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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 292143, 12 pages

http://dx.doi.org/10.1155/2013/292143

## Parallel Rank of Two Sandpile Models of Signed Integer Partitions

Dipartimento di Matematica, Universitá della Calabria, Via Pietro Bucci, 87036 Arcavacata di Rende, Italy

Received 4 December 2012; Accepted 24 January 2013

Academic Editor: Filomena Cianciaruso

Copyright © 2013 G. Chiaselotti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce the concept of fundamental sequence for a finite graded poset *X* which is also a discrete dynamical model. The concept of fundamental sequence is a refinement of the concept of parallel convergence time for these models. We compute the parallel convergence time and the fundamental sequence when *X* is the finite lattice of all the signed integer partitions such that , where , and when *X* is the sublattice of all the signed integer partitions of having exactly *d* nonzero parts.

#### 1. Introduction

The study of the order relation in particular subsets of integer partitions has been recently addressed from the point of view of the discrete dynamical models. The excellent survey [1] describes several subsets of integer partitions which have a lattice structure in terms of discrete dynamical models. Before going further, we borrow from [1] the following description of *discrete dynamical model* (briefly DDM): “At each (discrete) time step, such a model is in some state, which we call a *configuration*. Configurations are described by combinatorial objects, like graphs, integer partitions, and others, and we will not distinguish a configuration and its combinatorial description. A discrete dynamical model is then defined by an initial configuration and (some) evolution (rules) which (say) under which conditions the (configurations) may be changed, and which (describe) the new configurations one may obtain. These (rules) can generally be applied under a local condition, and (they imply) a local modification of the current configuration. Note that in the general case the evolution (rules) can be applied in several places in a configuration, leading to several configurations. If a configuration can be obtained from a configuration after one application of (some) evolution (rule), we say that is a successor of , or is a predecessor of , which is denoted by ." In this paper, we consider integer partitions; therefore, we use the term *configuration* as an equivalent to *integer partition*. There are two ways in which we can apply the evolution rules on a given configuration: a *sequential* way (in this case, we refer also to *sequential dynamic* of the DDM) and a *parallel* way (in this case, we refer also to *parallel dynamic* of the DDM). In the sequential way, we apply the evolution rules separately on each summand of the configuration. In the parallel way, we apply the evolution rules concurrently on all the summands of the configuration. When we start from the initial configuration and apply the evolution rules in a sequential way, generally we obtain an oriented graph which represents the Hasse diagram of a poset of integer partitions with an order relation . In this case, one usually proves that the relation is equivalent to saying that is covered by with respect to the partial order . Therefore, several concepts of the classical order theory find their interesting formulation in dynamic terms. The first implicit formulation of the covering rules of an integer partitions lattice as evolution rules of a DDM was given in the famous Brylawski paper [2]. In this paper, Brylawski proposed a dynamical approach in order to study the lattice of all the partitions of a fixed positive integer with the dominance order. However, the explicit identification of a specific set of integer partitions with a DDM begins in [3, 4]. In the DDM introduced by Goles and Kiwi in [4], denoted by , a configuration is an integer partition of the positive integer , and each summand of is identified with a pile of sand grains; hence, the name is *sandpile model* (briefly SPM). In this model, there is only one evolution rule.

*Rule 1 (vertical rule). *One grain can move from a column to the next one if the difference of height of these two columns is greater than or equal to 2.

In the model (introduced by Brylawski [2]), the movement of a sand grain respects Rule 1 and also the following.

*Rule 2 (horizontal rule). *If a column containing grains is followed by a sequence of columns containing grains and then one column containing grains, one grain of the first column can slip to the last one.

There are a lot of specializations and extensions of this model which have been introduced and studied under different names, different aspects, and different approaches. The can be also related to the self-organized criticality () property, introduced by Bak et al. in [5]. The study of such systems has been developed in an algebraic context [6], in a combinatorial game theory setting [4, 7, 8], in the theory of cellular automata [9–11], and in the setting of discrete dynamical systems [12–14].

There exists a wide literature concerning dynamical models related in several ways to the original model , [8, 15–29].

When we study the parallel dynamic of an SPM which also has a lattice structure, starting from its initial configuration (denoted by ), we always arrive after unit time steps to a unique final configuration (denoted by , that is, the maximum of the correspondent lattice. This nonnegative integer , denoted by , was introduced explicitly in [3, 4]. The number is relevant to the study of a DDM of integer partitions which is also a graded lattice because it represents a type of “parallel rank" of the graded lattice. The classical concept of rank can be identified with the number of sequential unit steps needed to reach starting from , and in [4] this number is denoted by . Our question is then as follows: is it possible to refine the concept of “parallel rank" in such a way that this refinement conducts us to have further information concerning the graded lattice? For example, can the parallel dynamic of the model, in some sense, characterize the symmetry of the corresponding lattice? It is very difficult, in general, to establish, for example, when a graded lattice has the Sperner property, or it is rank unimodal or rank symmetric. Can a refinement of the study of the parallel dynamic of an SPM that is also a graded lattice be helpful in addressing these problems? In this paper, we introduce and study some concepts related to the parallel dynamic of SPMs, and we study these concepts in the wider environment of the signed integer partitions, that is, integer partitions whose summands can be also negative. The signed integer partitions have been recently introduced by Andrews in [30] and further studied in [31] from an arithmetical point of view. The reason why we work in the signed integer partitions environment is not only formal. In the context of the signed integer partitions, it is easier to see the symmetry of the examined models. Almost all these models of the signed integer partitions are equipped with an involution order-reversing map that makes their self-duality immediately clear. Moreover, a signed integer partition is a generalization of a usual integer partition; therefore, all the classical results can be considered as particular cases of their analogues obtained with signed integer partitions. In the sequel, we assume that is a finite graded lattice of signed integer partitions with minimum and maximum . We denote by the rank of . Moreover, we also assume that is a generic *deterministic* DDM having distinct evolution rules . In our case, this means that at each position at most one rule is applied (however the same rule may be applied at several different positions). In our model , given two configurations , it results that is covered by (with respect to the partial order ) if and only if is obtained from with the application of some rule . Therefore, in the sequel, we can always identify with .

If and are two different configurations of , we say that is a *parallel successor* of , and we write or if is the configuration which is obtained with all the possible parallel applications of the rules on the parts of . If we apply in parallel times the rule on , for , in order to obtain , we write , and we also set and . We write if there exists such that . Let us note that there is a unique finite sequence of configurations in such that
The sequence in (1) is a chain of length in that we call *fundamental chain* of . It is clear that
and . We call *parallel rank* of the integer and *sequential rank* of the integer . We also call *fundamental sequence* of the integer sequence
We say that is *parallel unimodal* if the fundamental sequence of is a unimodal sequence; we also say that is *parallel symmetric* if the fundamental sequence of is a symmetric sequence. In this paper, we compute the parallel rank and the fundamental sequence in two cases. In the first case, when is the lattice of all the signed integer partitions of the form , where are two fixed integers and . In the second (more difficult) case, when is the sublattice of all the signed integer partitions of having exactly nonzero parts. The paper is structured as follows. In Section 2, we define the lattice and establish the evolution rules which determine its covering relations, and we compute the usual rank, the parallel rank, and the fundamental sequence of . In Section 3, we introduce a new integer parameter and define the sublattice . We establish three evolution rules which determine the covering relations in and determine the usual rank of this lattice. In Section 4, we compute the parallel rank and the fundamental sequence of for all the values of the integers parameters , , and . The complete results of our computations are listed in Tables 1 and 2.

#### 2. The Sandpile Model

In this section, we introduce the lattice of signed integer partitions , and we examine some of its basic properties. Let be two fixed nonnegative integers such that . We call -*signed partition* an -tuple of integers
such that(i),(ii),(iii).

In some cases, when and are clear from the context, we say simply *signed partition* instead of -signed partition. We denote by the set of all the -signed partitions. If is an -signed partition, we call *parts* of the integers , *nonnegative parts* of the integers , and *non-positive parts* of the integers . We set , and if is such that , we say that is an *-signed partition* of ; in this case, we write . We set and . In order to simplify the notations, in the numerical examples we write the integers of a generic -signed partition without the minus sign. For example, we write instead of . Sometimes, when we do not distinguish between nonnegative and nonpositive parts, we write an -signed partition without vertical bar . Also, we denote by the number of parts of that are strictly positive, with the number of parts of that are strictly negative, and we set . If and are two -signed partitions, we set if for all , if for all , and if and . On , we consider the partial order on the components that we denote by . As usual, we write if and . Moreover, we write (or ) if covers . Since is a finite distributive lattice (because its partial order is on the components), it is also graded with minimum element and maximum element .

Now let us describe the evolution rules which describe as a discrete dynamical model. In the sequel, if , we represent the sequence of the positive parts of as a not-increasing sequence of columns of stacked squares and the sequence of the negative parts of as a not-decreasing sequence of columns of stacked squares. We call *pile* a column of stacked squares and *grain* each square of a pile. For example, if , the configuration
(5)
is identified with the signed partition .

Let . We formally set , , and . If , we call the th-*plus pile* of , and if , we call the th-*minus pile* of . We call *plus singleton pile* if and *minus singleton pile* if . If , we say that has a *plus step* at if . If , we say that has a *minus step* at if .

*Remark 1. *The choice to set , , and is a formal trick to decrease the number of rules necessary for our model. This means that when we apply the next rules to one element , we think that there is an “invisible" extra pile in the imaginary place having exactly grains, an “invisible" extra pile with grains in the imaginary place to the right of and to the left of , and another “invisible" extra pile with grains in the imaginary place to the left of and to the right of . However, the piles corresponding, respectively, to , , and must not be considered as parts of .

##### 2.1. Evolution Rules

*Rule 1. *If has a plus step at , then one grain must be added on the th-plus pile
(6)

*Rule 2. *One grain must be deleted from the th-minus pile if has a minus step at (7)
We write (or ) to denote that is an -tuple of integers obtained from applying the Rule , for . We also set

Theorem 2. *If , then . *

*Proof. *We start to show that . Let and be the invisible pile in the place . We distinguish the two possible cases related to the previous rules.*Case **1.* Let us assume that , , and that has a plus step at . If , then . Since there is a plus step at , we have ; hence, , and this implies that . We must show now that covers in . Since and differ between them only in the place for and , respectively, it is clear that there does not exist an element such that . Hence, .*Case **2.* If and has a minus step at , we apply Rule 2 to on the minus pile , and we obtain , where . Since has a minus step at , we have , and therefore, because . As in the Case 1, we note that covers in because they differ between them only for a grain in the place .

We must show now that . Let and be two -signed partitions in . We prove at first that covers if and only if there exists exactly a place where and are different and that . In fact, if and if , it is immediate to see that covers . In order to prove the other implication, we assume that covers , and we proceed by contradiction. We must distinguish three cases.*Case **A.* and are equal in all their parts. In this case, , against the hypothesis.*Case **B.* There exist at least two places and , with , where and differ, with and . We consider the signed partition
Then, ; therefore, , and since , this is against the hypothesis that covers .*Case **C.* There exists exactly one place where and are different, but . This implies that . We consider now the signed partition
Let us observe then that and , and this is absurd because covers .

Therefore, if and are two -signed partitions such that , the condition that covers can occur just by one of the following possibilities:(i)there exists exactly an index , with , and so ,(ii)there exits exactly an index , with , and so .

According to the definition of , the thesis follows.

Now let us study the sequential and the parallel dynamic of the model . In such a context, we call *configuration* an element of . The initial configuration is . Each configuration converges, in sequential and in parallel, toward the unique fixed point because of the lattice structure of the model. Let us note that if is a configuration, when we use the evolution rules in parallel, on each column of , we can apply (due to the nature of Rules 1 and 2) at most one evolution rule; hence, our model is deterministic. Obviously, is independent of the order in which the sites are updated because is a finite distributive and hence also a graded lattice. From the previous theorem, we can easily obtain the rank function of .

Proposition 3. *Let one denote by the rank function of the graded lattice . Then, for each and . *

*Proof. *Let such that . From the previous theorem, is obtained from with an application of Rules 1 or 2. If we take then the function , we obtain and . Hence, is exactly the rank function of . The second part of the thesis is followed by .

For the sake of completeness, we recall that in [36] we have computed the rank of the sublattice of all the signed partitions of having positive and negative parts all distinct between them.

We recall now the concept of involution poset. An *involution poset* (IP) is a poset with a unary operation , such that(I1) , for all , (I2) if and if , then .

The map is called *complementation* of and the *complement* of . Let us observe that if is an involution poset, by (I1) it follows that is bijective, and by (I1) and (I2) it holds that if are such that , then . If is an involution poset and if , we will set . We note that if is an involution poset, then is a self-dual poset because from (I1) and (I2) it follows that if we have that , if and only if , and this is equivalent to say that the complementation is an isomorphism between and its dual poset .

Now, if , we set , and let us note that is still a signed partition in ; therefore, we can define a unary operation such that . Then, it is immediate to verify that is an involution poset. If , we also set , and we call the *transposed* of . Let us note that .

Proposition 4. *
We have:**
(i) **
(ii) . *

*Proof. *(i) It is sufficient to observe that can be identified with the set of all the ordered pairs , where is a decreasing string of length on the ordered alphabet and is a decreasing string of length on the ordered alphabet ; therefore, .

(ii) We define the map such that , for all . We prove then that is an isomorphism of posets. By part (i), it follows that ; therefore, is bijective since it is easily seen that it is injective. If , the condition follows at once from the conditions (I1), (I2) and because the transposed map is order-reversing and such that .

Theorem 5. *The parallel rank in is given by
**The fundamental sequence of is equal to*

*Proof. *By Proposition 4 (ii), we can consider . The initial configuration is . Since our model is deterministic, we can consider separately the action of the Rules 1 and 2. On the nonnegative part, Rule 1 is applied as in the sequence
of length and similarly Rule 2 on the nonpositive part as in the sequence
of length . It follows that the parallel rank of is equal to the maximum of the length of the two sequences (13) and (14) and that the fundamental sequence is the sum of the two sequences. The other parts follow easily by replacing with and conversely.

#### 3. The Submodel with Three Evolution Rules

In this section, we consider a submodel of which presents an evolution rule which permits us to move one single grain from the right part of a configuration into its left part. Let be a positive integer such that . We define as the set of all the -signed partitions such that . It is immediate to see that is a sublattice of . The lattice is related to some extremal combinatorial sum problems studied in [32–35]. The interesting problem is to determine some evolution rules which describe such a lattice as a discrete dynamical model. We consider then the following rules.

##### 3.1. Evolution Rules

*Rule **.* If the th-plus pile has at least one grain, and if has a plus step at , then one grain must be added on the th-plus pile(15)

*Rule *. If there are some minus singleton piles, then the first from the left of them must be shifted to the side of the lowest not empty plus pile(16)

*Rule **.* One grain must be deleted from the th-minus pile if has a minus step at and (17)

*Remark 6. *(i) In the previous rule, the lowest not empty plus pile can also be the invisible column in the place . In this case, all the plus piles are empty, and an eventual minus singleton pile must be shifted in the place .

(ii) We take implicitly for intended that the shift of one minus singleton pile into a plus singleton pile can be made if the number of plus piles (excluding ) with at least a grain is less than (otherwise we obtain a configuration that does not belong to ).

We write (or ) to denote that is an -tuple of integers obtained from applying Rule , for . We also set

Theorem 7. *If , then . *

*Proof. *We start to show that . Let and be the invisible pile in the place . We distinguish the three possible cases related to the previous rules.*Case **1.* Let us assume that , and that has a plus step at . If , then . It is clear that because . Since there is a plus step at , we have ; hence, , and this implies that . We must show now that covers in . Since and differ between them only in the place for and , respectively, it is clear that there does not exist an element such that . Hence, .*Case **2.* Let us assume that in there is a minus singleton pile , for some . Since , we can assume that , , for some . This means that has the following form: . Applying Rule to , we obtain , where . It is clear then that and since is obtained from with only a shift of the pile −1 to the left in the place . Let us note that the only elements such that , and are and , but and , hence are not elements of . This implies that covers in .*Case **3.* If and has a minus step at , we apply Rule to on the minus pile , and we obtain , where . Since has a minus step at , we have , therefore because and since implies . As in Case 1, we note that covers in because they differ between them only for a grain in the place . We now must show that . Let be a generic element of such that . If we show that there exists an element of such that and , we complete the proof of the theorem. Since , there is a place where the corresponding component of is an integer strictly bigger than the integer component of corresponding to the same place. We distinguish several cases.*Case A.* We assume that and for some . In this case, we can apply Rule in the place to obtain such that .*Case B.* We assume that and for some . In this case, we have , that is, ; therefore if , we can apply Rule in the place to obtain , otherwise we have . Iterating this procedure, to each step we can apply Rule in the place to obtain or it necessarily results that . Hence, if for no one we can apply Rule in the place , we necessarily arrive to the condition . Since , it must be ; therefore, we can apply Rule in the place to obtain , with because .*Case C.* We assume that . In this case, we can apply Rule in the place .*Case D.* We assume that , for some . In this case, we can apply Rule in the place .*Case E.* We assume that and , for some . Also in this case, we can apply Rule in the place .*Case F.* We assume that , for some . In this case, the number of negative parts of is strictly lower than the number of negative parts of , and since , it follows that there exists at least one index such that and . We choose then such index maximal, so that we have or and .

We suppose at first that . In this case, we have and ; therefore, we can apply Rule to move the “negative" grain from the place into the place , so that the -partition is such that and .

If and , we apply again Rule to move the “negative" grain from the place into the place , so that the -partition is such that and .

An analogous statement of the previous theorem was proven in [36] for the sublattice of all the signed integer partitions of having positive and negative parts all distinct between them. Below, we draw the Hasse diagram of the lattice by using the evolution Rules , and starting from the minimum element of this lattice, which is . We label a generic edge of the next diagram with the symbol if it leads to a production that uses the Rule , for (19)

#### 4. Sequential and Parallel Dynamics in

In this section, we study the parallel dynamic of the model . In such a context, we call *configuration* a generic element of . The initial configuration is . Each configuration converges, in sequential and in parallel, toward the unique fixed point because of the lattice structure of the model. Let us note that if is a configuration, when we use the evolution rules in parallel, on each column of we can apply (due to the nature of the Rules , and ) at most one evolution rule, hence our model is deterministic. We denote by and the number of time steps required to reach starting from the configuration , using, respectively, the sequential or the parallel updating scheme. Obviously is independent of the order in which the sites are updated because is a finite distributive, and hence also a graded lattice (let us note that is distributive because it is a sublattice of the distributive lattice . In the next proposition, we determine the rank function of .

Proposition 8. *The map such that is the rank function of . *

*Proof. *We denote by the rank function of the graded lattice . If and are two -signed partitions and , then . It follows that , for each . Let now be any saturated chain from to in the lattice . Let us assume that in this chain is obtained from with applications of , for some integer . To each step where we apply , there is the following situation: , for one only element . This means that , that is, . The integer is also the difference between the number of positive parts of and the number of positive parts of . Hence, the thesis follows.

By using the rank function, we can compute the sequential convergence time of any element as in the following:

In particular, if we denote by the rank of the lattice , then . In the next proposition, we compute this number.

Proposition 9. *
the following result holds:*

*Proof. *The minimal and maximal elements in are, respectively,
Then,
while the number of positive parts in and is, respectively,
The thesis follows then from simple arithmetic calculations and from Proposition 8, because .

The next theorem is the main result of this section. Here we determine the parallel rank of the lattice , for all the values of , , and . In the proof of the next theorem, we also determine the fundamental sequence of in almost all the values of , , and . We do not provide the construction of the fundamental sequence for all the parameters , , and because several cases have similar constructions. However, for completeness, in Tables 1 and 2 we provide a complete description of the parallel rank value and of the fundamental sequence for all the values , , and .

Theorem 10. (I)*If , .*(II)*If , .*(III)*If and , .*(IV)*If , .*(V)*If and , .*(VI)*If , . *(VII)*If and , . *(VIII)*If and , .*

*Proof. *In all cases, let us denote the sequence of configurations as in (1).*Case I **.* If , then and . Therefore,
Starting from , we obtain
where is repeated times. The next steps are
where is repeated times. Finally, we have
From the previous computations, it follows that the fundamental sequence of is
where the integer is repeated exactly times; hence, . Let us note that the sequence (29) is unimodal and symmetric. *Case II **.* In this case, and . Since , we have therefore,
with , where is repeated exactly times.

Since , by starting from we obtain
with , where is repeated exactly times.

Now, from we have
At this point we obtain
where is repeated times. Finally, we have
Thus, the parallel rank is equal to , and the fundamental sequence of is
and so in this case is parallel unimodal but not parallel symmetric.

In a similar way, we can compute in all other cases the parallel rank and the fundamental sequence. The details are left to the interested reader. Here are the results.*Case III ** and **.* In this case, , and the fundamental sequence is
It isunimodal and not symmetric.*Case IV **.* In this case, and the fundamental sequence is
It is unimodal, and if , it isnot symmetric.*Case V ** and *. In this case, . The fundamental sequence is
It is unimodal, and if , it is not symmetric. *Case VI *. In this case, . The fundamental sequence depends on the relations between , , and in the following way.(VI.1) If and , the fundamental sequence is
(VI.2) If , , and , the fundamental sequence is
(VI.3) If , and , the fundamental sequence is
(VI.4) If and , the fundamental sequence is
(VI.5) If , , and , the fundamental sequence is
(VI.6) If , , and , the fundamental sequence is

In those cases is parallel unimodal but not parallel symmetric.*Case VII ** and *. In this case, the parallel rank is . The fundamental sequence depends on the relations between , , and in the following way.(VII.1) If , the fundamental sequence is
In this case is parallel unimodal but not parallel symmetric.(VII.2) If , the fundamental sequence is
Therefore, in this case, , it is not parallel unimodal and it is not parallel symmetric. This is the only case in which the fundamental sequence is not unimodal (see Example 11).*Case VIII ** and *. In this case, the parallel rank is . The fundamental sequence depends on the relations between , , and in the following way.(VIII.1) If , the fundamental sequence is
(VIII.2) If and , the fundamental sequence is
(VIII.3) If and , the fundamental sequence is

In these cases is parallel unimodal but not parallel symmetric.

We summarize all the previous results in Tables 1 and 2.

*Example 11. * Let us consider the lattice with , , and . In this case, , and . The fundamental chain is the following:(50)

#### 5. Conclusions and Future Research Directions

In this work, we have introduced a refinement of the concept of parallel time convergence for a finite deterministic discrete dynamical model which also has a lattice structure. This new concept is the *fundamental sequence* of . In particular, the length of the fundamental sequence is exactly the parallel time of convergence from the minimum to the maximum of . In this paper, we have computed the fundamental sequence for two models whose configurations are signed integer partitions. The perspectives, for future research directions, are the following: to compute the fundamental sequence for more general models, to compare the properties of the fundamental sequence with the properties of other relevant combinatorial objects related to the order structure of (e.g., the Whitney number sequence and the rank of when is graded), and to characterize the models whose fundamental sequence has some specific property (e.g., unimodality or symmetry).

#### References

- É. Goles, M. Latapy, C. Magnien, M. Morvan, and H. D. Phan, “Sandpile models and lattices: a comprehensive survey,”
*Theoretical Computer Science*, vol. 322, no. 2, pp. 383–407, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - T. Brylawski, “The lattice of integer partitions,”
*Discrete Mathematics*, vol. 6, pp. 201–219, 1973. View at Google Scholar · View at MathSciNet - E. Goles, “Sand pile automata,”
*Annales de l'Institut Henri Poincaré A*, vol. 56, no. 1, pp. 75–90, 1992. View at Google Scholar · View at MathSciNet - E. Goles and M. A. Kiwi, “Games on line graphs and sand piles,”
*Theoretical Computer Science*, vol. 115, no. 2, pp. 321–349, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - P. Bak, C. Tang, and K. Wiesenfeld, “Self-organized criticality,”
*Physical Review A*, vol. 38, no. 1, pp. 364–374, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - D. Dhar, “The abelian sandpiles and related models,”
*Physica A*, vol. 263, no. 1–4, pp. 4–25, 1999. View at Publisher · View at Google Scholar - E. Goles and M. Margenstern, “Universality of the chip-firing game,”
*Theoretical Computer Science*, vol. 172, no. 1-2, pp. 121–134, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - E. Goles, M. Morvan, and H. D. Phan, “Lattice structure and convergence of a game of cards,”
*Annals of Combinatorics*, vol. 6, no. 3-4, pp. 327–335, 2002. View at Google Scholar · View at MathSciNet - G. Cattaneo, M. Comito, and D. Bianucci, “Sand piles: from physics to cellular automata models,”
*Theoretical Computer Science*, vol. 436, pp. 35–53, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - A. Dennunzio, P. Guillon, and B. Masson, “Sand automata as cellular automata,”
*Theoretical Computer Science*, vol. 410, no. 38–40, pp. 3962–3974, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - E. Goles and M. A. Kiwi, “One-dimensional sand piles, cellular automata and related models,” in
*Nonlinear Phenomena in Fluids, Solids and Other Complex Systems (Santiago, 1990)*, North-Holland Delta Series, pp. 169–185, North-Holland, Amsterdam, The Netherlands, 1991. View at Google Scholar · View at MathSciNet - J. Cervelle and E. Formenti, “On sand automata,” in
*Mathematical Foundations of Computer Science 2003*, vol. 2607 of*Lecture Notes in Computer Science*, pp. 642–653, Springer, Berlin, Germany, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cervelle, E. Formenti, and B. Masson, “Basic properties for sand automata,” in
*Mathematical Foundations of Computer Science 2005*, vol. 3618 of*Lecture Notes in Computer Science*, pp. 192–211, Springer, Berlin, Germany, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cervelle, E. Formenti, and B. Masson, “From sandpiles to sand automata,”
*Theoretical Computer Science*, vol. 381, no. 1-3, pp. 1–28, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - E. Formenti and B. Masson, “On computing the fixed points for generalized sandpiles,”
*International Journal of Unconventional Computing*, vol. 2, no. 1, pp. 13–25, 2005. View at Google Scholar - E. Formenti and B. Masson, “A note on fixed points of generalized ice piles models,”
*International Journal of Unconventional Computing*, vol. 2, no. 2, pp. 183–191, 2006. View at Google Scholar - E. Formenti, B. Masson, and T. Pisokas, “On symmetric sandpiles,” in
*Cellular Automata*, vol. 4173 of*Lecture Notes in Computer Science*, pp. 676–685, Springer, Berlin, Germany, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - E. Formenti, B. Masson, and T. Pisokas, “Advances in symmetric sandpiles,”
*Fundamenta Informaticae*, vol. 76, no. 1-2, pp. 91–112, 2007. View at Google Scholar · View at MathSciNet - E. Goles, M. Morvan, and H. D. Phan, “Sandpiles and order structure of integer partitions,”
*Discrete Applied Mathematics*, vol. 117, no. 1–3, pp. 51–64, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - M. H. Le and T. H. D. Phan, “Strict partitions and discrete dynamical systems,”
*Theoretical Computer Science*, vol. 389, no. 1-2, pp. 82–90, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - M. Latapy, “Partitions of an integer into powers,” in
*Discrete Models: Combinatorics, Computation, and Geometry (Paris, 2001)*, Discrete Mathematics and Theoretical Computer Science, pp. 215–228, 2001. View at Google Scholar · View at MathSciNet - M. Latapy, “Integer partitions, tilings of 2D-gons and lattices,”
*RAIRO—Theoretical Informatics and Applications*, vol. 36, no. 4, pp. 389–399, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - M. Latapy, R. Mantaci, M. Morvan, and H. D. Phan, “Structure of some sand piles model,”
*Theoretical Computer Science*, vol. 262, no. 1-2, pp. 525–556, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - M. Latapy and H. D. Phan, “The lattice structure of chip firing games and related models,”
*Physica D*, vol. 155, no. 1-2, pp. 69–82, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - M. Latapy and T. H. D. Phan, “The lattice of integer partitions and its infinite extension,”
*Discrete Mathematics*, vol. 309, no. 6, pp. 1357–1367, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - K. Perrot and E. Rémila, “Transduction on Kadanoff sand pile model avalanches, application to wave pattern emergence,” in
*Mathematical Foundations of Computer Science 2011*, vol. 6907 of*Lecture Notes in Comput. Sci.*, pp. 508–519, Springer, Heidelberg, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - K. Perrot, T. H. D. Phan, and T. van Pham, “On the set of fixed points of the parallel symmetric sand pile model,” in
*Automata 2011—17th International Workshop on Cellular Automata and Discrete Complex Systems*, Discrete Mathematics and Theoretical Computer Science, pp. 17–28, Discrete Mathematics and Theoretical Computer Science Proceedings, 2011. View at Google Scholar · View at MathSciNet - T. H. D. Phan, “Two sided sand piles model and unimodal sequences,”
*RAIRO—Theoretical Informatics and Applications*, vol. 42, no. 3, pp. 631–646, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - P. T. H. Duong and T. T. T. Huong, “On the stability of sand piles model,”
*Theoretical Computer Science*, vol. 411, no. 3, pp. 594–601, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - G. E. Andrews, “Euler's ‘de Partitio numerorum’,”
*Bulletin of the American Mathematical Society*, vol. 44, no. 4, pp. 561–573, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - W. J. Keith, “A bijective toolkit for signed partitions,”
*Annals of Combinatorics*, vol. 15, no. 1, pp. 95–117, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - G. Chiaselotti, “On a problem concerning the weight functions,”
*European Journal of Combinatorics*, vol. 23, no. 1, pp. 15–22, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - G. Chiaselotti, G. Infante, and G. Marino, “New results related to a conjecture of Manickam and Singhi,”
*European Journal of Combinatorics*, vol. 29, no. 2, pp. 361–368, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - G. Chiaselotti, G. Marino, and C. Nardi, “A minimum problem for finite sets of real numbers with nonnegative sum,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 847958, 15 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - G. Marino and G. Chiaselotti, “A method to count the positive 3-subsets in a set of real numbers with non-negative sum,”
*European Journal of Combinatorics*, vol. 23, no. 5, pp. 619–629, 2002.