Counting the compositions of a positive integer n using Generating Functions Start with, 1. x = 3 ), the number of compositions of 4.

Transcription

1 Coutg th compostos of a postv tgr usg Gratg Fuctos Start wth,... - Whr, for ampl, th co-ff of s, for o summad composto of aml,. To obta umbr of compostos of, w d th co-ff of (...) ( ) ( ) Hr for stac w obta (.) from products (. ), (. ), ad (. ). So co-ff of /(-) s, whch s umbr of two summad compostos of ),,,. Cotug wth th thr summad compostos w ow am (...) ( ) ( ) Oc aga w look at th was coms about aml, from products (.. ), (.. ), ad (.. ). So hr co-ff of s, whch accouts for th thr summad compostos,, ad (of ). Fall th co-ff of blow fucto s, (...) ( ) ( ) for o four summad composto (of ). Ths rsult tll us that th co-ff of ( ) s 8 ( ), th umbr of compostos of. I fact ths s also th co-ff of th abov qu. Gralzg th stuato w fd that th umbr of compostos of a postv tgr s th co-ff of th gratg fucto f ( )...(). ( ) But f w st /(-), th t follows that

2 f ( ) ( - ) ( ) ( ) [ ( ) ( ) ( )...]... So th umbr of tgr compostos of a postv tgr s th co-ff of f() ad ths s - as drvd th quato prvous sld. Lt us am th dtt -... Wh s rplacd b ths th rsult tlls - that for all blogg to Z, Whr do w us ths? Cosdr th spcal compostos of tgrs 6 ad 7, that rad sam lft to rght as rght to lft. 6 7 Ths ar paldroms for 6 ad 7. W fd that for 7 thr ar () () (-) paldroms. Thr s o paldrom wth o summad, 7. Thr s also o paldrom whr ctr summad s ad whr w plac o composto of o thr sd of ths summad (paldrom ). For th ctr summad w plac o of th two compostos of o th rght ad th match t o th lft, wth sam composto, rvrs ordr. (paldroms ad ) fall wh th ctr summad s, w put a gv composto of o th rght sd f ths ad match o lft sd wth sam composto, rvrs ordr. Thr ar - compostos of (paldroms,6,7,8). Th stuato s sam for paldroms of 6 cpt cas whr sg appars as ctr. So - for 6,

3 )Ctr summad 6 paldrom )Ctr summad (-) paldrom ) Ctr summad (-) paldrom v) sg at Ctr (-) paldrom So thr ar () (-) paldroms for 6. Now w look at th gral stuato. For thr s o paldrom. If k, for k blogg to Z, th thr s o paldrom wth ctr summad. for t k, thr ar t- paldroms of wth ctr summad -t. Hc th total umbr of paldroms of s (. k- ) ( k -) k (-)/ Now cosdr v, sa k for k blogg to Z. Hr thr s o paldrom wth ctr summad -s (o paldrom for ach of s- compostos of s). I addto thr ar k- paldroms whr a sg s at th ctr (o paldrom for ach of th k- compostos of k).i total, has (. k- k- ) ( k -) k / / Obsrv that for Z, has paldroms. Parttos of ItgrsPartto a postv tgr to postv summads ad skg th umbr of such parttos, wthout rgard to ordr. Ths umbr s dotd b p(). For ampl, p(): p(): p(): p(): p()7: W should lk to obta p() for a gv wthout havg to lst all th parttos. W d a tool to kp track of th umbrs of 's, 's,..., 's that ar usd as summads for kp track of 's : kp track of 's : kp track of k's : For ampl, p() s th coffct of. f ( ) ( )( ) ( ( ) ( I gral, P( ) ) ( k ) ( k 6 k ). ( ) ) grat th squc p(), p(),

4 Eampl:Fd th gratg fucto for th umbr of was a advrtsg agt ca purchas muts of ar tm f tm slots for commrcals com blocks of, 6, or scods. Lt scods rprst o tm ut. Th th aswr s th umbr of tgr solutos to th quato a b c wth a, b, c. Th assocatd gratg fucto f ( ) ( )( ad th coffct of )( 8 s ) s th aswr. Eampl: Fd th gratg fucto for p d (), th umbr of parttos of a postv tgr to dstct summads. Lt us cosdr parttos of 6: ) ) ) ) ) 6) 7) 8) 9) ) )6 Parttos 6,7,9 ad hav dstct summads, so P d (6)

5 For a k Z, Thr ar two posblts thr k s ot usd as a summad or t s. Ths ca b accoutd for b th polomal Cosqutl, th gratg fucto s P ( ) ( )( d for ach Z, p )( ( )( )...( wh 6, th coff of fucto for th squc p P () ( ( Now bcaus, -, w hav, d ) ( ) s th coff of ( ) ). ad p (). 6 - d ( )( )...( Cosdrg th parttos, w s that thr ar four parttos of 6 to odd summads, aml,, 6 ad th prvous ampl. W also hav p (6). lt p () dot th umbr of parttos of to odd (),p...)(...). -, (),p. - -, k...)( ) s. summads, wh. W df p (). Th gratg 7 P ( ) ( )( d P () p () p (), for all. )( )( From qualt of gratg fuctos, d 8... )... 6 d (),...s gv b ). Eampl: Partto to odd summads but ach such odd summads must occur a odd umbr of tms-or ot at all. Hr, for ampl, thr s o such partto of tgr, aml, thr ar o parttos of, thr ar two such parttos for tgr, aml ad. o partto for tgr aml. Th gratg fucto for th parttos dscrbd s gv b

6 f ( ) ( )( 9 )( ( )( ) ) k. k Usg Gratg fuctos, w wll also b abl to dal wth a sampl spac that s dscrt but ot ft. Eampl:Suppos that Braa taks a amato utl sh passs t. Furthr, suppos th probablt that sh passs th amatos o a gv attmpt s.8 ad th rsult of ach attmpt, aftr th frst, s dpdt of a prvous attmpt. If w lt P dot pass ad F dot fal, for a gv attmpt, th our sampl spac ma b prssd as {P, FP, FFP, FFFP,.}Whr, for ampl, Pr(FFP) s th probablt that sh fals th ams s twc bfor sh passs t, whch s gv b (.) (.8). I addto, th sum of probablts for th outcoms s Now suppos w wat to kow th probablt sh passs th am o a v umbrd attmpt. That s w wat Pr(A) whr A s th vt {FP, FFFP,.}. At ths pot w troduc th dscrt radom varabl Y whr Y couts th umbr of attmpts up to ad cludg th o whr sh passs th am. Th th probablt dstrbuto for Y s gv b Pr(Y) (.) - (.8),. So Pr(A) ca b dtrmd as follows: Pr(A) Pr( ) (.) (.8) (.8) (.) (.8) (.8)(.) (.) [(.) (.) (.)...] [ (.) (.)...] (.8)(.) (.) (.8)(.).96 6 Cotug wth Y, ow w d lk to fd E(Y), th umbr of tm sh pcts to tak am bfor sh passs t. To dtrm E(Y) w ll start wth th formula,

7 t t t... t takg th drvatv both sds, w fd that (-)(- t) - ( ) ( t) d dt t t t whr ths srs covrgs for t <. Thrfor, E(Y) (.8) (.8) t (.) [ (.) (.) (.)...] (.8) (.).Pr(Y ).8 (.8) (.) (.8) so sh pcts to tak am. tms bfor sh passs t. Fall,todtrmVar(Y),wfdfrst E(Y ). Todosomltpl b t thdffrt atdprvousrsult. th, t (- t) t t t t... Dffrt atboth sds,ow wgt, (- t) () t()( t)( ) t ( t) ( t) t t adthsaslocovrgsfor t <. d t dt( t) t...

8 [ ] 8 (.8)..) (-. (.8)... (.) (.) (.) (.8) (.) (.8) (.8) (.) ) Pr(Y ) E(Y So ow w hav, [E(Y)] ) E(Y Var(Y) Cosqutl,

9 Epotal Gratg Fuctos: Th gratg fuctos w hav dalt ow ar calld ordar Gratg fuctos, whch aros slcto problms whr ordr was rrlvat. Now lt us tur to th problms whr ordr s rlvat ad crucal. W sk a tool. To fd such a tool lt us cosdr th bomal thorm. For ach blogs to Z, ( )..., so ( ) s th ordar gratg fucto for th squc, Wh dalg wth,,,,...,,,,... ths w wrot that C(,r) rprstd th umbr of combatos of objcts tak r at a tm wth r. Cosqutl () gratd th squc C(,), C(,), C(,), C(,),.., C(,) Now for all r,! C(, r) P(,r), r!( - r)! r! whr P(, r) dots th prmutatos of objcts tak r at a tm.so, C(,) C(,) C(,) C(,)... C(, ) P(,) P(,) P(,) P(,)... P(, ).!!! O th bass of ths obsrvato W hav th followg dfto. ( ) For a squc a,a,a,a,a,a,...of ral umbrs, f() a a a a... a,!!! s calld th potal gratg fucto for th gv squc. Eg :Th Maclaura srs paso for s,... a!!!! so s th potal gratg fucto for th squc,,,,,... Th fucto s th ordar gratg fucto for th squc,,,,,,,...!!!! Eampl: I how ma was ca four lttrs of ENGINE b arragd? Th followg tabl shows lst of possbl slctos of sz from th lttrs E,N,G,I,N,E, alog wth umbr of arragmts thos lttrs dtrm.

10 E E N N!/(!!) E G N N!/! E E G N!/! E I N N!/! E E I N!/! G I N N!/! E E G I!/! E I G N! Lt us obta th soluto b usg potal g. fu. For th lttr E w us [( /!)], bcaus thr ar, or E s to arrag. Th umbr of dstct was to arrag two E s s (co-ff of th trm /).For th lttr N w us [( /!)], bcaus Thr ar, or N s to arrag. Th umbr of dstct was to arrag two N s s (co-ff of th trm /).Th arragmts for ach of th lttrs G ad I ar rprstd b ().Cosqutl, th potal gratg fucto s, f () [ (! )] ( ) th aswr s co - ff of! Cosdr two of th ght was whch th trm /! arss th paso of f () [ (! )] [ (! )]( )( ) (! )(! )( )( ) ) From th product whr! s tak from frst two factors tabl ( ) ad s tak from last two factors. Th (! )(! )( )( ) (!! )( )( ) (!!! ). (! ) Ad th co-ff of /! s!/(!!) whch s th umbr of was o ca arrag four lttrs E, E, N, N. ) From th product (! )( )( )( ) whr (! ) s tak from frst factor,s tak from scod factor ad s tak from last two factors. Hr ( /!)()()() /! (!/!)( /!) So th co-ff of /! s!/! Whch s th umbr of was th four lttrs E, E, G, I ca b arragd.i th complt paso of th f(), th trm volvg ad cosqutl!!!!!!!! /!, s!!!!!!!!!!!!!!!!! Whr th co-ff of /! Is th aswr ( arragmts) producd b th ght rsults th

11 Eampl: Cosdr th Maclaura srs pasos of ad !!!!!! add ths srs togthr w gt,...!!...!! subtract th srs w gt...!! Ths rsults hlp us followg ampls Eampl: A shp carrs 8 flags, ach of th colors rd, blu, wht ad black. of ths flags ar placd o a vrtcal pol ordr to commucat a sgal to othr shps. a) How ma of ths sgals us a v umbr of blu flags ad a odd umbr of black flags? Epotal gratg fucto, f() !!!!!! cosdrs all sgal mad up of flags,. Th last two factors rstrct to v o. of blu ad odd o. of black flags. Sc, f() ( ) ( )( ) ( ) ( ) ( )!! Th co-ff of /! f() lds (/)( ) sgals mad up of flags wth v o. blu & odd o. black flags b) How ma of th sgals hav at last wht flags, or o wht flags at all?

12 Epotal gratg fucto, f()......!!!!! ( )!! ( ) ( ) ( )!!! Hr th factor,...!!! rstrcts th sgals to thos that cota thr or mor of th wht flags, or o at all. Th aswr for th o. sgals hr s th co - ff of! f(). As w cosdr ach summad, w fd ( ) ( ) ), hr w hav a trm,!!! so th co - ff of s.! ( ) ). I ths, ordr to cosdr th trm!,! w d to cosdr th trm ( ) ( ) ( ) ()()!!! ( ) ad hr th co - ff of s ()().!!...

13 ( ) ),! w d to cosdr th trm ( ) ( ) ( )( )( ),!!! whr th co - ff of s ( )( )( ).! cosqutl, th umbr of flag sgals wth at last wht flags, or o at all,s Rsult of Rsult of Rsult of ( ) ( )( )( ),7,8. for ths last summad, ordr to gt trm,! Eampl: Compa hrs w mplos, ach of whom s to b assgd to o of th four subdvsos. Each subdvso wll gt at last o w mplo. I how ma was ca ths assgmts b mad? Callg th subdvsos A, B, C ad D, w ca quvaltl cout th lttr squcs whch thr s at last o occurrc of ach lttrs A, B, C, ad D. Th potal gratg fucto for ths arragmt s: f () ( ) 6 f ()... th aswr s th co - ff of f() :!!!! 6 ( ) ( ) ( ) ( ) ( )

14 Eampl: Dtrm th squcs gratd b followg potal gratg a)f (). ( ) so l : f ()! ths producs th squc,,,,... fuctos. b)f () ( 8) ( ) so l : f () 7 7!! th squc s wth,,,... whch s ( ) ( ).,,, 76,... c)f () so l : f ()! so th squc s,, ( ),,,,... d)f() so l : f() ! so squc s,( 9), 8,,... Summato Oprator,,,-,9,... ( 6 ), ( ) I ths scto w troduc a tchqu that hlps us to go from ordar gratg fucto for squc a, a,a, a,. to gratg fucto for th squc a, a a, a a a, a a a a,. 6

15 for f() f() (- a f(). ) [ a a a a...][...] a a f() so, grats (- ) (- ) Thus w rfr to as summato oprator. (- ) W kow that s th g. fu. for th squc,,,... - Appl th summato oprator, w gt, -. s th g. fu. for th squc,,, s th g. fu. for th squc,,, ( - ) a a ( a ) ( ) ( ) a a a a a a a a th squc a,( a a ), ( a a a ),......, Appl aga th summato oprator, w gt, f() cosdr, th fucto. (- ) Cosdr th g.fu., for th squc,,,,,... whch s g.fu. for th squc ( ),,,,...,,,,, Appl th summato oprator, w gt, ( ) whch s g. fu for,,,,.... th squc,,,,,... 7

16 Appl aga th summato fucto, w gt, ( ) ( ) whch s th g. fu.for th squc,,,,....,,, 9,... Ths suggsts that for, (k -) k Eampl: Fd a formula to prss as a fucto of. W start wth so s th g. fu. for th squc,,,,,... (- ) g()... th, - - dg() (-)(- ) ( )... ( ) d Rpatg ths tchqu w fd that, d dg() ( )... d d ( ) ( ) so ( ) grats,,,,... Appl summato oprator to ths, w gt, ( ) ( ). ( ) ( ) ( ) ths garats,,,,... ( ) Hc co - ff of s ( ) 8

17 9 ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( s, of ff so th co -... ca also b calcualtd as, ff But ths co - ( ) ( ) ( )( )( ) ( )( )( ) [ ] ( )( ) ( ) ( ) [ ] ( )( ) )!!(! )!!(! Eampl: Fd a formula for th sum of frst atural umbrs usg th gratg fucto for th squc,,, 6,,,. grats,,,6,,,... ) (- Th grats,,6,,,... ) (- th fucto Thus ) (- w hav,, for ) (- W kow that,

18 Now, k k co - ff of co - ff of co - ff of ( - ) ( - ) ( - ) ( ) ( ) ( ) ( )( ) (-)- - Summars (m objcts, cotars) Objcts Cotars Som Numbr Ar Ar Cotars of Dstct Dstct Ma B Empt Dstrbutos Ys Ys Ys m Ys Ys No!S(m,) Ys No Ys S(m,)S(m,)...S(m,) Ys No No S(m,) m No Ys Ys m (m ) No Ys No m No No Ys () p(m), for m No No No () p(m,)p(m,)...p(m,), <m p(m,) p(m.):umbr of parttos of m to actl summads