m
5
a m 5
b 4
+4 m
4
a m 4
b 3
d+6 m
4
a m 4
b 2
2
+3 m
3
a m 3
b 2
e+6 m
3
a m 3
bd+2 m
2
a m 2
bf+ m
3
a m 3
3
+2 m
2
a m 2
e+ m
2
a m 2
d 2
+ m
1
a m 1
g: (29)
Ifweseta=1,thetermwithexponentsb i
j
d k
e l
::: orrespondstothepartition
with i 1s, j 2s, k 3s, l 4s, et. Thus, for example, when n =6 he essentially
presentedthepartitionsintheorder
111111; 11112; 1113; 1122; 114; 123; 15; 222; 24; 33; 6: (30)
Heexplained howto list the partitions reursively, as follows (but in dierent
language related to his own notation): For k =1, 2, :::, n, start with k and
appendthe(previouslylisted)partitionsofn kwhosesmallestpart isk.
[Mysolution℄ wasordered tobepublished intheTransations,
not so muh as a matter relating to Play,
but as ontainingsome general Speulations
not unworthy tobeonsideredby theLoversof Truth.
| ABRAHAM DE MOIVRE (1717)
P. R.de Montmorttabulatedall partitions ofnumbers 9 into 6 parts
inhis Essay d'Analyse surles Jeuxde Hazard (1708), inonnetion with die
problems. His partitionswerelistedinadierentorderfrom (30);forexample,
111111; 21111; 2211; 222; 3111; 321; 33; 411; 42; 51; 6: (31)
Heprobablywas unawareofdeMoivre'spriorwork.
Sofaralmost noneoftheauthors we'vebeendisussingatuallydesribed
theproedures by whih theygenerated ombinatorial patterns. Wean only
infertheirmethods,orlakthereof,bystudyingtheliststhattheyatuallypub-
lished. Furthermore,inrareases suhasdeMoivre'spaperwhereatabulation
method was expliitly desribed, the author assumed that allpatterns for the
rstases1,2,:::,n 1hadbeenlistedbeforeitwastimetotakletheaseof
ordern. Nomethodforgenerating patterns\on they,"movingdiretlyfrom
one pattern to its suessor without looking at auxiliary tables, was atually
explained by any of the authors we have enountered, exept for Kedara and
Narayan
.
a. Today's omputerprogrammers naturallyprefer methods that are
morediretandneedlittlememory.
Roger Joseph Bosovih published the rst diret algorithm for partition
generation in Giornalede' Letterati (Rome, 1747), on pages 393{404 together
withtwofoldout tablesfaingpage404. His method,whihproduesforn=6
therespetiveoutputs
111111; 11112; 1122; 222; 1113; 123; 33; 114; 24; 15; 6; (32)
generatespartitionsinpreiselythereverseorderfromwhihtheyarevisitedby
Algorithm7.2.1.4P;andhismethodwouldindeedhavebeenfeaturedinSetion
7.2.1.4, exeptforthefatthat thereverseorderturns outtobeslightlyeasier
andfasterthantheorderthathehadhosen.
Bosovih published sequels inGiornalede'Letterati (Rome,1748), 12{27
and84{99,extendinghisalgorithmintwoways. First,heonsideredgenerating
onlypartitionswhosepartsbelongtoagivensetS,sothatsymbolimultinomials
withsparseoeÆientsouldberaisedtothemthpower. (Hesaidthatthegd
ofallelementsofS shouldbe1;infat,however,hismethodouldfailif12= S.)
Seond,heintrodued analgorithm forgeneratingpartitionsofnintomparts,
given m andn. Againhe was unluky: A slightly better way to do that task,
Algorithm7.2.1.4H,wasfoundsubsequently, diminishinghis hanesforfame.
Hindenburg's hype. The inventorof Algorithm 7.2.1.4Hwas Carl Friedrih
Hindenburg, who also redisovered Narayan
.
a's Algorithm 7.2.1.2L, a winning
tehniqueforgeneratingmultisetpermutations. Unfortunately,thesesmallsu-
essesledhimtobelievethathehadmaderevolutionaryadvanesinmathemat-
is|althoughhedidondesendtoremarkthatotherpeoplesuhasdeMoivre,
Euler,andLamberthadomelosetomakingsimilardisoveries.
Hindenburgwas a prototypial overahiever,extremely energeti ifnot in-
spired. HefoundedorofoundedGermany'srstprofessionaljournalsofmath-
ematis(published 1786{1789and 1794{1800),and ontributedlong artilesto
eah. He served several times as aademi dean at the University of Leipzig,
where hewas also theRetor in1792. If hehad beena better mathematiian,
Germanmathematis mightwellhaveourishedmore inLeipzigthaninBerlin
orGottingen.
Buthis rstmathematial work, Beshreibung einerganzneuen Art, nah
einem bekannten Gesetze fortgehende Zahlen durh Abzahlen oder Abmessen
bequemundsiherzunden(Leipzig:1776), amplyforeshadowed what was to
ome: His\ganzneue"(ompletelynew)ideainthatbookletwassimplytogive
ombinatorialsignianeto thedigitsofnumberswrittenindeimalnotation.
Inredibly,heonludedhismonographwithlargefoldoutsheetsthatontained
