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Enumerative Combinatorics: Volume 2: Second Edition
If F(x)and G(x)areformalpowerserieswith G(0) = 0,thenwehaveseen (afterProposition1.1.9)thatthecomposition F(G(x))isawell-definedformal powerseries.Inthischapterwewillinvestigatethecombinatorialramifications ofpowerseriescomposition.Inthissectionwewillbeconcernedwiththecase where F(x)and G(x)areexponentialgeneratingfunctions,andespeciallythe case F(x) = ex .
Throughoutthischapter K denotesafieldofcharacteristic0(suchas C withsomeindeterminatesadjoined).Wealsodenoteby Ef (x)theexponential generatingfunctionofthefunction f : N → K,thatis,
5.1.1Proposition. Givenfunctionsf , g : N → K,defineanewfunctionh : N → Kbytherule
whereXisafiniteset,andwhere(S,T)rangesoverallweakorderedpartitions ofXintotwoblocks,thatis,S ∩ T =∅ andS ∪ T = X . Then
Proof. Let#X = n.Thereare(n k )pairs(S, T )with#S = k and#T = n k,so
).
Fromthisequation(5.2)follows.
OnecouldalsoproveProposition 5.1.1 byusingTheorem3.18.41 appliedto thebinomialposet B ofExample3.18.3.
WehavestatedProposition 5.1.1 intermsofacertainrelationship(5.1) amongfunctions f , g and h,butitisimportanttounderstanditscombinatorialsignificance.Supposewehavetwotypesofstructures,say α and β,which canbeputonafiniteset X .Weassumethattheallowedstructuresdependonly onthecardinalityof X .Anew“combined”typeofstructure,denoted α ∪ β, canbeputon X byplacingstructuresoftype α and β onsubsets S and T , respectively,of X suchthat S ∪ T = X , S ∩ T =∅.If f (k)(respectively g(k)) arethenumberofpossiblestructuresona k-setoftype α (respectively, β), thentheright-handsideof(5.1)countsthenumberofstructuresoftype α ∪ β on X .Moregenerally,wecanassignaweight w( )toanystructure oftype α or β.Acombinedstructureoftype α ∪ β isdefinedtohaveweightequal totheproductoftheweightsofeachpart.If f (k)and g(k)denotethesumof theweightsofallstructuresona k-setoftypes α and β,respectively,thenthe right-handsideof(5.1)countsthesumoftheweightsofallstructuresoftype α ∪ β on X .
5.1.2Example. Givenan n-elementset X ,let h(n)bethenumberofwaysto split X intotwosubsets S and T with S ∪T = X , S ∩T =∅;andthentolinearly ordertheelementsof S andtochooseasubsetof T .Thereare f (k) = k! ways tolinearlyordera k-elementset,and g(k) = 2k waystochooseasubsetofa k-elementset.Hence n≥0
5.1.6Corollary (theExponentialFormula). Givenafunctionf : P → K, defineanewfunctionh : N → Kby
(0) = 1.
Then
Letussayabriefwordaboutthecomputationalaspectsofequation(5.5). Ifthefunction f (n)isgiven,thenonecanuse(5.4)tocompute h(n).However, thereisamuchmoreefficientwaytocompute h(n)from f (n)(andconversely).
5.1.7Proposition. Letf : P → Kandh : N → KberelatedbyEh(x) = exp Ef (x) (soinparticularh(0) = 1).Thenwehaveforn ≥ 0 therecurrences
Thecompositionalandexponentialformulasareconcernedwithstructures onaset S obtainedbychoosingapartitionof S andthenimposingsome“connected”structureoneachblock.Insomesituationsitismorenaturaltochoose a permutation of S andthenimposea“connected”structureoneachcycle. Thesetwosituationsareclearlyequivalent,sinceapermutationisnothingmore thanapartitionwithacyclicorderingofeachblock.However,permutations ariseoftenenoughtowarrantaseparatestatement.Recallthat S(S)denotes theset(orgroup)ofallpermutationsoftheset S.
5.1.8Corollary (theCompositionalFormula,permutationversion). Given functionsf : P → Kandg : N → Kwithg(0) = 1,defineanewfunction h : P → Kby h(#S) = π ∈S(S)
),#S > 0,(5.9) h(0) = 1, whereC1, C2, , Ck arethecyclesinthedisjointcycledecompositionof π Then
5.1.9Corollary (theExponentialFormula,permutationversion). Givena functionf : P → K,defineanewfunctionh : N → Kby h(#S) = π ∈S(S) f (#C1)f (#C2) f (#Ck ),#S > 0, h(0) = 1, wherethenotationisthesameasinCorollary 5.1.8.Then
Eh(x) = exp n≥1 f (n) xn n .
InChapter3.18(seeExample3.18.3(b))werelatedadditionandmultiplicationofexponentialgeneratingfunctionstotheincidencealgebraofthelattice offinitesubsetsof N.Thereisasimilarrelationbetween composition ofexponentialgeneratingfunctionsandtheincidencealgebraofthelattice n of partitionsof[n](orany n-set).Moreprecisely,weneedtoconsidersimultaneouslyall n for n ∈ P.RecallfromSection3.10thatif σ ≤ π in n,then wehaveanaturaldecomposition
σ , π
where |σ |= iai and |π |= ai.Let = ( 1, 2, ).Foreach n ∈ P, let fn ∈ I( n, K),theincidencealgebraof n.Supposethatthesequence f = ( f1, f2, ... )satisfiesthefollowingproperty:Thereisafunction(alsodenoted f ) f : P → K suchthatif σ ≤ π in n and[σ , π ]satisfies(5.10),then
If f = ( f1, f2, ... )where fn ∈ I( n, K)andeach f 1 n existsin I( n, K),then wewrite f 1 = ( f 1 1 , f 1 2 , ).
5.1.12Proposition. Supposefismultiplicativeandf 1 exists.Thenf 1 is multiplicative.
5.1.13Example. Let ζ , δ, µ ∈ M ( )havethesamemeaningsasabove,so ζµ = µζ = δ.Now
sobyTheorem 5.1.11 [exp Eµ(x)] 1 = x
)
+
ThuswehaveanotherderivationoftheMobiusfunctionof n (equation(3.37)).
5.1.14Example. Let h(n)bethenumberofwaystopartitiontheset[n],and thenpartitioneachblockintoblocksofoddcardinality.Weareaskingforthe numberofchains ˆ 0 ≤ π ≤ σ ≤ ˆ 1in n suchthatallblocksizesof π areodd. Define f ∈ M ( )by
Thenclearly h = f ζ 2,sobyTheorem 5.1.11, Eh(x) = Eζ (Eζ (Ef (x))) = exp
exp
1 = exp(e sinh x 1) 1.
Wehavediscussedinthissectionthecombinatorialsignificanceofmultiplyingandcomposingexponentialgeneratingfunctions.Threefurtheroperations areimportanttounderstandcombinatorially:addition,multiplicationby x (reallyaspecialcaseofarbitrarymultiplication,butofspecialsignificance), anddifferentiation.
5.1.15Proposition. LetSbeafiniteset.Givenfunctionsf , g : N → K,define newfunctionsh1, h2, h3, andh4 asfollows:
h1(#S) = f (#S) + g(#S)(5.14) h2(#S) = (#S)f (#T ), where #T = #S 1(5.15)
Then
Proof. Easy.
Equation(5.14)correspondstoachoiceoftwostructurestoplaceon S,one enumeratedby f andoneby g.Inequation(5.15),we“root”avertex v of S (i.e.,wechooseadistinguishedvertex v,oftencalledthe root)andthenplacea structureontheremainingvertices T = S −{x}.Equation(5.16)corresponds toadjoininganextraelementto S andthenplacingastructureenumeratedby f .Finallyinequation(5.17)wearesimplyplacingastructureon S androoting avertex.
function.Letusillustratetheseideasherebyinterpretingcombinatoriallythe formula Eh(x) = Ef (x)Eh(x)ofequation(5.8).Theleft-handsidecorresponds tothefollowingconstruction:takea(finite)set S,adjoinanewelement t,and thenplaceon S ∪{t} astructureenumeratedby h (or“h-structure”).Therighthandsidesays:chooseasubset T of S,adjoinanelement t to T ,placeon T ∪{t} an f -structure,andplaceon S T an h-structure.Clearlyif h and f arerelated by(5.4)(sothat h-structuresareuniquedisjointunionsof f -structures)then thecombinatorialinterpretationsof Eh(x)and Ef (x)Eh(x)areequivalent.
5.2.1Example. Thenumberofgraphs(withoutloopsormultipleedges)on an n-elementvertexset S isclearly2(n 2).(Eachofthe n 2 pairsofverticesmay ormaynotbejoinedbyanedge.)Let c(#S) = c(n)bethenumberof connected graphsonthevertexset S.Sinceagraphon S isobtainedbychoosingapartition π of S andthenplacingaconnectedgraphoneachblockof π ,weseethat equation(5.5)holdsfor h(n) = 2(n 2) and f (n) = c(n).HencebyCorollary 5.1.6,
5.2.2Example. Supposeweareinterestedinnotjustthenumberofconnected graphsonan n-elementvertexset,butratherthenumberofsuchgraphswith exactly k components.Let ck (n)denotethisnumber,anddefine
TherearetwowaystoobtainthisgeneratingfunctionfromTheorem 5.1.4 and Corollary 5.1.6.Wecaneitherset f (n) = c(n)and g(k) = tk in(3),orset f (n) = c(n)t in(5.5).Ineithercasewehave h(n) =
Againthesamereasoningworksequallyaswellforposets,digraphs,topologies, ... Ingeneral,if Eh(x)istheexponentialgeneratingfunctionforthetotal numberofstructuresonan n-set(whereofcourseeachstructureisaunique disjointunionofconnectedcomponents),then Eh(x)t alsokeepstrackofthe numberofcomponents,asin(5.23).Equivalently,if h(n)isthenumberof structuresonan n-setand ck (n)thenumberwith k components,then
≥0
whereweset ck (0) = δ0k and h(0) = 1.Inparticular,if h(n) = n! (thenumber ofpermutationsofan n-set)then Eh(x) = (1 x) 1 while ck (n) = c(n, k),the numberofpermutationsofan n-setwith k cycles.Inotherwords, c(n, k)isa signlessStirlingnumberofthefirstkind(seeChapter1.3);andweget n≥0 c(n, k) xn
