ISRN Combinatorics

Volume 2013 (2013), Article ID 615703, 7 pages

http://dx.doi.org/10.1155/2013/615703

Research Article

Sand Piles Models of Signed Partitions with Piles

C. Bisi,¹ G. Chiaselotti,² and P. A. Oliverio²

¹Dipartimento di Matematica e Informatica, Universitá di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy

²Dipartimento di Matematica, Universitá della Calabria, Via Pietro Bucci, Cubo 30B, 87036 Arcavacata di Rende, Italy

Received 3 October 2012; Accepted 4 November 2012

Academic Editors: J. F. Fang, N. A. Gordon, and M.-J. Jou

Copyright © 2013 C. Bisi 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.

Let be nonnegative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partitions that we denote by . A generic element of this model is a signed integer partition with exactly all distinct nonzero parts, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than . In particular, we determine the covering relations, the rank function, and the parallel convergence time from the bottom to the top of by using an abstract Sand Piles Model with three evolution rules. The lattice was introduced by the first two authors in order to study some combinatorial extremal sum problems.

1. Introduction

Discrete dynamical models whose configurations are integer partitions are also called Sand Piles Models and they have been deeply investigated. In these models an integer partition is treated as a sequence of piles of grains of sand and each singular grain as a single integer unit. An evolution rule in these models is a rule which describes how to move some particular grains of a configuration in order to obtain another configuration. The famous Brylawski paper [1] can be considered the first implicit study of an integer partitions lattice by means of two evolution dynamical rules which determine the covering relations of this lattice. In [1] Brylawski proposed a dynamical approach 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 Sand Piles Model begins in [2, 3].

In the Sand Piles Model introduced by Goles and Kiwi in [3], denoted by , a sand pile is represented by an ordered partition of an integer , that is, a decreasing sequence having sum , and the movement of a sand grain respects the following rule.

Rule (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, 1973 [1]), the movement of a sand grain respects Rule 1 and Rule 2, which is described as follows.

Rule (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.

The Sand Piles Model is a special case of the more general Chip Firing Game (), which was introduced by Spencer in [4] to study some “balancing game”. 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 () system introduced by Bak et al. in [5]. The study of such systems have been developed in an algebraic context ([6]), in a combinatorial games theory setting ([3, 7, 8]) and in the theory of cellular automata [9, 10].

In the papers [8, 11–20], several dynamical models related to have been studied. Almost all systems studied in the previous works have a linear topology and they have extended the classical models and to obtain more general models. An excellent survey on these topics is [21].

Let now and be non-negative integers less or equal than and the -set . In [22] the authors have introduced and studied a poset related to some extremal combinatorial sums problems (see also [23–26] for studies on these problems). Such a poset can be seen as a lattice of particular integer partitions with all distinct summands which can be positive or negative, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than . Such a poset is an involution poset; that is, it has an involution map that gives it some special symmetric properties, and it is also a lattice isomorphic (as noted in [27]) to the direct product , where is the lattice introduced by Stanley, [28], in order to solve an Erdös and Moser conjecture (here we denote by the dual lattice of ). The structure of is well known. For example, since is Peck, it follows that is Peck by [29]. This lattice contains an interesting sublattice that is the set of all the integer partitions of having exactly non-zero summands. Also this sublattice has been introduced in [22] and its structure at present is incompletely understood. For example, we know that it is a graded poset because it is a finite distributive lattice, but its rank function is unknown. In [30] Andrews introduced the concept of signed partition: a signed partition is a finite sequence of integers such that . In [30, 31] the signed partitions are studied from an arithmetical point of view.

In this paper we study the lattice as a Sand Piles Model of signed integer partitions with three evolution rules. The first of these rules is an outside adjunction rule on the “positive” piles. The second rule is a switching rule between “negative” piles and “positive” piles which allows to maintain constant the number of the piles. The third rule is an outside elimination rule on the negative “piles”. We prove that the covering relation in the lattice is uniquely determined from the three previous rules. The paper is articulated as follows. In Section 1 we recall some basic definitions and preliminary results, for example, the definition of and some of its properties. In Section 2 we explain how to see the signed partitions of as configurations of our Sand Piles Model and also we describe its evolution rules. In Section 3 we prove (Theorem 3) that the covering relation in the lattice is uniquely determined by three evolution rules of our Sand Piles Model. We determine the rank function of and we compute the rank of , that is, the sequential convergence time from the minimum to the maximum in . Finally, in Section 4 we give some estimates for the parallel convergence time in our model.

2. Definitions and Preliminary Results

If is a poset and , we write (or ) if covers . Now we briefly recall the definition of the lattice that we have introduced in [22] in a more formal context. In this paper we always denote with and two fixed non-negative integers such that . We call -string an -pla of integers

such that

(i);(ii);(iii);(iv)the unique element in (1) which can be repeated is .

If is a -string, we call parts of the integers , non-negative parts of the integers and non-positive parts of the integers . We set , and if is such that , we say that is a signed partitions of ; in this case we write . We set and . 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 . is the set of all the -strings. If and are two -strings, we set if for all , if for all and if and . On we consider the partial order on the components, that we denote by . To simplify the notations, in all the numerical examples the integers on the right of the vertical bar will be written without minus sign. Since is a finite distributive lattice it is also graded, with minimum element and maximum element .

We recall now the concept of involution poset (see [32, 33] for some recent studies on such class of posets). 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 follows that is bijective and by and 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 and 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 . In [22] has been shown that is an involution poset and its complementation map is the following:

where is the usual complement of in , and is the usual complement of in (e.g., in , we have that ). If is an integer such that , we set now . It's easy to see that is a sub-lattice of and obviously . In the sequel we always denote, respectively, by and the minimum and the maximum element of the lattice .

3. Evolution Rules

In this section we describe a discrete dynamical model with three evolution rules. In this model a configuration will be a generic element of . In the sequel, to comply with the terminology concerning the Sand Piles Models, 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 a column of stacked squares a pile and each square of a pile is called a grain. For example, if , the configuration

(3)

is identified with the partition . We denote by the configuration associated to , where is the Young diagram (represented with not-increasing columns) of the partition and is the Young diagram (represented with not-decreasing columns) of the partition with negative summands . Our goal is to define some rules of evolution that starting from the minimum of allow us to reconstruct the Hasse diagram of (and therefore to determine the covering relations in ).

Let . We formally set , and . If we call the -plus pile of , and if we call the -minus pile of . We call plus singleton pile if and minus singleton pile if . If we set and we call the plus height difference of in . If we set and we call the minus height difference of in . If , we say that has a plus cliff at if . If , we say that has a minus cliff at if .

Remark 1. The choice to set , and is a formal trick for 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 be not considered as parts of .

3.1. Evolution Rules

: If the -plus pile has at least one grain and if has a plus cliff at then one grain must be added on the -plus pile:

(4)

: If there is not a plus singleton pile and there is a minus singleton pile, then the latter must be shifted to the side of the lowest not empty plus pile:

(5)

: One grain must be deleted from the -minus pile if has a minus cliff at :

(6)

Remark 2. (i) Under the hypothesis in , the th minus pile must have at least 2 grains.

(ii) In 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 .

4. Covering Relations in

In this section we describe the covering relation in the lattice . The main result of this section is the Theorem 3. In the sequel we write (or ) to denote that is a -pla of integers obtained from applying , for .

Theorem 3. (i) If and for some , then and .

(ii) If and , then for some .

Proof. (i) Let , (the invisible pile in the place ) and . We distinguish the three possible cases related to the previous rules.

Case 1. Let us assume that , and that has a plus cliff at . If , then . It is clear that because . Since there is a plus cliff 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 not a plus singleton pile and that there is a minus singleton pile for some . Since , we can assume that , , for some . This means that has the following form: , where (otherwise has a plus singleton pile). Applying 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 cliff at , we apply to on the pile and we obtain , where . Since has a minus cliff at , we have , therefore because and since implies . As in the Case 1, we note that covers in because they differ between them only for a grain in the place .

(ii) As in (i), we take and (the invisible pile in the place ). Let a generic element of such that and . If we show that there exists an element of such that for some and we complete the proof. 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 . for some .

. . In this case we apply in the place to obtain such that .

. . Since and , then , therefore . Now, if , then and we can apply in the place . We can assume therefore . If we proceed as in A1 with in the place , otherwise, if and we proceed as before with in the place . Iterating, it follows that the cases to be examined are or and . In all these case we can apply in the place .

  . . In this case we just apply in the place .

  . for some .

  . . Then and we can apply in the place .

  . and . We apply in the place .

  . , and for some .

If then with , therefore we can not have such that and . We can assume then , and therefore . Since , it must be , hence and . Now, if , we apply in the place . We can suppose then that . Since , it must be , therefore, if then and we can apply in the place . Hence we can assume . If we apply then in the place because . Then we can suppose . Iterating this reasoning, we apply in some place , with , or we obtain for and the following forms:

with . We can apply therefore in the place .

  . , and for each .

In this case there is at least one place such that . We take this maximal, so that or and . Then, since , , and , it must be necessarily . We apply then shifting the negative grain from the place into a positive grain in the place , that is, we take with and and all other components unchanged.

Hence in all the previous cases we obtain an element , for some , such that .

Below we draw the Hasse diagram of the lattice by using the evolution rules starting to the minimum element of this lattice, which is . We label a generic edge of the next diagram with the integer if it leads to a production that uses , for :

(8)

Since is a finite distributive lattice it is also graded; in the next proposition we determine its rank function.

Proposition 4. The rank function of is .

Proof. We denote by the rank function of the graded lattice . It is easy to verify that for each (see also [22]). Let be any saturated chain from to . 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 exactly 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.

In the next proposition we compute the rank of the lattice .

Proposition 5. .

Proof. If , then and

So that . If and , then and . So that . If and , then and . Therefore . Finally, if and , then and . Therefore . The rank of is obviously , that we compute in all the previous cases applying the Proposition 4. The thesis follows then from simply arithmetic manipulations.

5. Dynamics of as Sand Piles Model

In this section we study the lattice as a discrete dynamical system. For the terminology concerning the discrete dynamical system we refer to [21]. 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 ) exactly one evolution rule, hence our model is deterministic. With the same notations of [3], we denote respectively by and the number of time steps required to reach starting from the configuration , using the sequential or the parallel updating scheme. Obviously is independent of the order in which the sites are updated because is a graded lattice. Moreover it is also clear that

We study now some properties of the dynamics in parallel. If and are two different configurations, 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 can apply in parallel times on , for , we set and . Let us note that can be only or . Obviously there is a unique finite sequence of configurations such that

The sequence in (11) is obviously a chain of length in that we call fundamental chain of . It is clear that . We also call fundamental sequence of the finite integer sequence

Remark 6. If is the fundamental chain of then

In the next result we compute the exact value of for a wide range of the integers parameters and in another case we provide a lower estimate.

Theorem 7. (i) If , .

(ii) If , .

(iii) If and , .

(iv) If and , .

(v) If and , .

Moreover, in all the previous cases the fundamental sequence of is symmetric and unimodal.

Proof. (i) If then and . In this case the first and the last rule which applies is always : in between all the rules apply in a very symmetric way, in view of their definition and of the symmetry of three parameters of . The number of rules which apply at each step follows this sequence: and the unimodularity and the symmetry of the sequence of parallel configurations is therefore straightforward. Moreover there exists a unique time in which the maximal number of rules apply; the string at which the rules apply is always of the following type:

  if is even: , where in both positive and negative part each number is the previous minus 2;   if is odd: , where in both positive and negative part each number is the previous minus 2.

Hence .

(ii) If then and . The first rule which applies is always and the last one is always . In this case there are several strings obtained with the maximal number of rules which is : the first one which appears from the bottom in the fundamental sequence of is:

 if is even, , where in both positive and negative part each number is the previous minus ;   if is odd, , where in both positive and negative part each number is the previous minus 2.

The number of steps with rules is exactly and the number of rules which apply at each step follows this sequence: : hence the unimodularity and the symmetry of the sequence of the parallel configurations is straightforward. Finally, .

(iii) With these parameters, and . The first rule from the bottom which applies is always and the last rule from the top which applies is always . In this case there are several strings obtained with the maximal number of rules which is still , and the number of rules which apply at each step follows this sequence: : therefore the unimodularity and the symmetry of the sequence of the parallel configurations follow such as the lower bound for .

(iv) With these parameters, and . In this case the first and the last rule which applies is always : in between all the rules . The number of rules which apply at each step follows this sequence: and the unimodularity and the symmetry of the sequence of parallel configurations is immediate. Moreover there exists a unique time in which the maximal number of rules apply. Hence .

(v) With these parameters, and . In this case the first and the last rule which applies is always . A difference with the previous case is that here there are several strings obtained with the maximal number of rules which is ; the number of string at which rules apply is exactly . The very interesting thing is that is independent of in this case, as in case . Obviously also in this case we have the unimodularity and the symmetry of the sequence of the parallel configurations.

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