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G. Chiaselotti, T. Gentile, G. Marino, P. A. Oliverio, "Parallel Rank of Two Sandpile Models of Signed Integer Partitions", Journal of Applied Mathematics, vol. 2013, Article ID 292143, 12 pages, 2013. https://doi.org/10.1155/2013/292143
Parallel Rank of Two Sandpile Models of Signed Integer Partitions
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.
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  describes several subsets of integer partitions which have a lattice structure in terms of discrete dynamical models. Before going further, we borrow from  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 . 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 , 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.
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 . The study of such systems has been developed in an algebraic context , 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].
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  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  and further studied in  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  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 .
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