Sequences Arising From Prudent Self-Avoiding Walks Shanzhen Gao, Heinrich Niederhausen Florida Atlantic University, Boca Raton, Florida 33431

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1 Seqences Arising From Prden Self-Avoiding Walks Shanzhen Gao, Heinrich Niederhasen Florida Alanic Universiy, Boca Raon, Florida Absrac A self-avoiding walk (SAW) is a seqence of moves on a laice no visiing he same poin more han once. A SAW on he sqare laice is prden if i never akes a sep owards a verex i has already visied. Prden walks di er from mos sb-classes of SAWs ha have been coned so far in ha hey can wind arond heir saring poin. Some problems and some seqences arising from prden walks are discssed in his paper. Keywords Self-avoiding walk, prden self-avoiding walk, generaing fncion, kernel mehod 1 Inrodcion A well-known long sanding problem in combinaorics and saisical mechanics is o nd he generaing fncion for self-avoiding walks (SAW) on a wo-dimensional laice, enmeraed by perimeer. A SAW is a seqence of moves on a sqare laice which does no visi he same poin more han once. I has been considered by more han one hndred researchers in he pass one hndred years, inclding George Polya, Tony Gmann, Laszlo Lovasz, Donald Knh, Richard Sanley, Doron Zeilberger, Mireille Bosqe-Mélo, Thomas Prellberg, Neal Madras, Gordon Slade, Agnes Diel, E.J. Janse van Rensbrg, Harry Kesen, Sar G. Whiingon, Lincoln Chayes, Iwan Jensen, Arhr T. Benjamin, and ohers. More han hree hndred papers and a few volmes of books were pblished in his area. A SAW is ineresing for simlaions becase is properies canno be calclaed analyically. Calclaing he nmber of self-avoiding walks is a common compaional problem [1], [8], [9]. In he pas few decades, many mahemaicians have sdied he following wo problems Problem 1 Wha is he nmber of SAWs from (0; 0) o (n 1; n 1) in an n n grid, aking seps from f"; #; ;!g? Donald Knh claimed ha he nmber is beween and for n = 11 and he did no believe ha he wold ever in his lifeime know he exac answer o his problem in However, afer a few years, Richard Schroeppel poined o ha he exac vale is 1; 568; 758; 030; 464; 750; 013; 14; 100 = [1], [10], [5]. I is sill an nsolved problem for n > 5 Problem Wha is he nmber f(n) of n-sep SAWs, on he sqare laice, aking seps from f"; #; ;!g? The nmber f(n) is known for n 71 [10], [11], [3], [1]. A recenly proposed model called prden self-avoiding walks (PSAW) was rs inrodced o he mahemaics commniy in an npblished manscrip of Préa, who called 1

2 hem exerior walks. A prden walk is a conneced pah on sqare laice sch ha, a each sep, he exension of ha sep along is crren rajecory will never inersec any previosly occpied verex. Sch walks are clearly self-avoiding [6], [13], [4], [7], []. We will alk abo some seqences arising from PSAWs in he following. Prden Self-Avoiding Walks De niions and Examples A PSAW is a proper sbse of SAWs on he sqare laice. The walk sars a (0; 0), and he empy walk is a PSAW. A PSAW grows by adding a sep o he end poin of a PSAW sch ha he exension of his sep - by any disance - never inersecs he walk. Hence he name prden. The walk is so carefl o be self-avoiding ha i refses o ake a single sep in any direcion where i can see - no maer how far away - an occpied verex. The following walk is a PSAW..1 Properies of a PSAW Unlike SAW, PSAW are sally no reversible. There is sch an example in he following gre. Each PSAW possesses a minimm bonding recangle, which we call box. Less obviosly, he endpoin of a prden walk is always a poin on he bondary of he box. Each new sep eiher in aes he box or walks (prdenly) along he border. Afer an in aing sep, here are 3 possibiliies for a walk o go on. Oherwise, only. In a one-sided PSAW, he endpoin lies always on he op side of he box. The walk is parially direced. A prden walk is wo-sided if is endpoin lies always on he op side, or on he righ side of he box. The walk in he following gre is a wo-sided PSAW.

3 3 Some Seqences Arising from One-sided PSAWs Seqence 1 Wha is he nmber (say f(n) ) of one-sided n-sep prden walks, aking seps from f "; ;!g? The generaing fncion eqals Also, X f (n) n = n = f(n) = f(n 1) + f(n ) = (1 p ) n + (1 + p ) n = 1 0 Xn+1 n k 0 k=0 k! 1 0 We obain seqence A of he On-Line Encyclopedia of Ineger Seqences. Seqence The nmber of one-sided n-sep prden walks, saring from (0; 0) and ending on y-axis, aking seps from f"; ;!g is 1 + b(n X 1)=cminfn k=1 X i=1 k;kg n k + 1 k 1 n k i i k i k We obain seqence A Seqence 3 Consider he nmber of one-sided prden walks saring from (0; 0) o (x; y), aking seps from f"; ;!g. The nmber of sch walks wih k + x righ! seps, k lef seps and y p " seps, is X minfy;k+xg i=1 y + 1 k + x 1 y + k i k + x i k If k = and x = y = n, we obain seqence A i

4 Seqence 4 The nmber of one-sided n-sep prden walks, from (0; 0) o (x; y), ( n x y is even) aking seps from f"; ;!g is minfy; X n+x y g i=0 y + 1 i n+x y 1 n+x y i n x+y i n x y If x = y = 3, we obain seqence A Seqence 5 Wha is he nmber of he one-sided n-sep prden walks, avoiding k or more consecive eas seps,! k? The generaing fncion eqals 1 + k 1 + k+1 If k =, we obain seqence A006356, coning he nmber of pahs for a ray of ligh ha eners wo layers of glass and hen is re eced exacly n imes before leaving he layers of glass. If k = 3, we obain seqence A (see also page 44 in [14]). Seqence 6 The nmber of one-sided n-sep prden walks, aking seps from f"; ;!; %g eqals 5 + p 17 p p 17! n 5 p 17 p 17 3 p! n 17 We obain seqence A Seqence 7 Wha is he nmber of one-sided n-sep prden walks, aking seps from f!; ; "; %; &g? The generaing fncion is We obain seqence A Seqence 8 Wha is he nmber of one-sided n-sep prden walks in he rs qadran, saring from (0; 0) and ending on he y-axis, aking seps from f"; ;!g? The generaing fncion is 1 3 ((1 + ) (1 ) p (1 4 ) (1 )) Seqence 9 Wha is he nmber of one-sided n-sep prden walks exacly avoiding =k, aking seps from f"; ;!g? 4

5 The generaing fncion eqals 1 + k + k k+1 k+ If k = 1; we obain seqence A Seqence 10 Wha is he nmber of one-sided n-sep prden walks exacly avoiding =k and " =k (boh a he same ime)? The generaing fncion is 1 + k + k k+1 k+ also, For k = 1, f(n) = n+ ( 1) bn=c + ( 1) b(n+1)=c =5; This is seqence A f(n) = f(n 1) f(n ) + f(n 3) wih f(1) = 1; f() = 3; f(3) = 7 4 Some Seqences Arising from Two-sided PSAWs Wha is he nmber of wo-sided, n-sep prden walks ending on he op side of heir box avoiding boh paerns, # (boh a he same ime), aking seps from f"; #; ;!g? Theorem 1 The generaing fncion (say T (; ) ) of he above wo-sided prden walks ending on he op side of heir box sais es 1 T (; ) = T (; ) ; (1) where cons he disance beween he endpoin and he norh-eas (NE) corner of he box. For insance, in he following gre, a walk akes 5 seps, and he disance beween he endpoin and he norh-eas corner is 3. So we can se 5 3 o con his walk. Oline of he proof of he heorem Case 1 Neiher he op nor he righ side has ever moved; he walk is only a wes sep. This case conribes 1 o he generaing fncion. 5

6 Case The las in aing sep goes eas. This implies ha he endpoin of he walk was on he righ side of he box before ha sep. Afer ha eas sep, he walk has made a seqence of norh seps o reach he op side of he box. Observe ha, by symmery, he series T (; ) also cons walks ending on he righ side of he box by he lengh and he disance beween he endpoin and he norh-eas corner. These wo observaions give he generaing fncion for his class as T (; ). Case 3 The las in aing sep goes norh. Afer his sep, here is eiher a wes sep or a bonded seqence of Eas seps. This gives he generaion fncion for his class as + T (; ) T (; ) Ping he hree cases ogeher, we ge he generaing fncion (1) for T (; ). Solve his generaing fncion for T (; ) sing he Kernel Mehod From 1 T (; ) = T (; ) ; we can ge (1 ) + 3 T (; ) = ()(1 )(1 + ) T (; ) (1 ) ( ) Se (1 ) ( + 3 ) = 0, hen here is only one power series solion for = q(1 3 3 ) 4 4 Le U be his solion, Se and replace by U From U = U() = q(1 3 ) 4 4 () (1 + )()(1 ) + T (; ) (1 ) () = 0; U T (; ) = (1 + U) (U ) (3) (1 ) + 3 T (; ) = ()(1 )(1 + ) T (; ) (1 ) ( ) ge T (; ) = ( )(1 )(1 + ) + T (; ) (1 ) ( ) (1 ) ( + 3 ) 6

7 Replace T (; ) by (3). Now T (; ) = (1 + )() + 3 (1 + U) (U ) (1 ) () (U ) (1 ) ( + 3 ) where U() has been de ned in (). Seqence 11 Noice ha T (; 1) is he generaing fncion of he nmber of wo-sided n-sep prden walks ending on he op side of heir box avoiding boh paerns, #, aking seps from f"; #; ;!g, hs T (; 1) = q (1 ) (1 ) (1 3 ) 4 4 (1 + ) ( ) (1 + 3 ) (1 3 ) = Seqence 1 Noe ha T (; 0) is he generaing fncion of he nmber of wo-sided n-sep prden walks ending a he norh-eas corner of heir box avoiding boh paerns, #, aking seps from f"; #; ;!g, so T (; 0) = (1 ) q (1 3 ) (1 3 ) = Seqence 13 Frhermore, T (; 1) T (; 0) is he generaing fncion of he nmber of wo-sided n- sep prden walks ending on he op side or righ side of heir box avoiding boh paerns, #, aking seps from f"; #; ;!g, hs T (; 1) T (; 0) = (1 ) q(1 3 ) (1 + 3 ) (1 3 ) = Open Problem 1 Wha is he nmber of wo-sided n-sep prden walks, ending on he op side of heir box, avoiding boh k, and # k (k > ) aking seps from f"; #; ;!g? The generaing fncion sais es 1 1 k k 1 T (; ) = k k 1 + T (; ); 7

8 where cons he disance beween he endpoin and he norh-eas corner of he box. For k = 3, i.e., T (; ) = T (; ) ( )T (; ) = ( )() + () T (; ) Se + (1 + 3 ) + ( 4 ) + ( 5 3 ) = 0, and solve for, as a power series of. We obained he rs one hndred erms for, beginning wih = Using his, we can ge many examples for he seqence. Open Problem Wha is he nmber of wo-sided n-sep prden walks, ending on he op side of heir box, exacly avoiding boh =, # =, aking seps from f"; #; ;!g? The generaing fncion is ( )T (; ) = 1 1 We do no have a solion o his eqaion. + T (; ) References [1] M. Bosqe-Mélo, A.J. Gmann and I. Jensen, Self-avoiding walks crossing a sqare, J. Phys. A 38 (005) [] M. Bosqe-Mélo, Families of prden self-avoiding walks, J. of Combinaorial Theory, Series A 117 (010) [3] A.R. Conway and A.J. Gmann, Sqare laice self-avoiding walks and correcions o scaling, Phys. Rev. Le. 77 (1996), [4] J.C. Dehridge and A.J. Gmann, Prden self-avoiding walks, Enropy 10 (008), [5] K.E. Donald, Science 194 (1976) 135. [6] E. Dchi, On some classes of prden walks. In Proceedings of he FPSAC 05, Taormina, Ialy, (005). 8

9 [7] T.M. Garoni, A.J. Gmann, I. Jensen and J.C. Dehridge, Prden walks and polygons, J. Phys. A 4 (009), [8] L.F. Gregory, Inersecions of Random Walks. Birkhäser (1996). [9] A.J. Gmann (Ed.), Polygons, Polyominoes and Polycbes, Springer (009). [10] J. Iwan, Enmeraion of self-avoiding walks on he sqare laice J. Phys. A 37 (004) [11] E.J. Janse van Rensbrg, Mone Carlo mehods for he self-avoiding walk, (Topical Review) J. Phys. A 4 (009), [1] M. Neal and S. Gordon, The Self-Avoiding Walk. Birkhäser (1996). [13] P. Préa, Exerior self-avoiding walks on he sqare laice, (1997) (manscrip). [14] R. P. Sanley, Enmeraive Combinaorics I, Wadsworh & Brooks/Cole (1986). 9