Sand Piles Models of Signed Partitions with 𝑑𝑑 Piles

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


Introduction
Discrete dynamical models whose con�gurations 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 con�guration in order to obtain another con�guration. e famous Brylawski paper [1] can be considered the �rst 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 �xed positive integer with the dominance order.
However, the explicit identi�cation of a speci�c 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 SPM , a sand pile is represented by an ordered partition of an integer , that is, a decreasing sequence 1 … having sum , and the movement of a sand grain respects the following 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, 1973 [1]), the movement of a sand grain respects Rule 1 and Rule 2, which is described as follows.
Rule 2 (horizontal rule). If a column containing 1 grains, is followed by a sequence of columns containing grains and then one column containing −1 grains, one grain of the �rst column can slip to the last one.
e Sand Piles Model SPM is a special case of the more general Chip Firing Game (CFG), which was introduced by Spencer in [4] to study some "balancing game". ere are a lot of specializations and extensions of this model which have been introduced and studied under different names, different aspects and different approaches. e SPM can be also related to the Self-Organized Criticality (SOC) system introduced by Bak et al. in [5]. e 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].
Let now and be non-negative integers less or equal than and [ the -set {1, … , . In [22] the authors have introduced and studied a poset ( ( , , related to some extremal combinatorial sums problems (see also [23][24][25][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 ( − ). e structure of ( is well known. For example, since ( is Peck, it follows that ( , is Peck by [29]. is 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 �nite distributive lattice, but its rank function is unknown. In [30] Andrews introduced the concept of signed partition: a signed partition is a �nite sequence of integers , … , 1 , −1 , … , − such that ≥ ⋯ ≥ 1 > 0 > −1 ≥ ⋯ ≥ − . 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. e �rst of these rules is an outside adjunction rule on the "positive" piles. e second rule is a switching rule between "negative" piles and "positive" piles which allows to maintain constant the number of the piles. e 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. e paper is articulated as follows. In Section 1 we recall some basic de�nitions and preliminary results, for example, the de�nition of ( , , and some of its properties. In Section 2 we explain how to see the signed partitions of ( , , as con�gurations of our Sand Piles Model and also we describe its evolution rules. In Section 3 we prove (eorem 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.
If ( , is a poset and , , we write (or ) if covers . Now we brie�y recall the de�nition of the lattice ( , that we have introduced in [22] in a more formal context. In this paper we always denote with and two �xed non-negative integers such that . We call ( , -string an -pla of integers such that (iv) the unique element in (1) which can be repeated is 0.
If is a ( , -string, we call parts of the integers , … , 1 , 1 … − , non-negative parts of the integers , … , 1 and non-positive parts of the integers 1 … − . We set ∑( ∑ 1 + ∑ − 1 , and if is such that ∑( , we say that is a signed partitions of ; in this case we write . We set + … 1 | and − | 1 … − . 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 . 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 �nite distributive lattice it is also graded, with minimum element 0 ⋯ 0 | 12 ⋯ ( − and maximum element ( − 1 ⋯ 21 | 0 ⋯ 0.
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 . 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 * . In [22] has been shown that ( ( is an involution poset and its complementation map is the following: It's easy to see that ( ( is a sub-lattice of ( ( and obviously | ( | = .
In the sequel we always denote, respectively, by 0 and 1 the minimum and the maximum element of the lattice ( .

Evolution Rules
In this section we describe a discrete dynamical model with three evolution rules. In this model a con�guration 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 notdecreasing 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 = 10 = = , the con�guration . .

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is identi�ed with the partition ( 3 1 0 0 0 | 0 0 1 3 = 33100 | 0113 (10 . We denote by ( = + ( ( the con�guration associated to , where + ( is the Young diagram (represented with not-increasing columns) of the partition ( … 1 and ( is the Young diagram (represented with not-decreasing columns) of the partition with negative summands ( 1 … . Our goal is to de�ne 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 = … 1 | 1 … ( . We formally set 0 = 0, +1 = + 1 and 0 = 0. If 0 ≤ ≤ + 1 we call the th-plus pile of , and if 0 ≤ ≤ we call the th-minus pile of . We call plus singleton pile if = 1 and minus singleton pile if = 1. If 1 ≤ ≤ + 1 we set Δ + ( = 1 and we call Δ + ( the plus height difference of in . If 1 ≤ ≤ we set Δ ( = | | | 1 | and we call Δ ( the minus height difference of in . If 1 < ≤ , we say that has a plus cliff at if Δ + ( 2. If 1 < ≤ , we say that has a minus cliff at if Δ ( 2.
Remark 1. e choice to set 0 = 0, +1 = and 0 = 0 is a formal trick for decrease the number of rules necessary for our model. is means that when we apply the next rules to one element ( we think that there is an "invisible" extra pile in the imaginary place + 1 having exactly + 1 grains, an "invisible" extra pile with 0 grains in the imaginary place to the right of 1 and to the le of | and another "invisible" extra pile with 0 grains in the imaginary place to the le of 1 and to the right of |. However the piles corresponding respectively to 0 = 0, +1 = + 1 and 0 = 0 must be not considered as parts of .

Evolution Rules
1 : If the th-plus pile has at least one grain and if has a plus cliff at + 1 then one grain must be added on the th-plus pile: . .

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2 : If there is not a plus singleton pile and there is a minus singleton pile, then the latter must be shied to the side of the lowest not empty plus pile: . .

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3 : One grain must be deleted from the th-minus pile if has a minus cliff at : Remark 2. (i) Under the hypothesis in 3 , the th minus pile must have at least 2 grains.
(ii) In 2 the lowest not empty plus pile can also be the invisible column in the place + 1. In this case all the plus piles are empty and an eventual minus singleton pile must be shied in the place .

Covering Relations in (
In this section we describe the covering relation in the lattice ( . e main result of this section is the eorem 3. In the sequel we write ′ (or ′ = ) to denote that ′ is a -pla of integers obtained from applying , for . 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.
. In this case we apply in the place to obtain ′ such that ′′ ⊒ ′ .
. . Since ′′ > 0 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 .
. ′′ < 0. en and we can apply in the place .
. ′′ 0 and . We apply in the place .
. ′′  . en 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 .

′′ 0,
and ≠ for each … . In this case there is at least one place … such that 0. We take this maximal, so that or < and > 0. en, since ′′ = 0, = − , ′′ ≥ > 0 and || || = || ′′ || = , it must be necessarily ′′ > 0. We apply then 2 shiing the negative grain from the place into a positive grain in the place , that is, we take ′ = 2 with ′ = and ′ = 0 and all other components unchanged.
Hence in all the previous cases we obtain an element ′ = , for some = 2 , such that ′ ⊑ ′′ .
Below we draw the Hasse diagram of the lattice 2 by using the evolution rules 2 starting to the minimum element of this lattice, which is 00 | 2 . We label a generic edge of the next diagram with the integer if it leads to a production that uses , for 2 : . .
Since is a �nite distributive lattice it is also graded� in the next proposition we determine its rank function.

Proposition 4. e rank function of is
Proof. We denote by the rank function of the graded lattice . It is easy to verify that = ∑ − ∑ 0 for each (see also [22]). Let ⋯ 0 be any saturated chain from 0 to . Let us assume that in this chain is obtained from 0 with applications of 2 , for some integer ≥ 0. To each step where we apply 2 , there is the following situation: ⊐ ⊐ − , for exactly one only element . is means that = , that is, = − . e integer is also the difference between the number of positive parts of and the number of positive parts of 0. Hence the thesis follows.
In the next proposition we compute the rank of the lattice .

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 con�guration a generic element of . e initial con�guration is 0. �ach con�guration converges, in sequential and in parallel, toward the unique �xed point because of the lattice structure of the model. Let us note that if is a con�guration, when we use the evolution rules in parallel, on each column of we can apply (due to the nature of the Rules 2 ) exactly one evolution rule, hence our model is deterministic. With the same notations of [3], we denote respectively by sec and par the number of time steps required to reach starting from the con�guration , using the sequential or the parallel updating scheme. Obviously sec 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 con�gurations, we say that ′ is a parallel successor of , and we write ′ or ′ = , if ′ is the con�guration 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 = 2 , we set = 2 and | | = 2 . Let us note that 2 can be only 0 or . Obviously there is a unique �nite sequence 0 of con�gurations such that 6 ISRN Combinatorics e sequence in (11) is obviously a chain of length in ( ( that we call fundamental chain of ( . It is clear that par (0 = . We also call fundamental sequence of ( the �nite integer sequence 0 1 … 1 .

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Remark 6. If ( 0 1 … is the fundamental chain of ( then In the next result we compute the exact value of par (0 for a wide range of the integers parameters and in another case we provide a lower estimate. Proof. (i) If = ( = then 0 = 0 … 0 | 1 … ( and 1 = 1 | 0 0. In this case the �rst and the last rule which applies is always 2 : in between all the rules 1 2 3 apply in a very symmetric way, in view of their de�nition and of the symmetry of three parameters of ( . e number of rules which apply at each step follows this sequence: 1 … ( 1 ( 1 … 1 and the unimodularity and the symmetry of the sequence of parallel con�gurations 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: ( 1 … 1 … 0 | 0 … 2 … , where in both positive and negative part each number is the previous minus 2; if is odd: ( 1 … 2 … 0 | 0 … 1 … , where in both positive and negative part each number is the previous minus 2.
Hence par (0 = 2 1. (ii) If = ( then 0 = 0 … 0 | 1 … ( and 1 = … ( 1 … 0 | 0 … 0. e �rst rule which applies is always 2 and the last one is always 1 . In this case there are several strings obtained with the maximal number of rules which is : the �rst one which appears from the bottom in the fundamental sequence of ( is: if is even, ( 1 ( 3 … 10 … 0 | 0 … 2 … , where in both positive and negative part each number is the previous minus 2; if is odd, ( 1 ( 3 … 20 … 0 | 0 … 1 … ( 2 , where in both positive and negative part each number is the previous minus 2.
e number of steps with rules is exactly ( 1 and the number of rules which apply at each step follows this sequence: 1 … ( 1 … ( 1 … 1: hence the unimodularity and the symmetry of the sequence of the parallel con�gurations is straightforward. Finally, par (0 = 1 = 1. (iii) With these parameters, 0 = 0 … 0 | 0 … ( 1 … ( and 1 = … ( 1 0 … 0 | 0 … 0. e �rst rule from the bottom which applies is always 3 and the last rule from the top which applies is always 1 . 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: 1 … ( 1 … ( 1 … 1: therefore the unimodularity and the symmetry of the sequence of the parallel con�gurations follow such as the lower bound for par (0 .
(iv) With these parameters, 0 = ( 1 0 | 1 … ( and 1 = ( 1 0 | 0 0. In this case the �rst and the last rule which applies is always 1 : in between all the rules 1 2 3 . e number of rules which apply at each step follows this sequence: 1 … ( 1 ( 1 … 1 and the unimodularity and the symmetry of the sequence of parallel con�gurations is immediate. Moreover there exists a unique time = in which the maximal number of rules apply. Hence par (0 = 2 1. (v) With these parameters, 0 = ( 1 0 | 1 … ( and 1 = ( 1 0 | 0 0. In this case the �rst and the last rule which applies is always 1 . 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 ( 2 1 . e very interesting thing is that par (0 = 1 is independent of in this case, as in case (ii . Obviously also in this case we have the unimodularity and the symmetry of the sequence of the parallel con�gurations.