Combinatorial and Automated Proofs of Certain Identities

Transcription

1 Cobiatorial ad Autoated Proofs of Certai Idetities The MIT Faculty has ade this article opely available Please share how this access beefits you Your story atters Citatio As Published Publisher Brereto, Justi, Aelia Farid, Marya Karib, Gary Marple, Alex Queo, ad Aula Tefera "Cobiatorial ad Autoated Proofs of Certai Idetities" Electroic Joural of Cobiatorics, Volue 18, Issue p14 Electroic Joural of Cobiatorics Versio Fial published versio Accessed Wed Sep 05 16:05:01 EDT 018 Citable Li Ters of Use Detailed Ters Article is ade available i accordace with the publisher's policy ad ay be subject to US copyright law Please refer to the publisher's site for ters of use

2 Cobiatorial ad Autoated Proofs of Certai Idetities Justi Brereto Massachusetts Istitute of Techology Marya Karib Waye State Uiversity Alex Queo Easter Michiga Uiversity Aelia Farid Colubia Uiversity Gary Marple Colorado State Uiversity-Pueblo Aalu Tefera Grad Valley State Uiversity Subitted: Apr 9, 011; Accepted: Ju 8, 011; Published: Ju 18, 011 Matheatics Subject Classificatio: 05A19 Abstract This paper focuses o two bioial idetities The proofs illustrate the power ad elegace i euerative/algebraic cobiatorial arguets, oder achieassisted techiques of Wilf-Zeilberger ad the classical tools of geeratigfuctioology Key Words ad Phrases: recurrece equatios, cobiatorial idetities, Zeilberger Algorith, WZ, geeratigfuctioology Dedicated to Doro Zeilberger o the occasio of his sixtieth birthday 1 Itroductio Cojecturig ad provig idetities of the for A B where A is su of ice ters such as bioial ad B is a closed for or a su of ice ters is aog aciet ad attractive atheatical probles There are several types of proof techiques fro the electroic joural of cobiatorics , #P14 1

3 various areas of atheatics that ca be used to prove such idetities Aog these techiques, we etio three here: euerative cobiatorics, geeratigfuctioology ad the the Wilf-Zeilberger WZ ethod Euerative cobiatorics deals with coutig the uber of certai cobiatorial objects It gives eaig ad uderstadig of such objects, ad provides a elegat ad creative way of verificatio May probles that arise i applicatios have a relatively siple cobiatorial iterpretatio For exaple the bioial coefficiet couts the uber of differet ways of selectig objects fro a set of objects Eve though this ethod gives cobiatorial iterpretatios, it is at ties challegig to fid cobiatorial descriptios for idetities that have ultiple paraeters or ivolve o-itegral values For a itroductio o cobiatorial arguet techiques, see, aog others, [1, ] Aother classical ethod for provig idetities is the geeratigfuctioology techique which is tie-tested sice Euler It uses foral series expasios ad is flexible eough to be applied to ay situatios Foral series expasios satisfy additive ad ultiplicative properties aig the oe of the ost widely used ethods for provig idetities However, this approach ivolves tedious algebraic aipulatios ad lacs to give eaig of the idetities A classical boo that discusses the geeratigfuctioology ethod is [7] The WZ ethod is oe of the ost recet, efficiet, coputer-assisted/autoated revolutioary techique for provig idetities Aog its powerful features are its istat geeratio of elegat ad short proofs ad its ability to validate idetities for o- itegral values of free paraeters such as, real, coplex, eve ideteriates as well as broader geeralizatios For a superb expositio of the WZ ethod see, aog others, the boos [5, 4] which are devoted to this ad other ethods For a abridged 10 iutes itroductio to the WZ proof style see [6] I this paper we preset proofs of two idetities usig the above etioed ethods The paper is orgaized as follows I Sectio we state the ai idetities I Sectio 3 we provide cobiatorial proofs I Sectio 4 we preset coputer geerated proofs of the ai idetities I Sectio 5 we provide proofs of the idetities usig geeratig fuctios Throughout this paper we deote the set {, + 1, +, } for Z by N We use the covetio that a b 0 for b < 0 or a < b For a foral power series Fx 0 f x, we deote the coefficiet of x by [x ]Fx Mai Idetities Theore 1 If, N 0 the r r r r0 + r r0 r 1 the electroic joural of cobiatorics , #P14

4 Theore If N 0 ad N 1, the Rear The sus i Theore 1 are called Delaoy ubers i the literature 3 Cobiatorial Proofs of the Mai Idetities 31 Proof of Theore 1 We show that both sides of equatio 1 cout the uber of ways of forig two teas with ad positios, respectively, fored by people of oly three atioalities Aerica, Caadia, ad Mexica such that the total uber of Caadias is, ad tea oe does ot have ay Aericas Aswer 1: For 0 r, there are r ways to put r Mexicas o tea oe The reaiig r positios o tea oe will the be filled by Caadias There are r ways to put r Caadias o tea two We the decide positio by positio whether each of the reaiig r positios will be filled by a Aerica or a Mexica There are r ways to do this Therefore, the are r r r ways to a for two teas with r Mexicas o tea oe Thus r0 r r r ways to for the two teas Aswer : For 0 r, there are r ways to put r Aericas o tea two Of the reai + r positios, choose positios to be filled by Caadias There are + r ways to do this We the assig Mexicas to the reai r positios Thus + r r0 ways to for the two teas Hece the idetity follows r 3 Proof of Theore We first divide both sides of equatio by ad split up the su o the right to get a equivalet idetity: We brea the proof of this idetity ito the followig iterestig leas + the electroic joural of cobiatorics , #P14 3

5 Lea 1 If N 0 ad N 1, the Proof Suppose we have + tiles divided ito two groups, the first group cotaiig the first ad the secod cotaiig the last We color each of the first tiles gree The we radoly color each of the reaiig tiles either yellow or blue, with each color equally liely The for the tiles, colored yellow or blue, we radoly choose ot to chage the color, or repait it red, with each actio beig equally liely There are 3 possible fial colorigs of the + tiles, but 4 ways to pic the colors of all + tiles ad the previous colorigs of the red tiles, each of which are equally liely evets We clai that if we radoly color the + tiles this way, ad radoly select oe of the o-red tiles fro all + tiles, the the probability that there are o blue tiles showig ad that the tile we selected is gree is equal to both + ad There are ways to pic the set of red tiles Each of the tiles ca be red or ot red, givig a 1 probability of becoig red Suppose that there are yellow ad blue tiles thus red tiles The the probability that there are o blue tiles ad we pic a gree tile is There are + ways of picig red tiles Therefore, suig over, the probability that there are o blue tiles ad we pic a gree tile is + Now we calculate the followig probabilities: the probability that there are o blue tiles ad the probability that we pic a gree tile give that there are o blue tiles For each o-gree tile, there is a 1 chace the tile eds up red, ad a 1 chace that it is blue if it is ot red So each tile has a 1 chace of becoig blue, ad a 3 chace of ot 4 4 beig blue So clearly the probability that there are o blue tiles is 3 4 Note that, as show above, for ay idividual tile the probability that it is yellow is 1, 4 the probability that it is blue is 1, ad the probability that it is red is 1 So if we ow 4 oly that a tile is ot blue, the the probability that the tile is red is 3 Let p deote the probability that the tile we pic is gree Clearly p 0 1 Now cosider the followig algorith of radoly selectig a o-red tile We radoly select oe of the + tiles If it is ot red it is our tile ad the algorith teriates If the electroic joural of cobiatorics , #P14 4

6 the tile is red we throw it out ad repeat the algorith o the reaiig + 1 tiles This algorith has at ost +1 steps so it always teriates by choosig a o-red tile At every step of the algorith, each o-red tile is equally liely to be chose, so this algorith radoly selects a o-red tile with each o-red tile equally liely We use this algorith to recursively calculate p O the first step there is a chace we pic a gree tile There is a chace that + + we pic a o-gree tile, ad a chace that tile is red Repeatig the algorith with o-gree tiles, we get p + + Let q 1 p The q 0 0, ad + 3 p 1 Thus q p p p q 1 It is clear by iductio that q Therefore, the probability that we pic a gree tile give that there are o blue tiles is , 1 ad the probability that there are o blue tiles ad that we pic a gree tile is Hece, We ow see to evaluate 1 Assue 1, whe 0 we ca directly calculate the desired idetity Suppose we have + tiles divided ito two groups, the first group cotaiig the first ad the the electroic joural of cobiatorics , #P14 5

7 secod cotaiig the last We color each of the first tiles gree The we radoly color each of the reaiig tiles either yellow or blue, with each equally liely Select radoly a subset with a eve uber of eleets fro the blue/yellow tiles uder the followig coditios: all such subsets are equally liely, leave the color of every tile i the selected subset uchaged ad chage the color of the reaiig tiles i the blue/yellow tiles to red We clai that if we radoly color the tiles this way, ad the radoly select oe of the o-red tiles, the probability that there are o blue tiles showig ad that the tile we selected is gree is equal to ad First, give tiles, there are 1 ways to pic a eve uber of tiles to eep blue or yellow this is why we require 1 If tiles reai blue or yellow, the probability that there are o blue tiles ad that we pic a gree tile is Whe we su this + probability over all, we get that the probability that there are o blue tiles ad we pic a gree tile is Now we prove the followig lea: Lea After the above procedure, if there are o blue tiles, ad we pic a rado o-gree tile, the probability that we get a red tile is Proof: Let r deote the desired probability r equals the expected value of the uber of red tiles Thus, r Hece, where a r 1 Now usig the idetity A B A B A a B 1 1 A B we get j 1 j j a 1 a, the electroic joural of cobiatorics , #P14 6

8 By reversig the order of suatio, a j j0 j j j0 j j j Now for j 1, j j j 1 Therefore, a j1 j j j0 j j j0 j j + 1 By the bioial theore, it follows that a Thus, r Lea 3 Uder the restrictio that the uber of blue or yellow tiles is eve, the probability that there are o blue tiles ad that we pic a gree tile is Proof This probability is equal to the product of probability that there are o blue tiles ad the probability that we pic a gree tile, give that there are o blue tiles The probability that there are o blue tiles is 1 1 a Now let p 1 be the probability that we pic a gree tile give that there are o blue tiles The total probability that we pic a gree tile ad there are o blue tiles is p 1 Clearly p 1 0 1, but also ote that p sice whe 1 we ust have 1 red tile ad 0 blue or yellow tiles We use the sae algorith as i the first lea to radoly pic a gree or yellow tile We pic a rado tile: if it is ot red we are doe; if it is red, we toss it out ad repeat the algorith O the first step of the algorith oe of the followig happes: we pic a gree tile with probability, we pic a red tile with r chace ad repeat the + + algorith; otherwise we have piced a yellow tile ad the algorith teriates Thus, we have p r + p the electroic joural of cobiatorics , #P14 7

9 Let q 1 1 p 1 By Lea, we have p p p p 1 1 Therefore, q q q 1 1 Fro p we ow q 1 1 0, ad a easy iductio iplies 1 + q Hece, the probability that there are o blue tiles ad that we pic a gree tile is p This proves that the electroic joural of cobiatorics , #P14 8

10 Cobiig all leas, we get Lea Proof: We clai both quatities cout the uber of ways to pic at least +1 itegers fro the first + itegers If we su over the uber of objects piced, we get + +1 Reversig the order of suatio yields If we su over the placeet of the + 1-th relatively sallest iteger i the set we get the electroic joural of cobiatorics , #P14 9

11 Reversig the order of suatio yields Therefore, Now we have Therefore, This copletes the proof of Theore Autoated Proofs 41 Proof of Theore 1 Sice the suads o the left ad right sides of equatio 1 vaish outside the itervals [0, ] ad [0, ], the idetity is equivalet to r + r 3 r r r r Z r Z Suppressig the free paraeter, let us deote the left ad right sides of equatio 3 by S ad T, respectively Let F 1, r deote the suad of S, ie, F 1, r r r r Applyig Zeilberger s algorith to F1, we get the recurrece equatio: +F 1 +, r+3++5f 1 +1, r +1F 1, r G 1, r+1 G 1, r 4 where + 1 G 1, r + r r r 1 r Suig both sides of equatio 4 over all itegers r yields + S S S 0 the electroic joural of cobiatorics , #P14 10

12 Now let F, r deote the suad of T Applyig Zeilberger s algorith to F, we get the recurrece equatio: +F +, r+3++5f +1, r +1F, r G, r+1 G, r where G, r r r r + + r r Suig both sides of this recurrece equatio over all itegers r yields + T T T 0 Therefore, S ad T satisfy the sae liear recurrece relatio I additio, S0 T0 1 ad S1 T1 + Thus S T for all N 0 4 Proof of Theore + Let F, ad f F, Applyig Zeilberger s algorith to F, we get the recurrece equatio: + + 1F + 1, + 1F, We su both sides fro 0 to to get + + 1f + 1 F + 1, f Usig the rsolve coad i Maple, we fid that Now let T, 1 f 1!! +! ad t T, We iput T, ito Zeilberger s algorith to get the followig recurrece: 4T +, T + 1, + 1T, the electroic joural of cobiatorics , #P14 11

13 We su both sides fro 0 to 1 to get 4t + T +, + T +, + 1 T +, t + 1 T + 1, + 1 T + 1, + 1t T, Usig the rsolve coad i Maple, we fid that t 1!! +! Therefore, f t for all N Rear Those recurreces pertiet to Sectio 4 are autoatically geerated by the Maple pacage EKHAD which is freely available fro [3] Alteratively, oe ca also use the built-i SuTools pacage i Maple that cotais progras that ipleet Zeilberger s algorith 5 Proofs with Geeratigfuctioology 51 Proof of Theore 1 We show that the left ad right sides of equatio 1 represet the coefficiet of x i the expasio of Fx x + x + 1 Now Fx x x r x r r Therefore, [x ]Fx r0 But Fx ca also be writte as Fx 1 + x + 1 x + 1 Therefore, [x ]Fx r0 r r r r r r r0 x j0 Hece the idetity follows + j x j the electroic joural of cobiatorics , #P14 1

14 5 Proof of Theore Equatio is equivalet to Suppressig the free paraeter, let p ad q deote the left ad right sides of equatio 5, respectively Let Px 0 p x ad Qx 0 q x We eed to show that Px Qx Pascal s relatio iplies that p p 1 That eas Px xpx + 1 x 1 1 x, hece 0 1 Px 1 x1 x Swappig the order of suatio i Qx as 0 we get Qx 1 + x x x + 1 x x 1 x +1 1 x x 1 x +1 1 x Px Rear The expasio Acowledgeets 1 1 y y has bee utilized repeatedly We would lie to tha the aoyous referee for providig us with costructive coets ad suggestios ad for providig us a geeratigfuctioology proof of Theore This wor was supported by the Natioal Security Agecy NSA uder Grat No H , which fuds the Suer Udergraduate Research Istitute i Experietal Matheatics SURIEM at Michiga State Uiversity MSU The authors would lie to tha NSA ad MSU for their geerous support the electroic joural of cobiatorics , #P14 13

15 Refereces [1] A T Bejai ad J J Qui, Proofs that Really Cout: The Art of Cobiatorial Proof, The Matheatical Associatio of Aerica, 003 [] M Bóa, A Wal Through Cobiatorics: A Itroductio to Eueratio ad Graph Theory, d ed, 006 [3] EKHAD, a MAPLE pacage by Doro Zeilberger, zeilberg/ [4] W Koepf, Hypergeoetric Suatio: A Algorithic Approach to Suatio ad Special Fuctio Idetities, AMS, 1998 [5] M Petovše, H S Wilf ad D Zeilberger, A B, A K Peters, Wellesley, Massachusetts, 1996 [6] A Tefera, What is a Wilf-Zeilberger Pair?, Notices of the Aerica Matheatical Society, Vol 57, No 4, 010, pp [7] H S Wilf, geeratigfuctioology, Acadeic Press, d ed, 1993 the electroic joural of cobiatorics , #P14 14