DIAGONAL CHECKER-JUMPING AND EULERIAN NUMBERS FOR COLOR-SIGNED PERMUTATIONS

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1 DIAGONAL CHECKER-JUMPING AND EULERIAN NUMBERS FOR COLOR-SIGNED PERMUTATIONS Niklas Eikse Heik Eiksso Kimmo Eiksso iklasmath.kth.se heikada.kth.se Kimmo.Eikssomdh.se Depatmet of Mathematics KTH SE Stockholm Swede NADA KTH SE Stockholm Swede IMa Mäladales högskola Box 883 SE Västeås Swede Submitted: Septembe ; Accepted: Jauay Abstact. We itoduce colo-siged pemutatios to obtai a vey explicit combiatoial itepetatio of the q-euleia idetities of Beti ad some geealizatios. I paticula we pove a idetity ivolvig the golde atio which allows us to compute uppe bouds o how high a checke ca each i a classical checkejumpig poblem whe the ules ae elaxed to allow also diagoal jumps. 1. Itoductio This pape has two themes. Fist we will peset a otio of colo-siged pemutatios that is siged pemutatios whee the plus sig comes i p diffeet colos ad the mius sig comes i q diffeet colos. These pemutatios give a vey explicit combiatoial itepetatio of the q-euleia idetities of Beti [2] ad also povide some geealizatios of them. Secod we will teat a vaiat of a old checke-jumpig poblem: Whe playig checkes o the squae gid give that the lowe half-plae is full of checkes ad the uppe half-plae is empty how high up i the uppe half-plae ca we place a checke by playig the game? This poblem was solved log ago (see Belekamp Coway ad Guy s etetaiig book o combiatoial games [1]). H. Eiksso ad Lidstöm geealized checke-jumpig to highe dimesios [5]. I his M.Sc. thesis Eikse [4] chaged the ules to allow also diagoal checke-jumpig moves theeby obtaiig a hade poblem fo which we will hee peset a uppe boud o how high the checke ca be placed Mathematics Subject Classificatio. Pimay: 05A30; Secoday: 05A99. Key wods ad phases. Euleia umbes colo-siged pemutatios checke-jumpig pagoda fuctio. N.E.: Patially suppoted by the Swedish Natual Sciece Reseach Coucil. K.E.: Patially suppoted by the Swedish Natual Sciece Reseach Coucil ad the Swedish Foudatio fo Stategic Reseach. 1

2 2 the electoic joual of combiatoics 7 (2000) #R3 The coectio betwee the two themes is the followig idetity: Let be a positive itege ad let σ deote the golde atio ( 5 1)/2. The (1) m1 m (2m +1) 1 σ m+2 σ 2 b k σ k whee the umbes b k ae defied by b 11 1b k 0fok 0adfok>ad the ecuece elatio The fist few values of b k ae: k1 b k (2( k)+1)b 1k 1 +(2k +1)b 1k. k The idetity (1) is eeded fo computatio of the uppe boud i the checkejumpig poblem. Zeilbege (pesoal commuicatio) told us how (1) could be poved usig iductio. Ou ow seach fo a combiatoial poof led us to the idea of colo-siged pemutatios. 2. Pelimiaies o Euleia ad q-euleia umbes Let [] deote the itege set {1...}. Siged pemutatios o [] defie a goup deoted by B. The subgoup cosistig of siged pemutatios with evey value plus-siged is isomophic to the symmetic goup S. We will descibe a siged pemutatio π by its sequece π 1...π of values. Two statistics o pemutatios will be impotat: the umbe of descets ad the umbe of mius sigs. A descet is a positio i whee the pemutatio value deceases: π i 1 >π i. Whe coutig descets we always have the covetio that π 0 0. Letd(π) deotetheumbe of descets of a siged pemutatio π. The umbe of mius sigs (esp. plus sigs) i a siged pemutatio π B is deoted by N(π) (esp. P (π)). I othe wods N(π) def #{i [] :π i < 0} ad P (π) def N(π). The well-kow Euleia umbes A k cout pemutatios i S with k descets. Similaly Euleia umbes of type B ae the umbes B k of siged pemutatios i B with k descets. These umbes defie the Euleia polyomials: A (x) def A k x k ad B (x) def B k x k.

3 the electoic joual of combiatoics 7 (2000) #R3 3 Refiig also by the statistic N(π) Beti [2] defied a q-aalog to these polyomials called the q-euleia polyomial: B (x; q) def B k (q)x k def π B q N(π) x d(π) whee the polyomials B k (q) ae called q-euleia umbes. Foq 1 they educe to the Euleia umbes of type B adfoq 0 they educe to the odiay Euleia umbes. 3. Colo-siged pemutatios Fom the defiitio of the q-euleia umbes it is clea that if the value of q is a oegative itege the so is B k (q). Thee is a simple combiatoial itepetatio of q ot metioed by Beti: If we colo each mius sig by oe of q colos the B k (q) istheumbeofsuchq-colo siged pemutatios i B with k descets. (Coloig does ot affect what is meat by a descet.) This itepetatio allows a obvious geealizatio amely coloig the plus sigs by oe of p colos. Defie the pq-euleia umbe B k (p q) astheumbeofpq-colo siged pemutatios i B with k descets. A fist obsevatio is that B k (p q) (p + q)! sice thee is a choice amog p plus sigs ad q mius sigs fo evey value of a usiged pemutatio. The pq-euleia umbes ae easily expessed i tems of Beti s q-euleia umbes. Lemma 3.1. The pq-euleia umbes ae elated to the q-euleia umbes by B k (p q) p B k ( q p ). Poof. Sice this is a polyomial idetity it suffices to veify it fo all iteges p ad q such that q is divisible by p. Give oe colo fo plus sigs ad q colos fo mius p sigs split evey colo ito p diffeet subcolos. This gives p possible subcoloigs fo evey oigial coloig. This simple elatioship gives diect pq-aalogues of all Beti s theoems fo q- Euleia umbes i Sectio 3 of [2] dealig with ecueces geeatig fuctios polyomial idetities log-cocavity ad uimodality. Usig the geeatig fuctio we obtai the followig explicit fomula fo the pq-euleia umbes.

4 4 the electoic joual of combiatoics 7 (2000) #R3 Theoem 3.2. The pq-euleia umbes B k (p q) satisfy k +1 B k (p q) ( 1) j ((p + q)(k j)+p) j j ( 1) j+1 ((p + q)(k j)+p). j jk+1 Poof. The poof follows the poof i Comtet [3] p. 243 fo odiay Euleia umbes. We shall ow peset ew combiatoial poofs of two of the pq-aalogues of Beti s theoems ad the poceed with a futhe geealizatio. Theoem 3.3. The pq-euleia umbes satisfy the ecuece elatio B k (p q) (p(k +1) + qk)b 1k (p q) +(p( k)+q( k +1))B 1k 1 (p q) ad the bouday coditios B 00 1ad B k 0fo k<0 ad k>. Poof. The bouday coditios ae obvious. The fist tem of the ecuece couts ways to iset ± ito a siged [ 1]-pemutatio with k descets such that the umbe of descets is uchaged. Eithe the ± must go ito a old descet o a + goes last; i all this gives p(k +1) + qk possibilities. The secod tem has a simila itepetatio. Theoem 3.4. As a polyomial i m m + k (p(m +1)+qm) B k (p q) fo positive iteges. I ode to make the idea as clea as possible we shall peset the combiatoial poof of Theoem 3.4 i a seies of thee cases of iceasig geeality Splittig the hypecube usiged case. Fo p 1 ad q 0 Theoem 3.4 specializes to the followig well-kow idetity fo Euleia umbes (cf. Comtet [3] p. 243): m + k (2) (m +1) A k. Fo oegative iteges m the left-had side ca be itepeted as the umbe of gid poits i the -dimesioal hypecube {(x 1...x ) [0m] }. We would the like to fid a simila itepetatio of the ight-had side. Let us divide the hypecube ito! simplices as follows: To evey pemutatio π S coespods a simplex defied by the iequalities (3) 0 x π1 x π2... x π m.

5 the electoic joual of combiatoics 7 (2000) #R3 5 The symbol is defied i the followig way: { x πk <x πk+1 if π k >π k+1 i.e. a descet; x πk x πk+1 x πk x πk+1 if π k <π k+1. It is easy to see that these simplices ae all disjoit ad cove the hypecube. Example 3.5. Let m 2. x 2 6 The pictue shows a lage simplex 0 x 1 x 2 2 coespodig to π 12 ad a smalle simplex 0 x 2 <x 1 2 coespodig to π 21. We emak that apat fom the ules about which egio bouday poits belog to the divisio of the hypecube is doe by the well-kow hypeplae aagemet of type A. The size of a simplex is the umbe of itegal solutios to the iequalities. If k of the iequalities ae stict the by a stadad agumet thee ae m+ k itegal solutios. Sice thee ae A k pemutatios of [] withk descets we have a itepetatio of the ight had side of (2) Splittig the hypecube siged case. Fo p q 1 Theoem 3.4 specializes to the Euleia idetity of type B: m + k (4) (2m +1) B k. We ow itepet the left-had side as the umbe of gid poits i the -dimesioal hypecube {(x 1...x ) [ m m] }. The biomial pat of the ight-had side still sigifies the size of a simplex but the umbe of simplices of each size has ow chaged. Fo a siged pemutatio π B defie the meaig of x πi to be sg(π i )x πi. This gives a well-defied meaig fo siged pemutatios to the simplexdefiig iequalities (3) (ow coespodig to the so called B-aagemet) ad so explais the idetity i a simila way as befoe. Example 3.6. Let m 2. x 2 - x 1 - x 1 6

6 6 the electoic joual of combiatoics 7 (2000) #R3 Fo istace the smallest simplex is defied by 0 < x 1 < x 2 2 coespodig to π 1 2. Obseve that a cosequece of the iequalities (3) is that i the simplex coespodig to a siged pemutatio π the sig of the coodiate x πi is always the sig of π i Splittig the hypecube colo-siged case. The geeal idetity with p colos of plus sigs ad q colos of mius sigs is m + k (5) (p(m +1) +qm) B k (p q). The left-had side is the size of a hypecube with side legth p(m +1)+qm. We obtai this by lettig the age of evey coodiate be [ m m] with p choices of colo fo each oegative value ad q choices of colo fo each egative value. The simplex coespodig to a colo-siged pemutatio π is defied by (3) togethe with the defiitio of the colo of x πi to be the colo of the sig of π i. Example 3.7. Let m 2 ad let p 1ad q 2. colo 1 colo 2 colo 2 colo 1 Fo a colo-siged pemutatio π the colo of the sig of x πi is the colo of the sig of π i Diffeet colos fo diffeet positios. We ca obtai othe tiagles of umbes by allowig a diffeet umbe of sigs to some of the pemutatio values. Say that we have p colos of plus sigs ad q colos of mius sigs fo the fist values ad p esp. q colos fo the emaiig values. We would the get a elatio like this: Theoem 3.8. (p (m +1)+q m) (p(m +1)+qm) m + k b k. The umbes b k ae detemied by the ecuece i Theoem 3.3 usig p q up to ad theeafte cotiuig with p q.

7 the electoic joual of combiatoics 7 (2000) #R3 7 Fially we make a ote that all the above polyomial idetities ca be taslated ito powe seies idetities by the followig well-kow fomula fo geeatig fuctios: (6) A(m) m + k b k m0 A(m) x m b k x k (1 x) Deivig idetity (1). With p 0 ad q 1fo 1 while p 1 ad q 1fohighe Theoem 3.8 gives the followig idetity: m + k m(2m +1) 1 b k with the umbes b k defied by b 10 1b 11 0 ad fo >1 by the stadad ecuece fo p q 1. Idetity (6) yields m(2m +1) 1 x m b k x k (1 x). +1 m0 If we ow take x σ ( 5 1)/2 ad use the fact that 1 σ σ 2 we obtai idetity (1) as pomised i the itoductio. 4. Playig diagoal Pegs i Z Peg solitaie o Pegs is oe of the most famous boad games fo oe peso. It is played o a boad with 33 holes 32 of which ae occupied by a peg ad the objective is to get id of all pegs but oe. (Detailed desciptios ad some iteestig geealizatios ca be foud i Belekamp Coway ad Guy [1] ad i Eikse [4]). A move ca be made wheeve thee ae two adjacet pegs ext to a empty hole all i the same ow o colum. The the oute peg may jump ove the middle peg ito the empty hole theeby emovig the middle peg. With white pegs ad black holes it looks like this: gg - g Oe ca of couse play Pegs o othe boads tha the taditioal oe. I [1] the followig situatio is studied: Use Z 2 as ou boad with the etie lowe half plae (icludig y 0) filled with pegs ad the uppe half plae empty. How fa up ito the uppe half plae ca you put a peg usig Pegs moves? The aswe is that although thee is a ifiite umbe of pegs to use oe ca oly each the fouth ow! The eachability poblem was geealized to Z 2 by H. Eiksso ad Lidstöm [5]: Theoem 4.1 (Eiksso ad Lidstöm [5]). Whe playig Pegs o Z statig with pegs i all holes with coodiates (x 1 x 2... x ) such that x 0 it is impossible to play a peg ito ay hole with x 3 1.

8 8 the electoic joual of combiatoics 7 (2000) #R3 Eiksso ad Lidstöm also pove that the boud is shap: a good playe ca always each level x 3 2. They emak that allowig L-moves of the followig kid will ot icease the age. e e - e I the peset pape we will study how fa pegs ca each whe diagoal moves ae allowed such as e e The pagoda fuctio. We will use the stadad tool fo this kid of poblems: the pagoda fuctio itoduced i [1]. A pagoda fuctio is a fuctio P (x) that assigs a eal value to each positio o the boad with the popety that the sum of the values of all peg positios ca eve icease by a legal move. I ou case the boad is Z ad the iitial peg distibutio is a filled halfspace. The pagoda fuctio is ow used as follows. Let f( k) def P (x). x:x k The if we fid a value of k such that f( k) P ( 0) this implies that it is impossible to each k levels above the peg-filled halfspace sice the pagoda fuctio caot icease. Thus ou fist task is to fid a good pagoda fuctio. I the aalysis of the classical case whee o diagoal moves ae allowed [1 5] the pagoda fuctio was take to be P classic (x) σ x x wheeσ ( 5 1)/2. It is easy to veify that this is a pope pagoda fuctio usig the equatio σ 2 + σ 1. Whe diagoal moves ae allowed too the above fuctio is o loge a pope pagoda fuctio. We will istead defie (7) P (x) def σ max xi fo which the pagoda popety is easily veified. We have P ( 0) 1 so if f( k) 1 fo some ad k the we caot each the kth level i Z. Calculatig f( k) will thus povide us with a uppe limit o the age of the pegs Computatio of f( k). Computatio of the fuctio f( k) is ot tivial ad we will appoach the poblem i seveal steps. To begi with we ca fid a ecuece ivolvig a ifiite sum. Lemma 4.2. The umbes f( k) satisfy the ecuece f( k) 4 f( 1l)+(2k +1)f( 1k) with the iitial values lk+1 f(1k)σ k 2. e

9 the electoic joual of combiatoics 7 (2000) #R3 9 Poof. Let us fist veify the iitial values. We fist obseve that i Z 1 we get a sigle lie of σ σ 2 σ 3... Simple calculatios give f(1k) σ l σk 1 σ σk 2. lk Fo the ecuece we stat with the case 2. Figue 1 shows the σ-logaithm of the Pagoda fuctio. (The zeo is of couse at the oigi.) Fist we calculate f(2 1): O the ight side of the middle colum we have diagoally a sum equivalet to f(1 1) uig fom (1-1) two f(1 2)s uig fom (1-2) ad (2-1) two f(1 3)s etc. The same goes fo the left side ad the middle colum will poduce a f(1 1) givig a total of f(2 1) 3f(1 1) + 4f(1 2) + 4f(1 3) Figue 1. The σ-logaithm of the Pagoda fuctio Calculatig f(2 2) is doe i a aalogous way: We get ice diagoals o the ight ad the sides if we fist emove the thee middle colums. The total weight will the be 5f(1 2) + 4f(1 3) + 4f(1 4) +... A iductive agumet the shows that the ecusio is valid fo evey f(2k) povig the fomula coect fo 2. Highe dimesios ca be dealt with i the same mae. Fo example i ode to compute f(3 1) use the vetical plae i the middle (x 2 0) as a divide ad coside the ight ad left sides. O each we have oe f(2 1) two f(2 2)s two f(2 3)s etc. Togethe with the dividig plae which equals f(2 1) we get f(3 1) 3f(2 1) + 4f(2 2) + 4f(2 3) +... ad the agumet obviously applies to all f( k). Fom this ecusio we ca wite f as a liea fuctio of k makig it easy to fid the smallest value of k such that f( k) 1 fo ay give povided the value of f(j 1) is kow fo all j. Lemma 4.3. The fuctio f( k) is give by the fomula 1 f( k) f( 1) 2(k 1) f(j 1). Hece i paticula f is a liea fuctio of k. The smallest value of k fo which f( k) 1 is clealy k f( 1) 1 () j1 f(j 1) j1

10 10 the electoic joual of combiatoics 7 (2000) #R3 Poof. Fist we simplify the ecusio a bit. Usig k 1 istead of k we obtai f( k 1) 4 f( 1l)+(2k 1)f( 1k 1) lk ad subtactig the secod ecusio fom the fist gives f( k) f( k 1) (2k 3)f( 1k) (2k 1)f( 1k 1). Let us ow assume that f( k) ca be witte i the fom f( k) f( 1) (k 1)h() whee h does ot deped o k ad is to be detemied. We the substitute this ito the expessio above ad extact h(). I ode to be able to use the above fomula ou ext step is to calculate f( 1). Lemma 4.4. The value of all holes with x 1 i Z is give by f( 1) σ 2 m1 m (2m +1) 1 σ m. Poof. We wish to compute the total value of the holes with x 1. This value is equal to the value of the holes with x 1. Hece if we let g() bethevalueofall holes i Z the we ca subtact the total value of the hypeplae x 0 ad divide the diffeece by two to obtai f( 1). But the value of the hypeplae is g( 1). Thuswehavepovedthat g() g( 1) (8) f( 1) 2 The holes i Z with value σ m fo a fixed m 1 costitute the oute shell of a -dimesioal hypecube ceteed aoud the oigi with side legth 2m + 1The umbe of such holes is theefoe (2m +1) (2m 1). Usig (8) we ow obtai f( 1) m1 m1 ((2m +1) (2m 1) ) ((2m +1) 1 (2m 1) 1 ) 2 (2m +1 1)(2m +1) 1 (2m 1 1)(2m 1) 1 (m(2m +1) 1 (m 1)(2m 1) 1 )σ m m1 m(2m +1) 1 σ m m1 2 m(2m +1) 1 σ m+1 m0 m (2m +1) 1 σ m+2 m1 σ m σ m

11 the electoic joual of combiatoics 7 (2000) #R3 11 Fially we obseve that this ifiite sum is computable via idetity (1) which was poved i Sectio 3.5. Hece we have a way of computig k (). The esults ae collected i the below theoem. We ae ow eady to state the theoem which follows diectly fom the lemmas. Theoem 4.5. Suppose that the half-space x 0 is filled with pegs ad the est of Z is empty. The the age of the pegs is bouded eve if we allow diagoal moves. Uppe bouds fo small values of ca be foud i Table 1. Table 1. Ueachable levels of Diapegs i Z ueachable level k f( k ) f( k 1) Lowe bouds. Whe Eiksso ad Lidstöm [5] poved that the uppe bouds could be attaied i the case whee diagoal moves ae ot allowed they showed how a successful costuctio i two dimesios could be used iteatively to solve the poblem i highe dimesios. Such a scheme does ot seem likely to succeed whe diagoal moves ae allowed sice evey time the dimesio is iceased a ew type of diagoal move is itoduced. We have o otivial geeal lowe bouds. Howeve usig ad hoc methods we have costucted a game i Z 2 that eaches level 8 so hee the uppe boud is shap! It is possible that the bouds ae shap fo all but completely ew costuctio ideas would be eeded to pove such a esult. Refeeces [1] E.R.BelekampJ.H.CowayR.K.GuyWiig ways Academic Pess Lodo 1982 vol [2] F. Beti q-euleia Polyomials Aisig fom Coxete Goups Euop. J. Combiatoics 15 (1994) [3] L. Comtet Advaced Combiatoics D. Reidel Publishig Compay. Dodecht [4] N. Eikse Pegs Pebbles Peies ad Piles a study of some combiatoial games Maste thesis KTH [5] H. Eiksso ad B. Lidstöm Twi jumpig checkes i Z d Euop. J. Combiatoics [6] R. Staley Eumeative Combiatoics vol. 1 Cambidge Uivesity Pess Cambidge [7] D. Zeilbege pesoal commuicatio 1999.