Tackling Sequences From Prudent Self-Avoiding Walks

Transcription

1 Taclng Sequences From Pruden Self-Avodng Wals Shanzhen Gao, Keh-Hsun Chen Deparmen of Compuer Scence, College of Compung and Informacs Unversy of Norh Carolna a Charloe, Charloe, NC 83, USA Emal: sgao3@unccedu, chen@unccedu Absrac A self-avodng wal SAW s a sequence of moves on a lace no vsng he same pon more han once A SAW on he square lace s pruden f never aes a sep owards a verex has already vsed Pruden wals dffer from mos subclasses of SAWs ha have been couned so far n ha hey can wnd around her sarng pon Some neresng problems and sequences arsng from pruden wals of one-sded and wo-sded are dscussed n hs paper A few mehods such as compuaonal, ernel, generang funcon, recurrence relaon and consrucve mehod are appled o our sudy Several open problems are posed Keywords: Self-avodng wal, pruden self-avodng wal, generang funcon, ernel mehod, neger sequence I INTRODUCTION A well-nown long sandng problem n combnaorcs and sascal mechancs s o fnd he generang funcon for self-avodng wals SAW on a wo-dmensonal lace, enumeraed by permeer A SAW s a sequence of moves on a square lace whch does no vs he same pon more han once I has been consdered by more han one hundred researchers n he pass one hundred years, ncludng George Polya, Tony Gumann, Laszlo Lovasz, Donald Knuh, Rchard Sanley, Doron Zelberger, Mrelle Bousque-Mélou, Thomas Prellberg, Neal Madras, Gordon Slade, Agnes Del, EJ Janse van Rensburg, Harry Kesen, Suar G Whngon, Lncoln Chayes, Iwan Jensen, Arhur T Benamn, and ohers More han hree hundred papers and a few volumes of boos were publshed n hs area A SAW s neresng for smulaons because s properes canno be calculaed analycally Calculang he number of self-avodng wals s a common compuaonal problem [], [], [3] In order o presen our problems and resuls clearly and effcenly, we nroduce some noaons n he followng Eas sep: E or or, 0, x-sep You can see more n he able below: 0,, 0, 0, N E NE S, 0,,, W SW NW SE : or more han consecuve seps : consecuve seps avodng : no or more han consecuve seps avodng : no consecuve seps, bu can have more han or less han consecuve seps x : he larges neger no greaer han x, floorx x : s he smalles neger no less han x, celngx [x n ]fx denoes he coeffcen of x n n he power seres expanson of a funcon fx [x m y n ]fx, y denoes he coeffcen of x m y n n he power seres expanson of a funcon fx, y n r he number of combnaons of n hngs r a a me n n! r n r!r! n n r n n r r n n r r r r In he pas few decades, many mahemacans have suded he followng wo classcal problems: Classcal Problem Wha s he number of SAWs from 0, 0 o n, n n an n n grd, ang seps from {,,, }? Donald Knuh clamed ha he number s beween and for n and he dd no beleve ha he would ever n hs lfeme now he exac answer o hs problem n 975 However, afer a few years, Rchard Schroeppel poned ou ha he exac value s, 568, 758, 030, 464, 750, 03, 4, [4], [5], [6] I s sll an unsolved problem for n > 5 Classcal Problem Wha s he number fn of n-sep SAWs, on he square lace, ang seps from {,,, }? The number fn s nown for n 7 [4], [5], [7], [8] I s clear ha n fn 4 3 n fm n fmfn There exss a consan C such ha lm n fn/n nf n [fn]/n C C 64 up o 7 seps have been couned

2 C 638 up o 9 seps have been couned fn 638 n The number of SAWs/ he number of oal wals: 00 for n 0 for n A recenly proposed model called pruden self-avodng wals PSAW was frs nroduced o he mahemacs communy n an unpublshed manuscrp of Préa, who called hem exeror wals A pruden wal s a conneced pah on square lace such ha, a each sep, he exenson of ha sep along s curren raecory wll never nersec any prevously occuped verex Such wals are clearly self-avodng [9], [0], [], [], [3] We wll al abou some sequences arsng from PSAWs n he followng Each PSAW possesses a mnmum boundng recangle, whch we call box Less obvously, he endpon of a pruden wal s always a pon on he boundary of he box Each new sep eher nflaes he box or wals prudenly along he border Afer an nflang sep, here are 3 possbles for a wal o go on Oherwse, only In a one-sded PSAW, he endpon les always on he op sde of he box The wal s parally dreced A pruden wal s wo-sded f s endpon les always on he op sde, or on he rgh sde of he box The wal n he followng fgure s a wo-sded PSAW II PRUDENT SELF-AVOIDING WALKS: DEFINITIONS AND EXAMPLES A PSAW s a proper subse of SAWs on he square lace The wal sars a 0, 0, and he empy wal s a PSAW A PSAW grows by addng a sep o he end pon of a PSAW such ha he exenson of hs sep - by any dsance - never nersecs he wal Hence he name pruden The wal s so careful o be self-avodng ha refuses o ae a sngle sep n any drecon where can see - no maer how far away - an occuped verex The followng wal s a PSAW A Properes of a PSAW Unle SAW, PSAW are usually no reversble There s such an example n he followng fgure III SOME SEQUENCES ARISING FROM ONE-SIDED PSAWS Sequence Wha s he number say fn of one-sded n- sep pruden wals, ang seps from {,, }? The generang funcon equals f n n n 0 Also, fn fn fn n n [ 0 ] n [ n ] [ 0 We oban sequence A00333 of he On-Lne Encyclopeda of Ineger Sequences[5, A00333] Sequence ]

3 The number of one-sded n-sep pruden wals, sarng from 0, 0 and endng on y-axs, ang seps from {,, } s n / mn{n,} n n For he case 3 n he above heorem, here are 6 wals as follows: We oban sequence A3609[5, A3609] Sequence 3 Consder he number of one-sded pruden wals sarng from 0, 0 o x, y, ang seps from {,, } The number of such wals wh x rgh seps, lef seps and y up seps, s mn{y,x} y x x y If and x y n, we oban sequence A9578[5, A9578] Sequence 4 The number of one-sded n-sep pruden wals, from 0, 0 o x, y, n x y s even ang seps from {,, } s mn{y, nx y } 0 y nx y nx y n x y n xy If x y 3, we oban sequence A6376[5, A6376] Sequence 5 Wha s he number of he one-sded n-sep pruden wals, avodng or more consecuve eas seps,? The generang funcon equals If, If, we oban sequence [5, A006356]:, 3, 6, 4, 3, 70, 57, 353, 793, 78, 4004, 8997, 06, I also couns he number of pahs for a ray of lgh ha eners wo layers of glass and hen s refleced exacly n mes before leavng he layers of glass If 3, we oban sequence [5, A05967]:, 3, 7, 6, 38, 89, 09, 49, 53, 708, 6360, If 4, we oban sequence [5, A90360]:, 3, 7, 7, 40, 96, 9, 547, 306, 39, 39, 7448, Sequence 6 The number of one-sded n-sep pruden wals, ang seps from {,,, } equals n n 7 We oban sequence A055099[5, A055099] Sequence 7 Wha s he number of one-sded n- sep pruden wals, ang seps from {,,,, }? The generang funcon s 4 3 We oban sequence A6473[5, A6473] Sequence 8 Wha s he number of one-sded n-sep pruden wals n he frs quadran, sarng from 0, 0 and endng on he y-axs, ang seps from {,, }? The generang funcon s 3 4 Sequence 9 Wha s he number of one-sded n-sep pruden wals exacly avodng, ang seps from {,, }? The generang funcon equals If, we oban sequence A07806[5, A07806] Sequence 0 Wha s he number of one-sded n-sep pruden wals exacly avodng and boh a he same me?

4 also, The generang funcon s For, fn n n/ n/ /5, fn fn fn fn 3 wh f, f 3, f3 7 Ths s sequence A007909[5, A007909] IV SOME SEQUENCES ARISING FROM TWO-SIDED PSAWS Wha s he number of wo-sded, n-sep pruden wals endng on he op sde of her box avodng boh paerns, boh a he same me, ang seps from {,,, }? Theorem The generang funcon say T, u of he above wo-sded pruden wals endng on he op sde of her box sasfes u T, u u T,, where u couns he dsance beween he endpon and he norh-eas NE corner of he box For nsance, n he followng fgure, a wal aes 5 seps, and he dsance beween he endpon and he norh-eas corner s 3 So we can use 5 u 3 o coun hs wal Oulne of he proof of he heorem: Case : Neher he op nor he rgh sde has ever moved; he wal s only a wes sep Ths case conrbues o he generang funcon Case : The las nflang sep goes eas Ths mples ha he endpon of he wal was on he rgh sde of he box before ha sep Afer ha eas sep, he wal has made a sequence of norh seps o reach he op sde of he box Observe ha, by symmery, he seres T, u also couns wals endng on he rgh sde of he box by he lengh and he dsance beween he endpon and he norh-eas corner These wo observaons gve he generang funcon for hs class as T, Case 3: The las nflang sep goes norh Afer hs sep, here s eher a wes sep or a bounded sequence of Eas seps Ths gves he generaon funcon for hs class as u u T, T, Pung he hree cases ogeher, we ge he generang funcon for T, u Solve hs generang funcon for T, u usng he Kernel Mehod: From u we can ge T, u u T,, u u 3 u T, u u u T, u u Se u u 3 u 0, hen here s only one power seres soluon for u u Le U be hs soluon, U U Se u u T, u 0, and replace u by U: ge From T, U u u 3 u T, u U U 3 u u T, u u u u T, u u u 3 u T, u u u u 3 u Replace T, by 3 Now u T, u u 3 u U U u U u u 3 u where U has been defned n Sequence Noce ha T, s he generang funcon of he number of wo-sded n-sep pruden wals endng on he op sde of her box avodng boh paerns,, ang seps from {,,, }, hus T,

5 Sequence Noe ha T, 0 s he generang funcon of he number of wo-sded n-sep pruden wals endng a he norh-eas corner of her box avodng boh paerns,, ang seps from {,,, }, so T, Sequence 3 Furhermore, T, T, 0 s he generang funcon of he number of wo-sded n-sep pruden wals endng on he op sde or rgh sde of her box avodng boh paerns,, ang seps from {,,, }, hus T, T, Open Problem Wha s he number of wo-sded n-sep pruden wals, endng on he op sde of her box, avodng boh, and > ang seps from {,,, }? The generang funcon sasfes: u u u T, u u u u T,, where u couns he dsance beween he endpon and he norh-eas corner of he box For 3, u 3 3 u 3 4 u 4 u 4 5 u 3 u T, u u u 3 u 3 T, e, 3 u 4 u 5 3 u 3 4 u 4 T, u u u 3 u 3 T, Se 3 u 4 u 5 3 u 3 4 u 4 0, and solve for u, as a power seres of We obaned he frs one hundred erms for u, begnnng wh u Usng hs u, we can ge many examples for he sequence Open Problem Wha s he number of wo-sded n-sep pruden wals, endng on he op sde of her box, exacly avodng boh,, ang seps from {,,, }? The generang funcon s u u u 3 T, u u u T, I seems o us s no rval o solve hs generang funcon V SOME THEOREMS AND PROOFS Theorem The generang funcon of he number, say fn,, of he one-sded n-sep pruden wals, ang seps from {,, }, avodng or more consecuve eas seps, sasfes, and for, fn, n 0 0 n 0 0 n 0 0 n n n fn, n n n n Proof: Le F denoe he lengh generang funcon of he number of one-sded pruden wals, avodng or more consecuve eas seps We have he followng hree cases For he wals whch do no conan Norh seps, hey can be empy wal, wals wh only wes seps, wals wh only eas seps wh lengh a leas one and a mos, he conrbuons are,, respecvely For he wals obaned by concaenang a one-sded wal, a Norh sep, and hen a Wes wal, he conrbuon s F 3 For he wals obaned by concaenang a one-sded wal, a Norh sep, and hen a Eas wal wh a leas sep and a mos seps, he conrbuon s F Addng hese hree conrbuons gve he equaon F F F

6 Thus, F Now, le [ n ]F denoe he coeffcen of n n he power seres expanson of F [ n ] [ n ] 0 [ n ] 0 0 [ n ] 0 0 [ n ] 0 0 l0 n 0 0 n 0 0 n 0 0 l n n n l0 l l l l l ll n n n I s also he number of posve neger soluons o he equaon r x n Whou loss of generaly, we assume ha here are Eas seps, Wes seps and n Norh seps n a one-sded n-sep pruden wals, sarng from 0, 0 and endng on he y-axs We also assume ha > 0 snce here s only one such wal for 0 I s easy o see ha n / The n Norh seps provde n posons we can say n dfferen cells for Eas seps and Wes seps o be nsered Suppose ha we pu Eas seps no mn{n, } cells wh no empy cell Then here are ways of pung Eas seps no cells and n ways of choosng cells Now we dsrbue Wes seps no he remanng n cells, whch gve us n Therefore, we ge he number: n / mn{n,} n n Example: For n 4 n he above heorem, we have 7 such wals as follows: Theorem 3 The number of one-sded n-sep pruden wals, sarng from 0, 0 and endng on he y-axs, ang seps from {,, } s n / mn{n,} n n Proof: In our proof, we wll use he followng wo resuls whch could be found n some mahemacs boos such as [6]: The number of ways of pung n le obecs no r dfferen cells s n r n r n r I s also he number of nonnegave neger soluons o he equaon r x n The number of ways of pung n le obecs no r dfferen cells wh no empy cell s n r Theorem 4 The number, say fn, of generalzed one-sded n-sep pruden wals, ang seps from {,,, } equals n n n 3 n n 3 n n n wh generang funcon 0 n n,

7 Proof: Le P denoe he lengh generang funcon of generalzed one-sded pruden wals The conrbuon n P of wals ha do no conan Norh seps or Norheas seps horzonal wals s The conrbuon of wals obaned by concaenang a generalzed one-sded wal, a Norh sep or Norheas sep, hen a horzonal wal s P Addng hese wo conrbuons gves a lnear equaon for P : Therefore, P P P fn [ n ]P n n 3 n n 0 n n 3 n n 0 The second formula of fn can be easly derved from he lengh generang funcon Example: For n n he above heorem, we have 4 such wals: EN,NE, W N,NW,NNE,NEN,ENE,NEE, NEW, W NE, NN, W W, EE, NENE Theorem 5 The generang funcon of he number, say fn, of generalzed one-sded n-sep pruden wals, ang seps from {,,,, } s fn [ n ] [ n ] 4 m 3 m m m 0 m0 n [ ] 3 4 n 3 n n n 0 Proof: Le P denoe he lengh generang funcon of generalzed one-sded pruden wals The conrbuon n P of wals ha do no conan Norh seps or Norheas seps, or Norhwes sep horzonal wals s The conrbuon of wals obaned by concaenang a generalzed one-sded wal, a Norh sep or Norheas sep or a Norhwes sep, hen a horzonal wal s 3 P Addng hese wo conrbuons gves a lnear equaon for P, from whch we can ge P REFERENCES [] M Neal and S Gordon, The Self-Avodng Wal Brhäuser 996 [] LF Gregory, Inersecons of Random Wals Brhäuser 996 [3] AJ Gumann Ed, Polygons, Polyomnoes and Polycubes, Sprnger 009 [4] M Bousque-Mélou, AJ Gumann and I Jensen, Self-avodng wals crossng a square, J Phys A [5] J Iwan, Enumeraon of self-avodng wals on he square lace J Phys A [6] KE Donald, Scence [7] EJ Janse van Rensburg, Mone Carlo mehods for he self-avodng wal, Topcal Revew J Phys A 4 009, 3300 [8] AR Conway and AJ Gumann, Square lace self-avodng wals and correcons o scalng, Phys Rev Le , [9] E Duch, On some classes of pruden wals In Proceedngs of he FPSAC 05, Taormna, Ialy, 005 [0] P Préa, Exeror self-avodng wals on he square lace, 997 manuscrp [] JC Dehrdge and AJ Gumann, Pruden self-avodng wals, Enropy 0 008, [] TM Garon, AJ Gumann, I Jensen and JC Dehrdge, Pruden wals and polygons, J Phys A 4 009, [3] M Bousque-Mélou, Famles of pruden self-avodng wals, J of Combnaoral Theory, Seres A [4] R P Sanley, Enumerave Combnaorcs I, Wadsworh & Broos/Cole 986 [5] The Onlne Encyclopeda of Ineger Sequences 04, publshed elecroncally a hp://oesorg [6] John Rordan, An Inroducon o Combnaoral Analyss orgnally publshed: New Yor: John Wley 958, Dover Publcaons 00