Enumeration of Sequences Constrained by the Ratio of Consecutive Parts

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1 Eumeraio of Seueces Cosraied by he Raio of Cosecuive Pars Sylvie Coreel Suyoug Lee Carla D Savage November 13, 2004; Revised March 3, 2005 Absrac Recurreces are developed o eumerae ay family of oegaive ieger seueces λ =(,,λ ) saisfyig he cosrais: λ 2 λ 1 λ 0, a 1 a 2 a 1 a for a give cosrai seuece a =a 1,,a of posiive iegers They are applied o derive ew couig formulas, o reveal ew relaioships bewee families, ad o give simple proofs of he rucaed lecure hall ad ai-lecure hall heorems Nous développos des récurreces pour éumérer des familles de suies d eiers λ =(,,λ ) saisfaisa les coraies λ 2 λ 1 λ 0, a 1 a 2 a 1 a pour ue suie d eiers posiifs doée a =a 1,,a Ces récurreces permee de dériver de ouvelles formules déuméraio, de révéler de ouvelles relaios ere ceraies familles, e de doer des preuves simples des héorèmes des pariios Lecure Hall rouées e des composiios Lecure Hall rouées 1 Iroducio We cosider he problem of eumeraig oegaive ieger seueces λ =(,,λ ) saisfyig he cosrais: λ 2 λ 1 λ 0, (1) a 1 a 2 a 1 a CNRS PRiSM, UVSQ, 45 Aveue des Eas-Uis, Versailles, Frace (syl@prismuvsfr) Compuer Sciece, Norh Carolia Sae Uiversiy, Box 8206, Raleigh, NC 27695, USA (slee7@uiycsuedu) Research suppored i par by NSF gra DMS Compuer Sciece, Norh Carolia Sae Uiversiy, Box 8206, Raleigh, NC 27695, USA (savage@csccsuedu) Research suppored i par by NSF gras DMS ad INT

2 for a give cosrai seuece a =a 1,,a of posiive iegers We refer o a seuece λ =(,,λ ) of oegaive iegers as a composiio io oegaive pars If he pars of λ are oicreasig, he λ is a pariio Pariios ad composiios are commoly defied by he se of pars allowed, he umber of occurreces of a par, or he differece bewee cosecuive pars I coras, he composiios saisfyig (1) are cosraied by he raio of cosecuive pars ad we refer o hem as raio composiios Geeraig fucios are ow for raio composiios oly for some special cosrai seueces, a, icludig: a =1, 1,,1: ordiary pariios 1; a =1, 2, 4,,2 1 : Cayley composiios 8, 2, 14, 4; a =r 1,r 2,,r,1: Hicerso pariios 13; a =, 1,,1: lecure hall pariios 5, 6, 7, 15, 16, 3; a =1, 2,,: ai-lecure hall composiios 9; a =1, 2, 1, 2,, 2 ( 1) : oe-wo composiios 11; a =, 1, + 1: rucaed lecure hall pariios 10; a = +1,,,: rucaed ai-lecure hall composiios 10 I his paper, we iroduce a commo approach for he eumeraio of raio composiios by usig as a saisic a boud o he size of he firs par This geeralizes he eumeraio of ordiary pariios via he Gaussia polyomials I Secio 2, we derive a recurrece for he geeraig fucio of ay family of raio composiios wih firs par bouded We use his o derive ew couig formulas ad heir -aalogs Amog hese, we discover a family of polyomials wih several ieresig properies which arise i he eumeraio of lecure hall pariios I addiio, we fid a fucioal relaioship bewee he geeraig fucios for he raio composiios cosraied by a seuece a 1,a 2,,a ad hose cosraied by is reverse seuece a,a 1,a 1 This reveals for he firs ime he relaioship bewee, eg, lecure hall pariios ad ai-lecure hall composiios ad bewee Hicerso pariios ad Cayley composiios I Secio 3, we derive a differe recurrece for he eumeraio of raio composiios wih firs par bouded By allowig he boud o approach ifiiy, we ge a recurrece for he geeraig fucio of ay family of raio composiios wih firs par ubouded As oe coseuece, we discover ew easy proofs of he (rucaed) lecure hall ad (rucaed) ai-lecure hall heorems I coras o earlier proofs, where derivig a recurrece was a challege, here he recurrece is geeric ad he wor is moved eirely o sadard -series maipulaio i a iducio proof 2

3 2 Eumeraio of raio composiios wih firs par bouded To build a recurrece, we firs cosider he case where he cosrai seuece a i (1) saisfies a 1,,a =s,s 1,,s 1 for a ifiie seuece of posiive iegers {s i } For 0, le S be he se of composiios λ =(,λ 2,,λ ) saisfyig s λ 2 s 1 λ 1 s 2 λ s 1 0 (2) Le S (j,i) be he se of λ S wih js + i ad le S (j,i) () = λ λ S (j,i) Theorem 1 For 0, j 0, ad0 i s, S (j,i) () = 1 if =0or j = i =0,else S (j 1,s) () if i =0,else S (j,i 1) ()+ js+i S (j, is 1/s ) 1 () oherwise Proof The heorem is clearly rue for = 0 ad for j = i =0 Le(, j, i) saisfy>0, (j, i) > (0, 0) If i = 0, he j>0adjs +i = js =(j 1)s +s, so he heorem is rue Assume, he, ha 1 i s By defiiio, λ S (j,i) if ad oly if eiher λ S (j,i 1) λ S ad = js + i Bu (js + i, λ 2,,λ ) S ifadolyif(λ 2,,λ ) S 1 ad (js + i)/λ 2 s /s 1 Tha is, λ 2 s 1 (js + i) =js 1 + i s 1 s s So, sice λ 2 is a ieger, λ 2 js 1 + i s 1 s Noe, sice 1 i s, is 1 /s s 1,so(λ 2,,λ ) S (j, is 1/s ) 1 Remar 1 For ordiary pariios, P, io oegaive pars, {s i } = {1} ad is 1 /s = i, so he recurrece of Theorem 1 reduces o he recurrece P (j,i) () =P (j,i 1) he familiar recurrece for Gaussia polyomials ()+ j+i P (j,i) 1 (), The lecure hall pariios 5, L, are hose composiios, ecessarily pariios, saisfyig λ 2 1 λ 1 2 λ 1 or 0 (3) The L = S wih {s i } = {i} i (2) Sice i( 1)/ = i 1 we ge he followig from Theorem 1 3

4 Corollary 1 For 0, j 0, ad0 i, lel (j,i) () be he geeraig fucio for he lecure hall pariios λ L wih j + i L (j,i) = 1 if =0or j = i =0,else L (j 1,) () if i =0,else L (j,i 1) ()+ j+i L (j,i 1) 1 () oherwise For fixed >0, ay 0 ca be wrie uiuely i he form = j + i, where 0 i< So, we ge a ice couig formula for lecure hall pariios wih larges par a mos Theorem 2 For 0, j 0, ad0 i, he umber of lecure hall pariios i L wih firs par bouded by j + i is L (j,i) =(j +1) i (j +2) i Proof If = 0 he i =0,so(j +1) 0 0 (j +2) 0 =1 Ifi = j = 0, he (1) 0 (2) 0 =1 Le (, j, i) saisfy>0, (j, i) > (0, 0) ad assume he heorem is rue for (,i,j ) < (, i, j) If i = 0, he j>0ad by Corollary 1, L (j,0) (j) (j +1) =(j +1) (j +2) 0 Oherwise, by Corollary 1, L (j,i) (1) = L (j 1,) (1), which, by iducio, is (1) = L (j,i 1) (1) + L (j,i 1) 1 (1) = (j +1) i+1 (j +2) i 1 +(j +1) i (j +2) i 1 = (j +1) i (j +2) i I 5, i was show ha he geeraig fucio for he lecure hall pariios, L is: 1 L () = (; 2, (4) ) where (a; ) =(1 a)(1 a) (1 a 1 ) Le D be he se of pariios io disic pars ad le O be he se of pariios io odd pars The ses D ad O have geeraig fucios D() =( ; ) ad O() =(; 2 ) 1, respecively Sice lim L = D, he Lecure Hall Theorem (4) is a fiie versio of Euler s Theorem which says ha D() = O() The polyomial L (j) () =L (j,0) () ca be viewed as a -aalog of (j +1) ha ecapsulaes a furher fiiizaio of Euler s Theorem i he followig sese Corollary 2 The lecure hall polyomials L (j) () saisfy (i) L (j) (1) = (j +1), (ii) lim L (j) () =( ; ),ad (iii) lim j L (j) () =(; 2 ) 1 4

5 Proof The firs euaio follows from Theorem 2 The secod ad hird follow from he observaios ha lim L (j) = D ad lim j L (j) = L The ai-lecure hall composiios 9, A, are hose seueces saisfyig 1 λ 2 2 λ 1 1 λ 0 (5) I was show i 9 ha A has geeraig fucio A () =(, ) /( 2 ; ) The cosrai seuece for A, i he sese of (1), is 1, 2,,, he reverse of he cosrai seuece, 1,,1 for L We iroduce some oaio o describe he relaioship bewee heir geeraig fucios Le Sa 1,a 2,,a be he se of composiios λ =(,λ 2,,λ ) saisfyig a 1 λ 2 a 2 λ 1 a 1 λ a 0, (6) wih S (j,i) a 1,a 2,,a deoig hose wih ja 1 + i ad le S (j) a 1,a 2,,a = S (j,0) a 1,a 2,,a Theorem 3 The geeraig fucios for S (j) a 1,a 2,,a ad S (j) a,a 1,,a 1 saisfy: S (j) a 1,a 2,,a () = j(a 1+a 2 + +a ) S (j) a,a 1,,a 1 (1/) Proof The resul follows if we show ha λ S (j) a,a 1,,a 1 if ad oly if µ S (j) a 1,a 2,,a, where µ is defied by µ i = js i λ +1 i So, assume λ S (j) a,a 1,,a 1, ha is, ja ad a i λ i a i+1 λ +1 i for 1 i The for 1 i, λ +1 i a i a i+1 a i+1 a i+2 a 1 a a j = a i j, so µ i = ja i λ +1 i 0 Also, µ 1 = ja 1 λ saisfies µ 1 ja 1 To show µ S (j) a 1,a 2,,a, i remais o show a i+1 µ i a i µ i+1 : a i+1 µ i = a i+1 (ja i λ +1 i )=ja i a i+1 a i+1 λ +1 i ja i a i+1 a i λ i = a i µ i+1 The coverse is similar Remar 2 The proof of Theorem 3 also shows ha for 1 i, λ S (j) a,a 1,,a 1 if ad oly if µ S (j+1) a 1,a 2,,a adµ a i Corollary 3 Lecure hall pariios, L (j), wih firs par bouded by j ad ai-lecure hall composiios, A (j) wih firs par bouded by j have he followig relaioship: A (j) () = j(+1)/2 L (j) (1/) 5

6 Proof Observe ha A (j) Theorem 3 = S (j) 1, 2,,adL (j) This gives a couig formula for ai-lecure hall composiios = S (j), 1,,1 ad apply Corollary 4 The umber of ai-lecure hall composiios i A wih firs par bouded by j is A (j) (1) = (j +1) Proof By Theorem 3, A (j) (1) = L (j) (1) By defiiio, L (j) () = L (j,0) (), ad by Corollary 2, L (j,0) (1) = (j +1) (j +2) 0 As aoher example of he applicaio of Theorems 1 ad 3, we cosider Hicerso pariios H (for r = 2) ad Cayley composiios, C H is he se of composiios io oegaive pars saisfyig λ i 2λ i+1 ad C is he se of composiios io oegaive pars saisfyig λ i λ i+1 /2 So, H = S2 1, 2 2,,1 ad C = S1, 2, 4,,2 1 Le B() be he umber of biary pariios of, ie, he umber of pariios of io powers of 2 I is easy o chec ha B(0) = B(1) = 1, B(2) =B(2 2) + B(), ad B(2) =B(2 +1) Theorem 4 For 0 i<2 1, he umber of Hicerso pariios wih firs par a mos i is H (0,i) = B(2i); wihfirsparamos2 1 + i is H (1,i) = B(2 +2i); wihfirspara mos 2 is H (2,0) = B(2 +1 ) 1 Proof Use he recurrece of Theorem 1 wih he observaio ha sice {s i } = {2 i 1 } for Hicerso pariios, is i 1 /s i = i/2 The heorem follows by iducio usig he properies of B() Cayley s Theorem 8 says ha he umber of composiios i C wih posiive pars ad wih firs par 1 is eual o he umber of pariios of io pars from he se {1, 1, 2, 4,,2 2 } If we apply Theorem 3 ad Remar 2, we ge a geeralizaio ad reformulaio of Cayley s Theorem Theorem 5 For 0 i<2 1, he umber of Cayley composiios io posiive pars wih firs par 1 ad las par a leas 2 1 i is B(2i) The umber wih firs par a mos 2 ad las par a leas 2 i is B(2 +2i) Seig i =2 1 1 i Theorem 5 gives: Corollary 5 The umber of Cayley composiios io posiive pars wih firs par 1 is B(2 2); wihfirsparamos2isb(2 +1 2) These resuls ca be geeralized o r ary Hicerso pariios ad Cayley composiios For oher families of raio composiios, we ca expec Theorem 1 o be mos useful for seueces {s i } where is 1 /s has a ice form 6

7 As ges larger, solvig for H () ges harder This is i spie of he fac ha H has he ice geeraig fucio H () = =1 (1 2 1 ) 1 13 However, we will see i he () has a ice geeraig fucio whe j = 1, here is hope ha () will also ex secio ha if S (j) lim j S (j) I he ex secio we show how o ge a recurrece for he geeraig fucio of raio composiios whe he firs par uresriced 3 Eumeraio of raio seueces wih firs par ubouded We defie wo sligh variaios of he se S (j) a 1,a 2,,a below: P (j) a 1,a 2,,a ={λ S (j) a 1,a 2,,a λ 1}; R (j) a 1,a 2,,a ={λ S (j) a 1,a 2,,a <ja 1 } I P (j), all pars mus be posiive, whereas i R (j) pars ca be oegaive, bu he boud o he firs par becomes sric Theorem 6 For j 1, P (j) a 1,a 2,,a () = (a 1+a 2 + +a ) P (1) a +1,,a ()P (j 1) a 1,,a () Proof Le λ P (j) a 1,a 2,,a adle be he maximum idex such ha λ >a The ( a 1,λ 2 a 2,,λ a ) P (j 1) a 1,,a ad(λ +1,,λ ) P (1) a +1,,a Noe ha P (1) () ca be compued usig Theorem 1 as P (1) s,,s 1 () =S (1,0) s 1,,s () S (1,0) 1 s 2,,s () Thus, aig he limi as j i Theorem 6 gives a recurrece for couig he seueces λ i P a 1,a 2,,a wihou a resricio o he size of he firs par Theorem 7 For, P a 1,a 2,,a () = (a 1+a 2 + +a ) P (1) a +1,,a ()P a 1,a 2,,a () Remar 3 Theorem 6 ad is proof are valid wih all occurreces of P replaced by R We eed oly chage he saeme le be he maximum idex such ha λ >a o le be he maximum idex such ha λ a +1 Similarly, Theorem 7 holds wih P replaced by R 7

8 Theorem 7 ca be used o fid a explici form for he geeraig fucio i families where P (1) a 1,,a (orr (1) a 1,,a ) is ow Remar 4 For ordiary pariios io posiive pars, a i =1,soP a 1,a 2,,a () = (; ) 1 ad P (1) a +1,,a () =, ad he recurrece of Theorem 7 becomes 1 =, (; ) (; ) which cous pariios io posiive pars by summig over he umber,, of pars greaer ha 1 The pariios L, = P, 1,, +1 are called rucaed lecure hall pariios wih all pars posiive The composiios A, = R +1, +2,, are called rucaed ai-lecure hall composiios wih oegaive pars These were iroduced i 10 where heir geeraig fucios were show o be L, () = (+1 2 ) A, () = ( +1 ; ) ( 2 +1 ; ), (7) ( +1 ; ) ( 2( +1) ; ) (8) We ow show how Theorem 7 ca be used o give simple proofs of he (Trucaed) Lecure Hall ad (Trucaed) Ai-lecure Hall Theorems We begi by compuig P (1), 1,, +1() forl, ad R (1) +1, +2,,() fora, Lemma 1 For 1, he geeraig fucio for rucaed lecure hall pariios io posiive pars wih firs par less ha or eual o is P (1), 1,, +1() = (+1 2 ) Proof We show P (1), 1,, +1 is he se of pariios io disic pars from {1, 2,,}, which has he geeraig fucio claimed If λ P (1), 1,, +1, 1 / λ 2 /( 1) λ /( +1) > 0, so he pars of λ are disic ad bouded by Coversely, if λ i λ i+1 +1 for 1 i 1ad, he λ i+1 i, so λ i ( i) (λ i+1 +1)( i) λ i+1 ( i)+ i λ i+1 ( i)+λ i+1 λ i+1 ( i +1), ha is, λ i /( i +1) λ i+1 /( i), so λ P (1), 1,, +1 Lemma 2 For 1, he geeraig fucio for rucaed ai-lecure hall composiios io oegaive pars wih firs par less ha +1 is R (1) +1, +2,,() = 8

9 Proof If λ R (1) +1, +2,,, he for 1 i, λ i < + i ad λ i 1 λ i ( + i 1)/( + i), so λ i 1 λ i λ i (( + i 1)/( + i) 1) = λ i /( + i) > 1 Thus, λ i 1 λ i 0adλis a pariio io a mos pars of size a mos Coversely, ay such pariio is i R (1) +1, +2,, Corollary 6 ( ) L (j), =(j) = A (j), Proof By iducio I is obviously rue for j = 0 The for j>0, by Theorem 6 ad Lemma 2, ( )( ) L (j), (1) = (j 1) We use he classical ideiy ( ( )( ) = )( ) ad he biomial heorem ad ge he resul By Theorem 3, L (j), (1) = A(j), (1) Lemma 3 The geeraig fucio for rucaed lecure hall pariios saisfies for 1 ad L,0 () =1 L, () = (+1 2 ) ( ) L, (), Proof Sice L, = P, 1,, + 1, apply Theorem 7 wih Lemma 1 o ge ad simplify L, () = +( 1)+ +( +1) ( +1 2 ) This gives he Trucaed Lecure Hall Theorem: L, () Theorem 8 10 L, () = (+1 2 ) ( +1 ; ) ( 2 +1 ; ) Proof Le L, () = L, () (+1 2 ) 9

10 We show L, () = ( +1 ; ) ( 2 +1 (9) ; ) Subsiuig for L, () adl, () i he recurrece of Lemma 3 ad simplifyig gives L,1 = 1 ad for >1, as L, () = ( )+(+1 2 ) = L, (), (10) We mae use of (a rasformaio of) oe of he -Chu Vadermode ideiies 152 from 12: ( a; ) = a (+1 2 ) ( (c/a) +1 ; ) (c; ) (c +1 ; ) We do he subsiuio a = ad c = 2 ad ge ( +1 ; ) ( 2 +1 ; ) = ( )+(+1 2 ) ( +1 ; ) ( 2 +1 ; ), which shows ha L, () as give by (9) is he soluio o he recurrece (10) From his we also ge he Lecure Hall Theorem: Theorem 9 5 L () = ( ; ) ( +1 ; ) = 1 (; 2 ) Proof Seig = i Theorem 8 gives L,, which is he se of pariios i L wih all pars posiive So, L, () = (+1 2 ) L () The approach for rucaed ai-lecure hall composiios is similar Lemma 4 For, wih A (0), () =1 A, () = ( )+(+1 2 ) A +, () Proof Sice A, = R +1, +2,,, apply Theorem 7, usig Remar 3, he Lemma 2 ad simplify Now we ge a easy proof of he Trucaed Ai-lecure Hall Theorem 10

11 Theorem A, () = ( +1 ; ) ( 2( +1) ; ) Proof Le The usig Lemma 4, we ge ha as A, () = A, () = A, () ( )+(+1 2 ) + = A +, (), (11) We mae use of aoher rasformaio of he -Chu Vadermode ideiy 152 from 12: ( c/a; ) = (c/a) ( 2) ( a; ) (c; ) (c; ) Se c = 2( +1) ad a = +1 ad ge ( +1 ; ) ( 2( +1) ; ) = ( )+(+1 2 ) ( +1 ; ) ( 2( +1) ; ), showig A, () = ( +1 ;) saisfies he recurrece ( 2( +1) ;) Seig = i Theorem 10 gives he Ai-Lecure Hall Theorem Theorem 11 9 A () = ( ; ) ( 2 ; ) 4 Cocludig Remars The recurreces of Theorems 1 ad 7 provide simple compuaioal ools o ivesigae ay family of raio composiios More imporaly, hey supply he foudaio for a iducive proof of ay cojecured eumeraio resul Our experimes sugges ha couig formulas ad geeraig fucios will be possible for may oher families of raio composiios Oe paricular uesio of ieres is o characerize he subfamily of pariios io odd pars i {1, 3,,2 1} ha is coued by he polyomial L (j) () 11

12 Refereces 1 George E Adrews The heory of pariios Addiso-Wesley Publishig Co, Readig, Mass- Lodo-Amserdam, 1976 Ecyclopedia of Mahemaics ad is Applicaios, Vol 2 2 George E Adrews The Rogers-Ramauja reciprocal ad Mic s pariio fucio Pacific J Mah, 95(2): , George E Adrews MacMaho s pariio aalysis I The lecure hall pariio heorem I Mahemaical essays i hoor of Gia-Carlo Roa (Cambridge, MA, 1996), volume 161 of Progr Mah, pages 1 22 Birhäuser Boso, Boso, MA, George E Adrews, Peer Paule, Axel Riese, ad Voler Srehl MacMaho s pariio aalysis V Bijecios, recursios, ad magic suares I Algebraic combiaorics ad applicaios (Gößweisei, 1999), pages 1 39 Spriger, Berli, Mireille Bousue-Mélou ad Kimmo Erisso Lecure hall pariios Ramauja J, 1(1): , Mireille Bousue-Mélou ad Kimmo Erisso Lecure hall pariios II Ramauja J, 1(2): , Mireille Bousue-Mélou ad Kimmo Erisso A refieme of he lecure hall heorem J Combi Theory Ser A, 86(1):63 84, Arhur Cayley O a problem i he pariio of umbers Philosophical Magazie, 13: , 1857 repried i: The Colleced Mahemaical Papers of A Cayley, Vol III, Cambridge Uiversiy Press, Cambridge, 1890, ) 9 Sylvie Coreel ad Carla D Savage Ai-lecure hall composiios Discree Mah, 263(1-3): , Sylvie Coreel ad Carla D Savage Lecure hall heorems, -series ad rucaed objecs J Combi Theory Ser A, 108(2): , Sylvie Coreel ad Carla D Savage Pariios ad composiios defied by ieualiies Ramauja J, 8(3): , George Gasper ad Miza Rahma Basic hypergeomeric series, volume 35 of Ecyclopedia of Mahemaics ad is Applicaios Cambridge Uiversiy Press, Cambridge, 1990 Wih a foreword by Richard Asey 13 Dea R Hicerso A pariio ideiy of he Euler ype Amer Mah Mohly, 81: , Hery Mic A problem i pariios: Eumeraio of elemes of a give degree i he free commuaive eropic cyclic groupoid Proc Ediburgh Mah Soc (2), 11: , 1958/ Ae Ja Yee O he combiaorics of lecure hall pariios Ramauja J, 5(3): , Ae Ja Yee O he refied lecure hall heorem Discree Mah, 248(1-3): ,