Bidimensional Sand Pile and Ice Pile Models

Transcription

1 Bidimensional Sand Pile and Ice Pile Models Enrica Duchi, Robero Manaci, Ha Duong Phan, Dominiue Rossin To cie his version: Enrica Duchi, Robero Manaci, Ha Duong Phan, Dominiue Rossin. Bidimensional Sand Pile and Ice Pile Models. Aricle accepe pour publicaion dans le numero special du revue avec omie de lecure PureMahem <hal > HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 5 Jul 2007 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiues de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 BIDIMENSIONAL SAND PILE AND ICE PILE MODELS ENRICA DUCHI, ROBERTO MANTACI, HA DUONG PHAN, DOMINIQUE ROSSIN Absrac. In his paper we define an exension of he Sand Pile Model SPM and more generally of he Ice Pile Model IPM by adding a furher dimension o he sysem. By drawing a parallel beween hese unidimensional and bidimensional models we will find some common feauures and some differences. We will show ha, like for SPM(n), no all plane pariions are accessible in BSPM(n) saring from he iniial sae. However, i appears o be much more difficul o characerize he pariions ha are accessible in BSPM(n): we will be able o give some necessary bu no sufficien condiions for a pariion o be accessible. On he oher hand, we will show how several properies of he Ice Pile Model in one dimension can be generalized when one adds a second dimension.. Inroducion In his paper we inroduce he Bidimensional Sand Pile Model BSPM, ha is, a generalizaion of he Sand Pile Model SPM wih he addiion of a furher dimension. The SPM and some relaed models have been sudied in many differen domains. They were considered in he conex of ineger laices by Brylawski [3]. From he poin of view of physics, Bak, Tang, and Wiesenfeld used hem in order o illusrae he imporan noion of self organisaion criicaliy [2]. Moreover, Anderson e al. [], Spencer [9], and Goles and Kiwi [5] sudied hem from a combinaorial poin of view. SP M(n) is a discree dynamical sysem describing pilings of n granular objecs disribued on an array of columns. More precisely, each sae of he sysem can be described by using a l-uple s = (s, s 2,...s l ), where s i 0 is he number of grains in he column i. The sysem is iniially in he sae N = (n), ha is, all he grains are in he firs column. A each sep, he sysem evolves according o he following rule: () (s, s 2,...,s i, s i+,...,s l ) (s, s 2,...,s i, s i+ +,...,s l ) if s i s i+ >= 2 Because of he evoluion rule, we deduce ha each sae s = (s, s 2,...s l ) of he sysem saisfies s i s i+, for all i and l i= s i = n, where n is he oal number of grains, herefore each sae can be coded by a pariion of he ineger n. A pariion is said o be accessible in SPM if i can be obained by a seuence of applicaions of rule () (also called em ransiions) saring from he sae N. We denoe by SP M(n) he sysem whose se of configuraions is he se of all accessible pariions euiped wih he rule of evoluion (). We will omi he ineger n and simply wrie SPM when such noaion can be used wihou ambiguiy. Goles and Kiwi inroduced his model in [5] and proved ha SPM(n) has a uniue fixed poin, i.e. a configuraion in which no grain can fall under he rule (). Moreover, hey showed ha he order induced by SPM(n) on accessible pariions is a suborder of he L B (n) order, inroduced by Brylawski in [3]. This is he dominance order on all pariions of n, defined as follows: le a = (a, a 2,...,a k ) and b = (b, b 2,...,b ) be wo pariions, hen

3 a b Le us consider he rule: j a i i= j b j, i= j =,..., max(k, ). (2) (s,..., p +, p, p,...p, p...s }{{} l ) (s,...,p, p, p,..., p, p,...s }{{} l ) for any k. k k Goles and Kiwi showed ha he order L B (n) coincides wih he order defined as follows on he se of all pariions : a b b can be obained from a by a seuence of applicaions of rule () or (2) saring from he sae N. In paricular, rule () and rule (2) allow o reach all pariions of n from N and he L B (n) order is a laice. Laer, Goles, Morvan, and Phan [6] considered a generalisaion of SP M(n): he Ice Pile Model (IPM). More precisely, for any posiive ineger k, hey defined IPM k (n), obained by conserving rule () and modifying rule (2) as follows: (3) (s,..., p +, p, p,...p, p...s }{{} l ) (s,...,p, p,..., p,...s }{{} l ) for all k < k. k k These auhors proved ha he orders induced by IPM k (n) on he se of all pariions of n ha accessible from he iniial configuraion N are suborders of he laice L B (n), and ha hey form an increasing seuence of laices whose min is SPM(n) and whose max is L B (n). Indeed, IPM (n) IPM 2 (n)... IPM n (n), where IPM (n) and IPM n (n) correspond o SPM(n) and L B (n) respecively. Goles, Morvan, and Phan also gave necessary and sufficien condiions for a pariion o be accessible by IPM k (n) as well as an explici formula for he uniue fixed poin of IPM k (n). Coreel and Gouyou-Beauchamps [4] also sudied IPM k and compued asympoic bounds for he number of accassible configuraions in IPM k (n) by using he heory of pariions and of -euaions. Laapy, Manaci, Morvan, and Phan [8] exended SP M(n) o SPM( ), a naural exension of SPM(n) when one sars wih an infinie number of grains. By using wo differen approaches hey gave recursive formulae for SPM(n). In his paper we define an exension of SPM(n) and more generally of IPM k (n) by adding a furher dimension: he Bidimensional Ice Pile Model (BIPM). In order o do i, we place he grains on he verices of a bidimensional caresian ineger grid and we exend he previous rules so ha grains can fall or slide o he eas and o he souh, and in such a way ha he configuraions obained are coded by plane pariions. Definiion. A plane pariion of n is a marix a of inegers a i,j ha are nonincreasing from lef o righ and from op o boom, and such ha heir summaion is eual o n: a i,j a i+,j, a i,j a i,j+, for all i, j and a i,j = n. Definiion 2. BIPM k (n) is he dynamical sysem such ha: 2 i,j

4 (a) The sysem is iniially in he configuraion: N = n (b) A each sep, one grain can fall or slide o he eas or o he souh of is cell by applying locally one of he following rules o a submarix of he marix represening he configuraion: Eas ransiion:. (4) a i,j a i+,j a i,j+ a i,j+ a i,j+ a i,j a i,j+ + a i+,j if a i,j a i,j+ 2 a i,j+ a i,j+ a i,j a i+,j Souh ransiion: (5) a i,j a i,j+ a i+,j a i+,j a i,j a i+,j a i+,j + a i,j+ if a i,j a i+,j 2 a i,j a i,j+ a i+,j a i+,j Slide k (6) k {}}{ p + p p... p p p p p... p p.. p p p... p p p p p... p p k k {}}{ p p p... p p p p p... p p.. p p p... p p p p p... p p k wih k + k < k, where k and k is respecively he number of columns and of rows of he submarix. Observe ha, because he rules preserve he propery ha he rows and columns in he marix are non increasing, a newly obained configuraion is sill a plane pariion. Observe also ha a given configuraion may be obained applying differen seuences of rules o he iniial configuraion N. Noice ha for k = he applicable rules of he corresponding model BIPM (n) consis of Eas ransiions and of Souh ransiions only. We also observe ha BIPM (n) and BIPM n (n) represen a naural exension for SPM(n) and for L B (n) respecively, when one adds a furher dimension. For his reason, we decide o rename BIPM (n) by BSPM(n) and BIPM n (n) by BL B (n). 3

5 By drawing a parallel beween hese unidimensional and bidimensional models we will find some common feaures and some differences. We will show ha, like for SPM(n), no all plane pariions are accessible in BSPM(n) saring from N (see he example in Fig. ). However, i appears o be much more difficul o characerize he pariions ha are accessible in BSPM han i is in SPM: we will give some necessary bu no sufficien condiions for a pariion o be accessible Figure. A non accessible configuraion of BSPM wih n = 9. We will also show ha, like for SPM(n), he order induced by he rules of BSPM(n) on he se of all accessible configuraions is graded, ha is, if a and b are wo elemens of BSPM(n), and Γ and Γ are wo maximal chains having a as maximum and b as minimum, hen Γ = Γ. In oher words, if b is accessible from a hen all seuences of ransiions ha allow o obain b from a all have he same lengh. The number of seps needed in order o reach a given configuraion from N is eual o a uaniy ha we call he energy of he configuraion and for which we give an explici formula in Secion 2. Moreover, we will see ha all plane pariions are accessible in BL B (n), as i is he case in he corresponding unidimensional model L B (n). We will also find ha, unlike IPM k (n), he order induced by BIPM k (n) is no a laice. In paricular, his is he case for k =, as we will see by showing ha he sysem BSPM(n) may have more han one fixed poin. 2. Definiions and general resuls on he ordered srucure of BIPM k (n) Definiion 3. A parial order P is a pair (S, P ), where S is a se and P is a binary relaion on S such ha P is reflexive, anisymmeric, and ransiive. Definiion 4. Le P = (S, P ) and P = (S, P ) be wo parial orders. Then P is a suborder of P if S S and x, y S x P y if and only if x P y. Definiion 5. The relaion BIPMk (n) is he relaion defined as follows: for any pair (a, b) of accessible configuraions of BIPM k (n), a BIPMk (n) b b is obained from a by a seuence of applicaions of he rules of BIPM k (n). Proposiion. The relaion BIPMk (n) defines an order on he se of all accessible configuraions in BIPM k (n). Proof. I is immediae from he previous definiion ha BIPMk (n) is reflexive and ransiive. Furhermore, since BIPM k (n) ransiions are oriened (hey move grains owards eas and souh bu no owards he wes and norh), hen BIPMk (n) is also anisymmeric. 4

6 Figure 2. An example of evoluion in BSPM wih n = 5. While SPM(n) is a suborder of L B (n) we have ha BSPM(n) is no a suborder of BL B (n). Here is a counerexample: le ( ) 3 2 a = 2 0 and b = ( we have ha a and b are accessible in BL B (n) and in BSPM(n). Moreover we have ha b is obained from a by applying he slide rule of BL B (n). Bu b can no obained from a by BSPM rules. Then BSPM is no a suborder of BL B. Figure 2 shows he srucure of he order obained by applying he rules of BSPM and saring wih n = 5. Le P = (S, P ) be a finie parial order. For any x, y S, an elemen z S is said o be an upper bound or a lower bound of x, y, respecively, when x, y P z or x, y P z. Le us denoe by sup(x, y) and inf(x, y), respecively, he smalles upper bound and he greaes lower bound of x and y if hey exis. Definiion 6. Le P = (S, P ) be a parial order. Then P is a laice if for any x, y P, sup(x, y) and inf(x, y) exis. In paricular his implies ha a laice has a uniue absolue minimum and hence, if he order associaed wih he configuraion space of he sysem is a laice, hen he sysem has a uniue fixed poin (i.e. is converging). Remark. BSPM(n) is no a laice, he example of Figure 2 shows i. 5 )

7 Moreover, we can also show ha BSP M(n) does no have a local laice srucure, in he sense ha no any inerval is a laice. Here here is a counerexemple: le ( ) ( ) ( ) ( ) a = ; b = ; c = ; d = We have ha c and ( d are ) boh obained from boh a and b using BSPM rules. Moreover 2 he pariion e = is obained from c and from d. Then a and b do no have a 2 infimum in he inerval [(6), e]. 3. Energy and Accessibiliy of configuraions Definiion 7. Le a be a plane pariion, hen is energy E(a) is defined as follows: E(a) = i,j a i,j (i + j ) Noice ha each ime we apply an eas or a souh ransiion o a pariion a, he energy E(a) increases by one. Le us verify his saemen in he case of an eas ransiion: suppose we apply rule (4) o a cell (i, j) of a, hen in he summaion expressing he energy, he erms a i,j (i + j ) + a i,j+ (i + j) are replaced by (a i,j )(i + j ) + (a i,j+ + )(i + j), which shows ha he energy increases by one. This implies ha he order induced by rules (4) and (5) of he BSPM(n) model is graded, ha is, a configuraion a is always reached from he configuraion N by applying he same number of hese rules. Such number is he difference E(a) E(N) = E(a) n. On he oher hand, he energy may increase by more han one uni when one applies he slide rule. Definiion 8. A pariion is said o be accessible wih respec o a se of rules Σ (or Σ- accessible) if i can be obained from N = (n) by applying a seuence of rules of Σ. Proposiion 2. All plane pariions of n are accessible in BL B (n). Proof. Le a be a plane pariion such ha a N = (n). We wan o show ha here exiss a pariion a whose energy is sricly smaller han ha of a and such ha a BLB (n) a. By ieraing his process, we will evenually obain he uniue pariion having minimal energy, ha is, N. Le us say he value conained in he cell (, ) of a is p. Then we have he following cases: i. The cell (, 2) conains he value p. Then consider he larges recangle whose op lef corner is (, ) and whose cells all conain he value p. If his recangle only conains he cells (, ) and (, 2) hen a is obained by a reverse applicaion of he Eas ransiion moving one grain from (, 2) o (, ). If his recangle conains more han wo cells hen a is obained by a reverse applicaion of he slide rule. ii. The cell (, 2) conains he value p and 0. Le r be he value conained in he cell (, 3). Here we disinguish wo cases: r, hen a is obained by a reverse applicaion of he eas ransiion moving one grain from (, 2) o (, ); r =, hen apply he same argumen as in case i. o he recangle having he op lef corner in (, 2). 6

8 iii. The cell (, 2) conains he value = 0. In his case we can apply symmeric argumens han han hose in case i. and case ii. by focusing on he he cell (2, ) insead of he cell (, 2). In each of hese cases we move grains oward wes or norh, hen he energy decreases. The same resul is no rue for BSPM(n). We give now a necessary condiion for a plane pariion o be accessible in BSPM. Proposiion 3. Each pariion conaining he suare marix is non accessible in BSPM(n). (wih > 0) as submarix Proof. Le us ake a plane pariion a of n conaining a leas one suare. We wan o show ha, by applying backwards he rules of BSPM(n), we can no rever o he saring configuraion having all n grains in he cell (, ). We consider he firs ime ha one of such rules modifies he value of one of he cells of he suare. There are wo possibiliies: a reverse Eas or Souh ransiion moves a grain from a cell adjacen o he suare o a cell wihin he suare. a reverse Eas or Souh ransiion moves a grain from a cell wihin he suare. In eiher cases, i is easy o check ha a leas wo cells wihin he suare would no respec he decreasing condiion over he rows and columns. Therefore, a pariion needs o avoid he paern in order o be accessible in BSPM(n). We wan o show ha his is he uniue forbidden paern. Definiion 9. Le M be a recangular marix whose enries depend on a se of parameers,..., k. We say ha M is a forbidden paern wih respec o a se of rules Σ if, regardless of he choice of he values for he parameers,..., k, no Σ-accessible configuraion conains M as submarix. is he uniue forbidden paern for BSPM-accessible con- Theorem. The paern figuraions. Proof. Firs, i is sraighforward o see ha any paern wih only one row (respecively, one column) is no forbidden. Indeed, i is possible o creae such a paern on he second row (respecively, column) of a plane pariion having a sufficienly large number of grains disribued on he firs row (respecively, column). r Le us show now ha no oher paern of size 2 by 2 is forbidden. If is any paern s differen from r, hen >. We show ha he plane pariion can be obained s from anoher plane pariion having smaller energy. There are wo cases: eiher > r or r r >. In he firs case, can be obained from r +, in he second case, i can be s s 7

9 obained from + r. By ieraing his process we can consruc a seuence of plane s pariions whose energy sricly decreases, his implies ha his process evenually allows o obain he iniial configuraion N = (n) and herefore r canno be forbidden. s Suppose now ha here exis oher forbidden paerns having more han wo rows or more han wo columns and ha do no conain he paern and le S be he se of such forbidden paerns having minimal area. If S is no empy, hen here mus exis a paern M S such ha no grain in M can be moved by a reverse BSP M-ransiion. Oherwise, he same energy argumen used in he case of paerns of size 2 by 2 would show ha i would be possible o obain he iniial configuraion by applying a seuence of reverse BSPM-ransiions o any elemen of S. Le M be such a paern and le a be a plane pariion conaining M as a submarix: M = (a i,j ) r i s,u j v. As firs sep, we prove ha all inegers in he firs row of M are disinc. Le us consider he wo columns v and v of a and le a r,v =. If a r+,v were smaller han, hen one could move a grain (by using an inverse ransiion) from a r,v o a r,v. Hence a r+,v mus be eual o. Because M can no conain, hen a r,v mus be greaer han, le us denoe i by 2. Now if a r+2,v <, one could move a grain from a r+,v o a r,v. Hence a r+2,v mus be eual o. Because M does no conain, hen a r+,v mus be greaer han. Furhermore, if i were smaller han 2, hen one could move a grain from a r,v o a r,v. Hence, a r+,v mus be eual o 2 and herefore a r,v 2 mus be greaer han 2, or M would conain he suare By ieraing his process, we obain he desired sep, ha is, ha all he enries of he firs row of M are disinc. Therefore, if he paern M is included in a pariion of a sufficienly large ineger n i is possible o bring ino he firs row of he submarix M an arbirary number of grains coming from cells of he pariion locaed eas of M, using reverse BSP M-ransiions. All he paerns obained his way mus be forbidden. Oherwise, if i exised an accessible plane pariion conaining one of hem, hen i would be possible o obain an accessible pariion conaining M. This shows ha all paerns having he same shape as M and saisfying: he firs row conains oally arbirary values, all oher rows are eual o hose of M are forbidden. This implies ha he paern M obained from M by removing he firs row would be forbidden as well, which conradics he minimaliy of he area of M. 8

10 The avoidance of he paern, however, does no compleely characerize accessible pariions in BSPM(n). For insance, a pariion such as he one showed in Figure 3 is non accessible, even if i does no conain he suare. a, a 2, wih > 0. Figure 3. A non accessible configuraion in BSPM. In oher erms, an accessible configuraion canno conain hree adjacen cells on he firs row having he same value and having hree empy cells jus o he souh of hem (he same remark can be ransposed o he columns, of course). We would like o noe ha is no a forbidden paern, in he sense ha accessible configuraions do no need o avoid i when i is placed a a differen locaion. For example he following pariion is accessible in BSPM(8): We have deermined several oher of hese submarices ha canno be found a specific locaions of an accessible configuraion, e.g.: a, a 2, a 3, wih >. However, he avoidance of hese submarices a heir respecive forbidden locaions does no compleely characerize he accessible pariion of BSP M(n) eiher. For insance, he following plane pariion: is no accessible in BSPM(9) even if i does no conain any of he menioned submarices. 4. Fixed poins in BSPM In his secion we sudy he configuraions of BSP M(n) o which no ransiion rule can be applied. 9

11 Definiion 0. A pariion a is said o be sable wih respec o a se of rules Σ if none of he rules of Σ can be applied o a. Definiion. A pariion is said o be a fixed poin wih respec o a se of rules Σ if i is accessible and sable wih respec o he rules of Σ. Definiion 2. A sable pariion a is said o be smooh if a i,j a i,j+ and a i,j a i+,j for all i, j. A smooh pariion is obviously a fixed poin bu here are also non-smooh fixed poins in BSPM(n). In Figure 4 here is an example of non smooh fixed poin in BSPM(n): Figure 4. A non smooh fixed poin of BSPM wih n = 35. Definiion 3. For a posiive ineger l, we will call l-h diagonal of a marix (or of he corresponding pariion) he se of all cells having indices (i, j) wih i + j = l. Remark. The energy of a plane pariion a is defined in such a way ha he inegers on he l-h diagonal conribue o he sum E(a) wih a weigh l. Noaion. For a posiive ineger l, le S(l) = l i= i = l(l+) and T(l) = l 2 i= i(l i+) = l j j= i= i = l(l+)(l+2). Then for each n N here exiss a uniue riple (k, m, ) of non 6 negaive inegers such ha: k+ n = T(k) + S(m) + wih 0 S(m) < i = i= (k + )(k + 2) 2 and 0 < m +. Example. If n = 03, he uniue decomposiion is 03 = T(7)+S(5)+4 where T(7) = 84 and S(5) = 5. Definiion 4. Le n be an ineger and le n = T(k) + S(m) +. A pyramidal pariion of n is a pariion obained by aking he following seps (see Figure 5) in his order: Main saircase. For l =, 2,..., k, fill all he cells of he l-h diagonal of he pariion wih value k l +. Tha is, he cell (i, j) conains k i j + 2 grains for i + j k +. Addiional riangle. If m 0, hen choose any of he cells of he (k m + 2)-h diagonal (hese cells conain he value m ). Le us denoe (i 0, j 0 ) he coordinaes of his cell. Take he sub-pariion having his cell as op-lef angle. Then add o each cell of he firs m diagonals of his sub-pariion, ha is, o all cells (i, j) wih i 0 i i 0 + m and j 0 j j 0 + m. 0

12 Addiional row or column. If 0, hen ake one of he wo cells (i, j ) or (i 2, j 2 ) on he (k +2)-h diagonal and having eiher j = j 0 or i 2 = i 0 (one of hese cells may no exis if i 0 = or j 0 = ). Then add o he values in he firs cells ha are eiher souh of (i, j ) or eas of (i 2, j 2 ), respecively, depending on wheher he cell you choose is (i, j ) or (i 2, j 2 ). (The explici values of he indices of his cell can be compued as follows : since i + j = k + 2, wih j = j 0, and i 2 + j 2 = k + 2, wih i 2 = i 0, hen (i, j ) = (k j 0 + 4, j 0 ) or (i 2, j 2 ) = (i 0, k i 0 + 4)) main saircase wih he addiional riangle wih he addiional column Figure 5. A pyramidal pariion of n = 03, where n = T(k) + S(m) + wih k = 7, m = 5, and = 4. Remark. By consrucion, pyramidal pariions are smooh fixed poins. Proposiion 4. The number of pyramidal pariions of n = T(k) + S(m) + is given by: 2(k m) + 2 if m, > 0 k m + 2 if m > 0 and = 0 if m = 0 Proof. If m > 0 and > 0, here are (k m+2) choices for he posiion of he op-lef angle of he addiional riangle on he (k m + 2)-h diagonal. For (k m) of such choices, i is possible o obain wo pyramidal pariions (one having an addiional row, and one having an addiional column), bu for he wo exremal cells of he diagonal, only one pyramidal pariion can be obained (in one case, one can only add he addiional column and in he oher case, one can only add he addiional row). If m > 0 and = 0, hen a pyramidal pariion is deermined only by he choice of he posiion of he addiional riangle. If m = 0 hen he only pyramidal pariion is he saircase. Proposiion 5. Every pyramidal pariion of n is accessible in BIPM k (n) for all k. Proof. Observe ha we jus need o prove i for BIPM (n), i.e. for BSPM(n). Le p be a pyramidal pariion of n = T(k) + S(m) +. Then we have he following cases: The pariion p consiss of he main saircase only. Tha is, m = 0. We wan o show ha p is accessible:

13 i. Sar wih placing he T(k) grains in he cell (, ). ii. By applying BSPM s Eas ransiions, consruc he pariion : ( ) T(k) k(k ) k k This is possible because ( T(k) k(k ) k k ) 2 is an accessible configuraion of SPM(n). iii. For j = 2,...,k, consruc he j h row as follows: move one grain from he cell (, ) o he souh unil i reaches he cell (j, ). Then move i o he eas as far as possible. Repea he previous operaion unil he cells (j, ), (j, 2),..., (j, k j + ) conain he values k j +, k j,...,. The pariion p consiss of he main saircase and of he addiional riangle. Tha is, m 0 and = 0. Le i be he row index of he cell of he (k m + 2)- h diagonal where he verex of he addiional riangle lays. In oher erms, he cell conaining he verex of he addiional riangle is (i, k m + 3 i). Sar by consrucing he main saircase as described in he previous case. A he end of his sep, he cell (, ) will conain he value k +S(m) and all he ohers will conain he values of he main saircase. To consruc he l-h row of he addiional riangle, for l =, 2,..., m, do as follows: Le one grain fall from he cell (, ) o he eas unil i reaches he cell (, k m + 3 i ) (noe ha his cell is on he column immediaely preceding he one conaining he verex of he addiional riangle), hen move i o he souh unil i reaches he cell (i + l, k m + 3 i ). Then move i o he eas as far as possible. Repea he previous operaion unil he cells (i+l, k m+3 i), (i+l, k m+4 i),..., (i+l, k +2 i l+) conain he values m l +, m l,...,. The pariion p consiss of he main saircase, of he addiional riangle, and of he addiional row or column. Tha is, m 0 and 0. Sar by consrucing he main saircase and he addiional riangle as in he previous cases. Le us suppose he addiional row sars a he cell (i, m + i + ). Le one grain fall from he cell (, ) o he eas unil i reaches he cell (, m + i + ), hen move i o he souh unil i reaches he cell (i, m + i + ). Then move i o he eas as far as i can. Repea he previous operaion unil he cells (i, m + i + ), (i, m + i + 2),...(i, m + i) conain he values,,...,. The consrucion is analoguous if an addiional column needs o be added insead of an addiional row. I is easy o verify ha all he described moves are permied under he se of rules of BSPM. 5. Pyramidal Pariions and Energy We wan o show ha pyramidal pariions of n correspond o he smooh fixed poins of BSPM(n) having minimal energy. 2

14 Proposiion 6. The energy of pyramidal pariions of n = T(k) + S(m) + is given by: k(k + ) 2 (k + 2) (3k m + 4)(m + )m (2k + 3 ) Proof. The conribuion of he main saircase is: k i(k i + ) 2 = k(k + )2 (k + 2). 2 i= If p > 0, he conribuion of he addiional riangle is: m (3k m + 4)(m + )m i(k m + i + ) =. 6 i= Finally if > 0, he conribuion of he addiional column or row is: (2k + 3 ) (k + i + ) =. 2 i= Noaion. Le us denoe by d i (a) he sum of he enries in he i-h diagonal of a marix a. Lemma. Le a be a smooh pariion of n = T(k) + S(m) + and le p be a pyramidal pariion of n. We have he following for all s: s s d j (p) d j (a). j= Proof. We give a proof by recurrence on he ineger s. For s =, since he cell (, ) of p conains k, we mus show ha a, k. Le us suppose we have a value a leas eual o k + in his cell. This means ha he wo cells on he second diagonal conain a value greaer han or eual o k, oherwise a would no be smooh. For he same reason he cells of he hird diagonal of a conain a value a leas eual o k and so on. Bu hen he sum of he values of he pariion would be a leas k+ i= i(k i + 2) = T(k + ). Bu T(k + ) > n, since k is he larges number such ha T(k) n, which is a conradicion. For s, le us suppose ha i i d j (p) d j (a) (7) (8) j= j= j= for i =,...s. Then we wan o prove ha s s d j (p) d j (a). j= j= Le us suppose (7) is no rue, ha is, we have: s s d j (p) < d j (a), j= j= 3

15 herefore, since by he recurrence hypohesis s j= d j(p) s j= d j(a), we have necessarily d s (p) < d s (a). Conseuenly, here exis some enries on he s-h diagonal of a ha are larger han he corresponding enries of p. This is he argumen we are going o use in he proof in order o obain a conradicion. We disinguish wo cases: (a) The s-h diagonal of he pyramidal pariion p is enirely in he main saircase (see Figure 6). s h diagonal k m m m m m m m m m m Figure 6. Case in which he s-h diagonal is enirely in he main saircase of p. Recall ha he enries of he s-h diagonal of p are all eual o k s +. Le us call he ineger k s +, in order o simplify noaions in he remainder of he proof. The s-h diagonal of a conains hen values ha are larger han. The proof of his lemma is paricularly echnical and compuaional. Therefore, for sake of clariy, we have decided o provide firs a complee proof in he simple case where only one enry of he s-h diagonal of a is larger han he corresponding enry of p and i conains he value +. We will hen provide he proof in he general case. Le us hen suppose we are in he simples case: euaion (8) holds wih only one erm a i0,j 0 of d s (a) such ha a i0,j 0 > and in paricular a i0,j 0 = +. Le us denoe by a he smooh pariion having he enries in he firs s diagonals eual o hose of he pariion a and whose rows and columns decrease as fas as possible (ha is, by one uni) in he remaining diagonals. More precisely, he values of he cells of a ha are souh of he s-h diagonal are defined as follows (see Figure 7,(b)): The cells (i, j) wih i i 0 and j j 0 form a perfec saircase of heigh + whose highes poin is placed in (i 0, j 0 ); If i < i 0 and j > s + i, hen a i,j = a i,j ; if j < j 0 and i > s + j, hen a i,j = a i,j ; 4

16 k m m m m m m m m m m (a) k m + m m m m m m m m m (b) (a) Figure 7. The pyramidal pariion p on he lef and he pariion a on he righ. Le us denoe by a and a respecively he sum of all he elemens of a and a. Then we have a a = n, because a is smooh and herefore he rows and he columns of a decrease a leas as fas as hose of a. Figure 7,(b) illusraes ha we can wrie a as: a = s j= d j(a ) + T( + ) ( + ) + (s )S( ) = s j= d j(a) + T() + S() + (s )S( ), where he erm T(+) comes from he riangular region wih a hicker border in he figure and he erm (s )S() comes from he remaining rows and columns. The erm (+ ) is a correcing erm ha is necessary because he conribuion of he cell (i 0, j 0 ) is included boh in s j= d j(a ) and in T( + ). Similarly, we can compue p as (see Figure 7, (a)): s p = d j (p) + T() + S(m) + + (s )S( ), j= where > m, because we are supposing ha he s-h diagonal is enirely in he main saircase. Since S() > S(m) + and s j= d j(a) > s j= d j(p) hen a a > p and we ge a conradicion, since a = p = n. Le us now give he proof in he case where more han one cell of he s-h diagonal conains a value greaer han. Suppose here are u cells conaining he values j i, wih j i 0 for i =,...,u and v cells conaining he values + l i, wih l i for i =,..., v. Since we are supposing ha d s (a) > d s (p), a leas one cell conains a value larger han and, for he same reason, we have v i= l i u i= j i > 0. We consruc again a new smooh pariion a, having he enries in he firs s diagonals eual o hose of he pariion a and whose cells placed souh of he s-h diagonal conain values ha we are going o define nex. 5

17 Le (i 0, j 0 ) be he souhmos cell on he s-h diagonal such ha a i0,j 0 > p i0,j 0 =, and suppose his cell conains he value + l, hen he values of he cells of a ha are souh of he s-h diagonal are defined as follows: The cells (i, j) wih i i 0 and j j 0 form a perfec saircase of heigh + l whose highes poin is placed in (i 0, j 0 ); If i < i 0 and j > s + i, hen a i,j = a i,j ; if j < j 0 and i > s + j, hen a i,j = a i,j. Like in he simple case, we have a a = n, because a is smooh and herefore he rows and he columns of a decrease a leas as fas as hose of a. Recall ha p = = = s d j (p) + T() + S(m) + + (s )S( ) j= s d j (p) + T( ) + S( ) + S(m) + + ss( ) j= s d j (p) + T( ) + S(m) + + ss( ). j= Noe ha S( + l i ) > S() + l i and S( j i ) S() j i for all > 0, for all l i > 0 and for all j i 0. We will use his fac when we now compue a and we compare i o p. a = = = = s v u d j (a) + T( + l ) ( + l ) + S( + l i ) + S( j i ) j= i=2 s d j (a) + T( + l ) + S( + l ) + j= s d j (a) + T( + l ) + j= s d j (a) + T( + l ) + j= i= v S( + l i ) + i=2 v S( + l i ) + i= u S( j i ) i= v [S( ) + l i ( )] + i= s v d j (a) + T( + l ) + ss( ) + ( )( l i j= i= i= u S( j i ) i= u [S( ) j i ( )] i= u j i ). Now we use he fac ha v i= l i u i= j i > 0, as well as he fac ha when l > 0, one has T(+l ) T( )+S( ) and he fac ha, by recurrence hypohesis, s j= d j(a) > s j= d j(p). 6

18 a > s d j (p) + T( ) + ss( ) + S( ) + ( ) j= s d j (p) + T( ) + ss( ) + S(m) + = p, j= he las ineualiy being jusified by he fac ha S( ) S(m) and. We have hen ha a a > p, which is a conradicion. (b) The s-h diagonal of he pyramidal pariion p is no enirely in he main saircase (see Figure 8). In his case we have o disinguish beween cells of he main saircase ha are eual o and hose ha are eual o + bu oherwise we can apply he same argumens as before. s h diagonal k m m m m + m + m + m + m + m + m Figure 8. Case in which he s-h diagonal is no enirely in he main saircase of p. Proposiion 7. Le a be a plane pariion and le us denoe by l is las non zero diagonal. Then is energy E(a) = k, a k,(k + ) can be rewrien as l l d j (a) j= l i= i d j (a) = ln j= 7 l i= i d j (a) j=

19 Proof. We have ha a k, (k + ) = k, = = l jd j (a) j= l i= l ( i= = l( l d j (a) j=i l i d j (a) d j (a)) j= l d j (a)) j= = ln j= l i d j (a) i= l i d j (a). I is sraighforward o verify ha he ranges of he indices i and j in hese sums can be modified o make hem eual o hose of he claim of he proposiion wihou modifying he values of he sums hemselves. Proposiion 8. Le a be a smooh pariion of n and le p be a pyramidal pariion of n. If i j= d j(p) = i j= d j(a) for each i, hen a is a pyramidal pariion. Proof. For simpliciy we prove he resul for he simples case where p consiss of he main saircase only. The oher cases can be proved by using he same argumens. Since d (a) = d (p) hen a, = p,. Moreover d 2 (a) = d 2 (p), ha is i= j= a,2 + a 2, = p,2 + p 2, = 2(p, ). Le us suppose now ha a,2 > p,2, hen from he previous euaion we mus have ha a 2, < p 2, = p,, bu his is impossible since a, = p, and a is smooh. Therefore a,2 = a 2, = p,. By ieraing he same argumen on he remaining diagonals we obain ha a = p. Proposiion 9. In he se of all accessible smooh pariions, pyramidal pariions have minimal energy. Proof. Le p be a pyramidal pariion of n, by consrucion p is smooh. We wan o show ha i has minimal energy among all smooh pariions, ha is, E(p) E(a) for all smooh pariion a. From Proposiion 7, we have ha E(p) = ln l i i= j= d j(p). Then from Lemma we have he resul. Proposiion 0. A smooh pariion having minimal energy among all smooh pariions is pyramidal. 8 j=

20 Proof. Le a be a smooh pariion of n wih minimal energy among all smooh pariions. In Proposiion 9 we proved ha pyramidal pariions all have minimal energy. Therefore for any pyramidal pariion p one has E(p) = E(a). Using Proposiion 7 his means ha: (9) nl l i= i d j (p) = nl j= l i= i d j (a), where l and l denoe he las non zero diagonal of p and a respecively. Noe ha l l, oherwise, if l < l hen l j= d j(a) = n, while l j= d j(p) < n which conradics Lemma. Ideniy (9) can be rewrien as l (0) n(l l) + ( i= i d j (p) j= i d j (a)) j= j= l i=l i d j (a) = 0 Observe ha n(l l) l i i=l j= d j(a) >= 0 because for all i, he erm i j= d j(a) is smaller han or eual o n and he sum l i=l ( i j= d j(a)) has (l l) erms. From Lemma we have ha he remaining erm l i= ( i j= d j(p) i j= j= d j(a)) of he lef-hand i j= d j(p) side of euaion (0) is also non negaive. Therefore boh erms n(l l) l i=l and l i= ( i j= d j(p) i j= d j(a)) mus be eual o 0. However, l i= ( i j= d j(p) i j= d j(a)) = 0 implies i d j (p) j= i d j (a) = 0 for all i j= because, by Lemma, all erms i j= d j(p) i j= d j(a) are greaer han or eual o 0. We deduce ha i j= d j(p) = i j= d j(a) for all i and conseuenly ha l = l. By using Proposiion 8, his implies ha a is pyramidal. From hese wo proposiions we can deduce he following herorem. Theorem 2. The se of smooh pariions wih minimal energy is he se of pyramidal pariions. 6. Longes and shores maximal chains in BL B (n) Our goal in his secion is o deermine he minimal and he maximal number of ransiions of BL B (n) ha allow o reach a fixed poin of BL B (n) saring from he iniial configuraion. Definiion 5. Le P = (S, P ) a parially ordered se and le x, x 2,...,x m be elemens of S. We say ha x, x 2,...,x m form a chain if x P x 2 P... P x m. The ineger m is called he lengh of he chain. Definiion 6. We say a chain Γ is maximal if Γ canno be included as sub-chain in any oher chain. 9

21 Remark. Noe ha if a chain Γ = x, x 2,...,x m is a maximal chain of BIPM k (n), hen x is he iniial configuraion N = (n) and x m is a fixed poin of BIPM k (n). The lengh of he shores and of he longes maximal chain in BIPM n (n) = BL B (n) are in fac he minimal and he maximal number of applicaions of BL B (n) rules ha allow o reach a fixed poin of BL B (n) saring from he iniial configuraion. Remark. We recall ha all plane pariions are accessible in BL B (n). Furhermore, i is easy o see ha a plane pariion a = (a i,j ) having a, > canno be a fixed poin. Conseuenly, he fixed poins of BL B (n) are precisely he plane pariions whose pars are all eual o. Definiion 7. We define wo special fixed poins of BL B (n) as follows. Le P R be he plane pariion defined by P R (, j) = for j n and le P C be he plane pariion defined by P C (i, ) = for i n. The fixed poin P R is obviously represened by a one-row marix, while P C is represened by a one-column marix. Noe ha, if n 3, he longes and he shores maximal chains in BL B (n) have he same lengh and his lengh is n. The nex proposiion considers he cases where n 4. Proposiion. For n 4, he shores maximal chains in BL B (n) have lengh 2n 5 and more precisely: If P is any fixed poin of BL B (n) wih P P R and P P C, hen i is possible o consruc a maximal chain of lengh 2n 5 ending wih P. If P = P R or P = P C, hen he shores maximal chain ending wih P has lengh 2n 4. Proof. We firs prove ha a leas 2n 5 ransiions are necessary o obain a fixed poin P from he iniial configuraion. We will coun how many ransiions are needed o move a grain form he cell (, ), where i is inially placed, o an empy cell (i, j) (, ). Le us observe ha one (Eas or Souh) ransiion is needed o move one grain from he cell (, ) o he cell (, 2) or o he cell (2, ). Le (i, j) be a cell no in {(, ), (, 2), (2, )}. While he cell (, ) conains more han 2 grains, one needs a leas wo ransiions o move one grain from his cell o he cell (i, j). I is clear indeed ha one ransiion would no be sufficien o move a grain from he cell (, ) o he cell (i, j). A he end of his process, when all cells excep one have been filled and here are only 2 grains lef in he cell (, ), hen i is only possible o apply one Slide ransiion, hus obaining a fixed poin. So, in oal, one needs a leas one ransiion o bring one grain o each of he cells (, 2), (2, ) and he las cell o be filled, and wo ransiions for each of he n 4 remaining cells, i.e., 3 + 2(n 4) = 2n 5 ransiions. We shall consruc now a seuence of 2n 5 ransiions allowing o obain P from N = (n). Le us consider firs he case where P is a fixed poin differen from P R and P C. Le r be he number of rows of P and for i r, denoe by j i he larges ineger such ha P(i, j i ) =. The firs ransiion (an Eas ransiion) ransfers one grain from he cell (, ) o he cell (, 2) and he second one (a Souh ransiion) ransfers one grain from he cell (, ) o he 20

22 cell (2, ). The las ransiion of he seuence is a Slide ransiion, moving one grain from he cell (, ) o he cell (r, j r ). The remaining 2n 8 ransiions are described as follows. For any of he cells (, j), wih 2 < j j i, we move a grain from cell (, ) o cell (, j) by applying wo ransiions : an Eas ransiion moving he grain from (, ) o (, 2), hen a Slide ransiion moving he grain from (, 2) o (, j). For any oher cell (i, j), we firs move a grain from cell (, ) o he cell (2, ) using a Souh ransiion, hen we move i from (2, ) o (i, j) using a Slide ransiion. The oal number of ransiions in he seuence is clearly 2n 5. I is easy o see ha a shores maximal chain ending wih P R orp C has lengh 2n 4. The following proposiion deals wih he lengh of longes maximal chains in BL B (n) and proves ha heir lengh is he same as he lengh of longes maximal chains in L B (n). Longes maximal chains in L B (n) were sudied by Greene and Kleiman in heir paper [7]. We refer he reader o his aricle for more deails on longes maximal chains in L B (n). Proposiion 2. Longes maximal chains in BL B (n) have he same lengh as longes maximal chains in L B (n). Proof. We will show firs ha he lengh of any longes maximal chain in BL B (n) is smaller han or eual o he lengh of a longes maximal chain in L B (n). Then we will show ha here exiss a fixed poin P such ha he longes maximal chain ending wih P has he same lengh as a longes maximal chain of L B (n). Le us firs consider he following map ϕ from BL B (n) o L B (n): for each plane pariion a, define ϕ(a) as he pariion obained from a by soring he pars of a in decreasing order. The map ϕ is clearly surjecive. Now, le a b be a ransiion in BL B (n). I is clear ha ϕ(b) is smaller han ϕ(a) by he dominance order in L B (n), so ϕ(b) can be obained from ϕ(a) by a ransiion in L B (see [3]). This implies ha every chain in BL B (n) can be mapped ono a chain of L B (n) having a leas he same lengh. Therefore he lengh of a longes maximal chain in BL B (n) is smaller han or eual o he lengh of a longes maximal chain in L B (n). On he oher hand, if we consider only plane pariions having one row, he evoluion of he sysem in BL B (n) is analoguous o he one in L B (n). So he lengh of a longes maximal chain ending wih P L is eual o he lengh of any longes maximal chain in L B (n). Acknowledgmens The auhors would like o hank Nicolas Desainville for he fruiful discussions on he opic of he paper. References [] R. Anderson, L. Lovász, P. Shor, J. Spencer, E. Tardos, and S. Winograd. Disks, ball, and walls: analysis of a combinaorial game. Amer. Mah. Monhly, 96, 989. [2] P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criicaliy. Phys. Rev. A, 38: , 988. [3] T. Brylawski. The laice of ineger pariions. Discree Mahemaics, 6:20 29, 973. [4] S. Coreel and D. Gouyou-Beauchamps. Enumeraions of sand piles. Discree Mahemaics, 3(256): ,

23 [5] E. Goles and M. A. Kiwi. Games on line graphs and sand piles. Theoreical Compuer Science, 5:32 349, 993. [6] E. Goles, M. Morvan, and H.D. Phan. Sand piles and order srucure of ineger pariions. Discree Applied Mahemaics, 7:5 64, [7] C. Greene and D. J. Kleiman. Longes chains in he laice of ineger pariions ordered by majorizaion. European Journal of Combinaorics, 7: 0, 986. [8] M. Laapy, R. Manaci, M. Morvan, and H.D. Phan. Srucure of some sand piles models. Theoreical Compuer Science, 262( ), 200. [9] J. Spencer. Balancing vecors in he max norm. Combinaorica, 6:55 65,