On Some Generalizations via Multinomial Coefficients

Transcription

1 Bitish Joual of Applied Sciece & Techology 71: 1-13, 01, Aticle objast0111 ISSN: SCIENCEDOMAIN iteatioal wwwsciecedomaiog O Some Geealizatios via Multiomial Coefficiets Mahid M Magotaum 1 ad Najma B Pediama 1 1 Depatmet of Mathematics, Midaao State Uivesity-Mai Campus, Maawi City, Philippies 9700, Philippies Authos cotibutios This wo was caied out i collaboatio betwee all authos Autho MM desiged the idea of the study ad maaged the liteatue seaches Autho NP obtaied the esults ad wote the fist daft of the mauscipt Autho MM poof ead ad modified the mauscipt Authos MM ad NP caied out the evisio of the mauscipt ad the additioal liteatue seaches All authos ead ad appoved the fial mauscipt Aticle Ifomatio DOI: /BJAST/01/10 Editos: 1 Qig-We Wag, Depatmet of Mathematics, Shaghai Uivesity, PR Chia Reviewes: 1 Aoymous, Chia Aoymous, Filad Complete Pee eview Histoy: Oigial Reseach Aticle Received: 9 Novembe 014 Accepted: 30 Decembe 014 Published: 09 Jauay 01 Abstact This pape gives a bief discussio o the Multiomial coefficiets Usig this otio, we obtai geealizatios of the Vademode s ad the Chu Shih-Chieh s idetities fo the Biomial coefficiets, espectively This is doe though the use of two ow piciples i Combiatoics, amely, the Additio ad the Multiplicatio piciples Some examples of geeatig fuctios of a seuece ivolvig the multiomial coefficiets ae also deived ad peseted Keywods: Biomial coefficiets; Multiomial coefficiets; Geeatig fuctios; -aalogues 010 Mathematics Subject Classificatio: 0A10, 0A19, 0A1, 0A99 *Coespodig autho: mmagotaum@yahoocom

2 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast Itoductio Fo a set A of distict objects, a combiatio of A is simply a subset of A Moe pecisely, fo 0, a -combiatio of A is a -elemet subset of A see [1] Fom hee, the biomial coefficiets, ead as tae, is defied to be the umbe of -combiatios of the set A Appeaetly, the umbes play a impotat ole i eumeative combiatoics ad othe field of disciplie Oe may see the boos by Che ad Kho [1] ad Comtet [] fo a moe detailed discussio o the biomial coefficiets We ote that ca be expessed explicitly as!!!, 11! whee 1 1 is the fallig factoial of of ode ad! 1 31 The tem biomial coefficiets comes fom the fact that the umbes appea as coefficiets i the expasio of the biomial expessio x y as see i the well-ow Biomial theoem x y x y 1 Whe y 1, 1 becomes 0 x 1 x, 13 which is the odiay geeatig fuctio of the biomial coefficiets Othe basic popeties ad idetities ivolvig the biomial coefficiets ae the followig: 0 the tiagula ecuece elatio 1 1 ; 14 the idetities ; 1 ad 16 0 Combiatoially, the biomial coefficiets is itepeted as the umbe of ways to distibute idetical objects ito distict boxes such that each box ca hold at most oe object I 177, A T Vademode obtaied the ext idetity which is ow populaly ow as the Vademode s idetity give by m i0 m i i, 17 whee, m, ad ae positive iteges Othe ow esults ae the Chu Shih-Chieh s idetities 1 1, 18 1 fo all positive iteges ad with ; ad , 19

3 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 fo all positive iteges ad, discoveed by Chu Shih-Chieh i 1303 Euatios 19 ad 18 ae ofte efeed to as the Hocey-Stic idetities The euatios 17, 19 ad 18 ca be foud i [1] O the othehad, the Multiomial coefficiets 110 1,,, m is a idetity which geealizes the biomial coefficiets The multiomial coefficiets cout the umbe of ways to distibute distict objects ito m distict boxes such that 1 of them ae i box 1, of them ae i box,, ad m of them ae i box m, whee, m, i, i 1,,, m, ae o-egative iteges such that 1 m Usig this itepetatio, it is easy to show that the multiomial coefficiets satisfy the explicit fom Clealy, whe m i 111, 1, 1,,, m! 1!!! 1!! m! 111, 1, which is the biomial coefficiets The multiomial coefficiets 1,,, m ae ow to have the followig combiatoial itepetatios: the umbe of ways to patitio a -elemet set X ito m pats P 1, P,, P m such that P 1 1, P,, P m m with 1 m; ad the umbe of pemutatios of objects ot ecessaily distict tae all at a time This is euivalet to the umbe of ways to aage objects i a ow The study of the biomial ad the multiomial coefficiets as well as thei diffeet extesios ad applicatios is popula amog mathematicias eg [3, 4,, 6, 7, 8, 9, 10, 11, 1, 13, 14, 1] ad some of the efeeces theei Amog the ow popeties of the multiomial coefficiets ae the followig: the Multiomial theoem x 1 x x m 0 1,,, m 1 1,,, m x 1 1 x x m m, 11 fo positive iteges ad m ad m i1 i ; the idetities, 113 1,,, m α1, α,, αm whee {α1, α,, αm} {1,,, m}, 1 1 1,,, m 1 1,,, m 1, 1,, m 1 ; 114 1,,, m 1 ad m 11 1,,, m 0 1,,, m 3

4 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 Note that the biomial idetities 1, 1, 14, ad 16 ca be obtaied fom the multiomial idetities 11, 113, 114, ad 11, espectively whe m This motivates us to establish moe popeties ad idetities ivolvig the multiomial coefficiets that will geealize the esults i the biomial coefficiets I ode to achieve some of the mai esults of this pape, we will mae use of the Additio ad the Multiplicatio piciples stated as follow: Additio Piciple [1] Assume that thee ae 1 ways fo the evet E 1 to occu, ways fo the evet E to occu,, ways fo the evet E to occu, whee 1 If these ways fo the diffeet evets to occu ae paiwise disjoit, the the umbe of ways fo atleast oe of these evets E 1, E,, o E to occu is 1 i Multiplicatio Piciple [1] Assume that a evet E ca be decomposed ito odeed evets E 1, E,, E ad that thee ae E 1 to occu, ways fo the evet E to occu,, ways fo the evet E to occu The the total umbe of ways fo the evet E to occu is give by 1 i The Additio ad the Multiplicatio piciples ae two of the may fudametal tools used i povig combiatoial idetities Fo a moe detailed discussio o these piciples, see [1] The esults of this pape ae ogaized as follow: i sectio, a fomula that will geealize the Vademode s idetity i 17 is deived i tems of the multiomial coefficiets; i sectio 3, some idetities that will geealize the Chu Shih-Chieh s idetities i 19 ad 18 ae peseted; ad i sectio 4, the geeatig fuctios of a seuece ivolvig the multiomial coefficiets is examied i1 i1 Geealized Vademode s Idetity I this sectio, we will deive a geealizatio of the Vademode s idetity i 17 i tems of the multiomial coefficiets To achieve this, we fist let E be the evet of distibutig distict objects to distict boxes such that box 1 cotais 1 objects, box cotais objects,, ad box cotais objects so that E 1,,, Fo a o-egative itege, whee, evet E occus if the two succeedig evets E 1 ad E occu: E 1 : the evet of distibutig the fist objects to the boxes so that box 1 cotais at most 1 objects, box cotais at most objects,, ad box cotais at most objects That is, box 1 cotais 1 objects, box cotais objects,, ad box cotais objects, whee i i fo i 1,, ; ad E : the evet of distibutig the emaiig objects to the boxes so that 1 1 objects will be placed i box 1, objects will be placed i box,, ad objects will be placed i box Now, give a -ay seuece 1,,, of o-egative iteges with j1 j, the evet E1 occus i 1,,, ways while the evet E occus i 1 1,,, ways Moeove, by Multiplicatio piciple, the two succeedig evets occu i 1,,, 1 1,,, 4

5 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 ways Note that evet E occus fo ay -ay seuece 1,,, of o-egative iteges with j1 j Hece, by Additio piciple, E 1 1,,, 1 1,,, 1 We will state this esult i the followig theoem Theoem 1 Let ad be positive iteges such that j1 j The, 1,,, 1,,, 1 1,,, 1 whee the sum is tae ove all -ay seueces 1,,, of o-egative iteges, whee j1 j Rema 1 Whe i, we have This is exactly the Vademode s idetity i 17 Also, whe m m is a positive itege, the becomes m m, 3 1,,, 1,,, 1 1,,, 1 whee j1 j m 3 is actually idetical to the fomula which was ealie cosideed by Taube [3] ad Calitz [4] To illustate 3, we coside the followig basic poblem i distibutio Example Suppose that a college pofesso wated to fom 3 teams fom a goup of top female ad top male studets comig fom diffeet basic Math classes I how may ways ca this be doe if the said pofesso added a coditio that the fist team should have membes, the secod team should have 4 membes ad the thid team should have 1 membe oly? To solve this, ote that fom 3, we have, 4, ,, 3 1, 4, Obseve that i ode fo the coefficiet 1,4,1 3 to exist, we must have 1, 4, 3 1 Hece, the possible values of 1,, 3 fo which 1,, 3 is cofomable with 1,4,1 3 ae the followig: Futhemoe, we have, 4, 1 Thus, thee ae 10 ways,,, 1,, 3, 0, 1, 4, 0, 1, 3, 1 0, 4, 1,, 1 0,, 0, 3, 0 0, 1, 1 1, 4, 0 1, 0, 1 1, 3, 1 1, 1, 0 0, 4, 1, 0,

6 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast Geealized Chu Shih-Chieh s Idetities The theoems i this sectio cotai a geealized vesio of the Chu Shih-Chieh s idetities i 19 ad 18 i tems of the multiomial coefficiets Theoem 31 Let ad m be o-egative iteges Fo ay -ay seuece 1,,, of o-egative iteges with j1 j, m 1 i m i, 31 1,,,, m 1 i 1, i,, i, 0 1 i 1, i,, i, m i0 whee the ie sum is tae ove all -ay seueces i 1, i,, i of o-egative iteges with j1 ij i ad ij j fo j 1,, Rema 31 Whe 1, 31 becomes m 1 m i i This is idetical to the fist Chu Shih-Chieh s idetity i 19 i0 Poof of Theoem 31 Let S {1,,, m 1} with S m 1 ad let E be the evet of distibutig the elemets of S to 1 disjoit subsets S 1, S,, S 1 of S so that S 1 1, S,, S ad S 1 m 1 Hece E m 1 1,,,, m 1 We may also cout E as follows Note that evet E occus if ay of the 1 disjoit evets E 0, E 1, E,, E occu, whee E i : the evet of distibutig the elemets of S to 1 disjoit subsets as stated i E, whee 1,,, i / S 1 ad i 1 S 1 fo i 0, 1,,, Now, coside a -ay seuece i 1, i,, i of o-egative iteges with j1 ij i ad ij j fo j 1,,, we may decompose each of the evets E i s ito the followig evets: E i1 : the evet of distibutig the elemets 1,,, i to the 1 subsets so that i 1 of them is placed i S 1, i of them ae placed i S,, i of them ae placed i S, ad oe is placed i S 1 so that E i1 i i 1, i,, i, 0 E i : the evet of placig the elemet i 1 i S 1 so that E i 1 E i3 : the evet of fillig up the subsets with the emaiig elemets i, i 3,, m 1 so that S 1 1, S,, S ad S 1 m 1 That is, fom the emaiig m i elemets, we place 1 i 1 elemets i S 1, i elemets i S,, i elemets i S, ad m elemets i S 1 Thus, m i E i3 1 i 1, i,, i, m 6

7 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 Hece, by Multiplicatio piciple, the umbe of ways of distibutig the elemets of S to the 1 subsets so that S 1 1, S,, S ad S 1 m 1, whee 1,,, i / S 1 ad i 1 S 1 fo the -ay seuece i 1, i,, i of o-egative iteges with j1 ij i ad i j j fo j 1,, is 3 i m i E ij i 1, i,, i, 0 1 i 1, i,, i, m j1 Now, cosideig all -ay seueces i 1, i,, i of o-egative iteges with j1 ij i ad i j j fo j 1,,, by Additio piciple, E i i 1 i i i Applyig the Additio piciple, Hece, the poof is doe i i 1, i,, i, 0 E m i 1 i 1, i,, i, m E i i0 If S {1,,, m 1} is a set with S m 1 ad E is the evet of distibutig the elemets of S to 1 disjoit subsets S 1, S,, S 1 of S so that S 1 1, S,, S ad S 1 m 1, that is, E m 1 1,,,, m 1 the may also cout E i the followig mae: Note that E occus if ay of the m disjoit evets E 0, E 1, E,, E m1 occu, E i : the evet of distibutig the elemets of S to 1 disjoit subsets as stated i E, whee 1,,, i S 1 ad i 1 / S 1 fo i 0, 1,,, m 1 It ca be obseved that each evet E i occus if ay of the disjoit evets E i1, E i,, E i occu whee E ij : the evet of distibutig the elemets of S to the 1 disjoit subsets as stated i E, whee 1,,, i S 1 ad i 1 S j fo j 1,,, so that E ij m i 1,,, j 1,,, m i 1 Hece, by Additio piciple, E i Moeove, we have j1 E ij This esult is embedded i the ext theoem m i m i 1 1,,,, m i 1 1, 1,,, m i 1 m i 1,,, 1, m i 1 E E i i0, 7

8 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 Theoem 3 Fo o-egative iteges ad m, ad fo a -ay seuece 1,,, of oegative iteges with j1 j, m 1 1,,,, m 1 m1 i0 [ i 1 1 1,,,, i ] i 1 1,,, 1, i i 1 1, 1,,, i 3 Rema 3 Clealy, it ca be veified that whe 1 i 3, we ecove the secod Chu Shih- Chieh s idetity i 18 Note that the famous Pascal s Tiagle i Figue 1 ca be expessed via multiomial coefficiets as see i Figue Figue 1: Pascal s Tiagle 0, 0 1 0,0 1 0,1 1,0 3 0, 3 1,1 3, ,3 4 1, 4,1 4 3,0 4 0,4 1,3, 3,1 4,0 1,4,3 Figue : Pascal s Tiagle i multiomial coefficiets 3, 4,1,0 Notice that Figue 3 gives a simple illustatio of 3 sice 1 3 4, 3 1, 0 1, 1 1, 1, 3 is the case whe, m ad 1 Similaly, it ca be see i Figue 4 that , 4, 1 4, 0 3, 0, 0 1, 0 0, 0 which is pecisely 31, whee 4, m 0 ad 1 8

9 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 Figue 3: Illustatio of 3 Figue 4: Illustatio of 31 4 Geeatig Fuctios Let a a 0, a 1,, a, be a seuece of umbes The odiay geeatig fuctio fo the seuece a is defied to be the powe seies Ax 0 a x 41 Fo istace, the geeatig fuctio fo the seuece,,,,, 0, 0,, 0 1 whee is a o-egative itege is 1 x This is obtaied thought the use of the Biomial theoem which is the case y 1 i 1 give by 1 x x 0 Note that this coicides with 13 O the othehad, the expoetial geeatig fuctio fo the seuece a is defied to be the powe seies x a 0 a 1 1! x a! x a! x a! 4 0 9

10 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 Now, combiig 11 ad 13, we have 1 x 0 x! This meas that 1 x is the expoetial geeatig fuctio fo the seuece 0, 1,,,, 0, 0, Theoem 41 The expoetial geeatig fuctio fo the seuece a, whee a 1,,, 0 1,,, is a sum tae ove all -ay seueces 1,,, of o-egative iteges with j1 j, is 1 x x! x3 3! e x 43 Rema 41 It is easy to veify that whe 1 i 43, we get the expoetial geeatig fuctio of the seuece 1, 1, 1,, 1, which is e x Poof of Theoem 41 Clealy, we have e x e x e x e x x 1 x x 1 0 1! 0!! 0 x 1 x x ,,, 1!! x 3!! ,,, 1!! 1 3!! whee the sum is tae ove all -ay seueces 1,,, of o-egative iteges, whee j1 j Now, Thus, the poof is doe e x ,,, 0 1,,, x! x 1!,!,,!! x 1,,,! Rema 4 We ca also pove Theoem 41 usig the idetity i 11 That is, by multiplyig both sides of 11 with x ad summig up to ifiity yields! mx x! ,,, 1,,, m! m This is pecisely the esult i Theoem 41, 10

11 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 The ext theoem is deduced fom Theoem 41 Theoem 4 The odiay geeatig fuctio fo the seuece a, whee 1 a 1!!! 0 1,,, is a sum tae ove all -ay seueces 1,,, of o-egative iteges with j1 j, is 1 x x! x3 3! e x 44 Coclusio I this study, we have obtaied geealizatios fo some classical idetities ivolvig the Biomial coefficiets via Multiomial coefficiets The idetity obtaied i Theoem 1 is a geealizatio of the ow Vademode s idetity sice the latte is a paticula case whe the itege i euatio of Theoem 1 Also, the Chu Shih-Chieh s idetities i 18 ad 19 appeas to be paticula cases of the esults stated i Theoem 3 ad Theoem 31, espectively Some geeatig fuctios of seueces ivolvig the Multiomial coefficiets wee also ivestigated ad peseted 6 Recommedatios The authos ecommed the followig fo futhe eseach: 1 The biomial coefficiet satisfies the othogoality elatio 1 j j 1 j i j δ i, 61 j i j i ji ji { 0, i whee δ i is called oece delta 61 ca be obtaied usig the geeatig 1, i fuctio i 13 Maig use of 61, the ivese elatio f g g 1 f, 6 0 ca be obtaied It would be compellig to establish the othogoality ad ivese elatios fo the multiomial coefficiets A -aalogue of the multiomial coefficiet is ofte defied as [ ] []! 1,,, m [ 1]![ ]! [, 63 m]! whee []! i1 [i] is the -factoial of, [] 1 is the -itege, 1 [ ] [ ] []! 1, [ 1]![ 1]! is the -biomial coefficiets ad [ lim 1 1 1,,, m ] 0 1,,, m 64 11

12 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast is called as -multiomial coefficiets see Vioot [1] ad Waaa [9] It is ow that the -biomial coefficiets [ ] satisfy the -biomial ivesio fomula [ ] [ ] f g g 1 f see i [] ad the -biomial Vademode covolutio [ ] [ ] [ ] m m m 0 66 which was itoduced by Bede [] ad futhe studied by Evas [6] ad Sulae [7] Oe may ivestigate the possibility of establishig a -multiomial vesio of 6 ad 66 as well as the -aalogues of 31 ad 3 3 Cocio [13] defied the p, -biomial coefficiets as [ ] p i1 i1, 67 p i i p i1 whee p, ad obtaied its fudametal popeties It is easy to veify that 67 satisfies [ ] [] p! [] p![ ], 68 p! whee [] p p p p ad [] p! j1 [j]p Moeove, Ludow ad Rosege [14] used the p, -biomial coefficiets i 67 to descibe the magetizatio distibutio of the Isig Model It would be compellig to defie a p, -aalogue of the multiomial coefficiets ad study its popeties ad possible applicatios Acowledgmet The authos would lie to tha the efeees fo eviewig the mauscipt ad givig thei valuable commets which impoved the claity of the pape Competig Iteests The authos declae that o competig iteests exist Refeeces [1] Che C, Kho K Piciples ad Techiues i Combiatoics, Wold Scietific Publishig Co Pte Ltd, Sigapoe; 199 [] Comtet L Advaced Combiatoics, D Reidel Publishig Co, Dodetcht, The Neathelads; 1994 [3] Taube S O Multiomial Coefficiets, Ame Math Mothly 1963;70:

13 Magotaum ad Pediama; BJAST, 71, 1-13, 01; Aticle objast0111 [4] Calitz L Sums of Poducts of Multiomial Coefficiets, Elemete de Math 1963;18:37-39 [] Bede E A Geealized -biomial Vademode Covolutio, Discete Math 1971;1: [6] Evas R Bede s Geealized -biomial Vademode Covolutio, Aeuatioes Math1978;17: [7] Suate R A Geealized -Multiomial Vademode Covolutio, J Combiatoial Theoy, Seies A ;33-4 [8] Jiag Z Multiomial Covolutio Polyomials, Discete Math 1996;160:19-8 [9] Waaa SO The Adews-Godo Idetities ad -multiomial Coefficiets Available: axiv:-alg/960101v1; 1996 [10] Chu W Geeatig Fuctios ad Combiatoial Idetities, Glassic Mathematići 1998;333:1-1 [11] Masou T Combiatoial Idetities ad Ivese Biomial Coefficiets, Advaces i Applied Math 00;8:196-0 [1] Spugoli R Sums of Recipocals of the Cetal Biomial Coefficiets, INTEGERS: Electo J of Combi ad Num Theo 006;6, #A7 [13] Cocio R O p, -Biomial Coefficiets, INTEGERS: Electo J Combi ad Num Theo 008;8: #A9 [14] Ludow PH, Rosege A O the p,-biomial Distibutio ad the Isig Model Available: axiv: v1; 010 [1] Vioot CR A Eumeatio of Flags i Fiite Vecto Spaces, The Electo J of Combi 01;193: c 01 Magotaum ad Pediama; This is a Ope Access aticle distibuted ude the tems of the Ceative Commos Attibutio Licese which pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial wo is popely cited Pee-eview histoy: The pee eview histoy fo this pape ca be accessed hee Please copy paste the total li i you bowse addess ba wwwsciecedomaiog/eview-histoyphp?iid770&id&aid