內(nèi)容說明講稿

Exercises:Determinethenumberofwaystocolorthesquaresofa1-by-nchessboard,usingthecolorsred,white,andblue,ifanevennumberofsquaresaretobecoloredredandthereisatleastoneblueLethndenotethenumberofsuchcoloringswherewedefineh0=0.Thenhnequalsthenumberofn-permutationsofamultisetofthreecolors,eachwithaninfiniterepetitionnumber,inwhichredoccursanevennumberoftimesandatleastonebluesquare.Thustheexponentialgeneratingfunctionforh0,h1,h2,h3,…h(huán)n,…istheproductofred,whiteandblue x x x x x g(x)=(1+2!+4!+…)(1+x+2!+3!+4!+…)(x+2!+3! - =2(e+e)e(e-1)=2(e-e+e- n n 2 (2

-

n!+n!- 3n2n1

Hence,h=0ifn=0andh=3n2n

ifn=1, 7.Let????????equalthenumberofdifferentwaysinwhichthesquaresofa1-by-nchessboardcanbecolored,usingthecolorsred,white,andbluesothatnotwosquaresthatarecoloredredareadjacent.Findandverifyarecurrencerelationthat????????satisfies.ThenfindaformulaforR33Ifthefirstcelliscoloredred,thenthesecondonemustbecoloredwhiteorblue,thenumberofwaystocolortheremainder(n-2)cellswillbehn?2;Ifthefirstcellisn’tcoloredred,thenthenumberofwaystocolortheremainder(n-1)cellswillbehn?1.Sothenumberofwaystocolorthencellsishn=2hn?1+2hn?2Sothecorrespondingcharacteristicequationoftheaboverecurrencerelationisx2?2x?2=0.Thesolutionoftheequationisx1=√3+1,x2=?√3+hn=c1(1+√3)n+c2(1?Andh0=1,h1=

=2+√3,

=

2+

?2+ = (√3+1)n+ (?√3+n 25.Solvethefollowingrecurrencerelationsbyusingthemethodofgeneratingfunctionsasdescribedinsection7.5(a)????????=?????????????????,(????≥????);????????=????,????????= g(x)=h0+h1x+h2x2+?+hnxn+? ?4h0x2?…?4hn?2xn+? hn=4hn?2Thusg(x)?4x2g(x)=h0+ h0=1,h1=Sog(x) =1?1 1?=1 [2n?(?2)n]

1

4n

n

hn=4 ?(?2)17Determinethegeneratingfunctionforthenumberhnofbagsoffruitofapples,oranges,bananas,andpearsinwhichthereareanevennumberofapples,atmosttwooranges,amultipleofthreenumberofbananas,andatmostonepear.Thenfindaformulaforhnfromthegeneratingfunction.Theproblemisequivalenttofindingthenumberhnofnon-negativeintegralsolutionsofe1+e2+e3+e4=n,wheree1iseven,e2is0to2,e3isamultipleofthree,and0≦e4≦1,thenitsgeneratingfunctionisg(x)(1x2x4...)(1xx2)(1x3x6...)(1 (1n1n0(n

hnn forthe numberhn non-negativeintegralsolutionof????????????+????????????+????????+????????????=Supposef1=2e1,f2=5e2,f3=e3,f4=Sowef1+f2+f3+f4= g(x)=(1+x2+x4+?)(1+x5+x10+?)(1+x1+x2+?)(1+x7+x14+?) =1?x21?x51?x1?37.LetSdenotethemultiset{∞·e1,∞·e2,……∞·ek}.Determinetheexponentialgeneratingfunctionforthe????????,????????,????????，…????????…where????????=????andforn≥(c)????????equalsthenumberofn-permutationsofSinwhich????????occursatleastonce,????????occursatleasttwice,……????????occursatleastktimes.Letthetimese1occursben1wheren1≥thetimese2occursben2wheren2≥thetimesekoccursbenkwherenk≥We n1+n2+?+nk=

g(x)=?+ +??

+??

+?

(

k+=(ex?1)?ex?1

x?…(ex?1?

??

) (k?????????.Thegeneralterm ofasequenceisapolynomialinnof 3.Ifthefirstfourentriesofthe0throwofitsdifferencetableare1,-1,3,10,determine????????andaformulafor∑???? Answer:Example（267頁）,Theorem8.2.2（267））,Theorem（269Thedifferencetable - -4 6 -andthe0thdiagonalelementsequalto1,-2,6,-

nk= +6n+ ?3n+

?

?

?????????????.LetXbeap-elementsetandletYbeak-elementset.Provethatthenumberoffunctionsf:X→????whichmapXontoYequals????!????(????,????)=??#ThenumberoffunctionsequalsthenumberofpartitionsofXintoY.SinceYhaskdifferentelements,inotherwords,thenumberoffequalsthenumberofpartitionofpelementsintokdistinguishableboxesandeveryoneofthekboxesisnotempty.Hence,thenumberequalsk!S(p,k)=S#(p,????????????.Let????????,????????,…????????bedistinctpositiveintegers,andlet????????????????(????????,????????,…????????)equalthenumberofpartitionsofninwhichallpartsaretakenfrom????????,????????,…????????.Define????????=????.Showthatthegeneratingfunctionfor????????,????????,????????…????????…is?(???????????????Answer:Theorem8.3.1（283頁）證明過程Weknowtheexpressionaboveequalstheproduct(1+xt1+?+t1+?)(1+xt2+?+xa2t2+?)(1+xt3+?+xa3t3+?)Atermxnariseinthisprojectbychoosingatermxa1t1fromthefirstxa2t2fromthesecond,xa3t3fromthethird,andsoon,witha1t1++a3t3+?+amtm=n.Thuseachpartitionofncontributes1tothecoefficientofxnequalsthenumberqnofpartitionofn.Hence qnxn= (1?xtkSothegeneratingfunctionforq0,q1,q2…qn…is (1?xtk排列問題用指數(shù)生成函數(shù)13.LetfandgbethepermutationsinExercise1.Letc=(R,B,B,R,R,R)beacoloringof1,2,3,4,5,6withthecolorsRandB.Determinethefollowingactionsonc:??????????Asc=(R,B,B,R,R,R),f

12345 64215

andg

13

3462

,thencan- 12345 - =43

.For25

,1->4,2->3…then4colorR,3colorB…1B,assameasforf-1*c=(RRBRRg*c=(RRRRB20.UseTheorem14.2.3todeterminethenumberofinequivalentcoloringsofthecornersofatrianglewhichisisoscelesbutnotequilalwiththecolorsredandblue.Withpcolors(cf.Exercise4)1 Firstly,wecandrawagraphasThepermutationgrouphastwo

So,accordingtothetheoremofBurnside,we23+N =2Ifweusepcolorsforcoloring,wecansimilarlyhavethep3+N226.Howmanydifferentnecklacesaretherewhichcontain4redand3bluebeads? Consideringthecolorproblemofregularheptagon,thepermutationgroupis:G[I,,2,3,4,5,6,,,......].And i(i0,.... sevenrotations,i(i1,...,7)aresevenreflections.ThuswecanP(z,.....,z)1(z76z7z 1PG(rb,r2b2,r3b3,r4b4,r5b5,r6b6,r7=1[rb76(r7b7)7(rb)r2b23]Sothecoefficient r4

1

73]

4 Sothereare4differentkindsof注意：關(guān)于項鏈翻轉(zhuǎn)問題的爭論，這里注意項鏈?zhǔn)强梢苑D(zhuǎn)的，詳見第三版書563頁例子。42.Letnbeaprimenumber.DeterminethenumberofnecklacesthatcanbemadefromnbeadsofkdifferentFirstly,weknowthatthepermutationg