Abstract

Let n, d, and r be three integers such that . Chiaselotti (2002) defined as the minimum number of the nonnegative partial sums with d summands of a sum , where are n real numbers arbitrarily chosen in such a way that r of them are nonnegative and the remaining are negative. Chiaselotti (2002) and Chiaselotti et al. (2008) determine the values of for particular infinite ranges of the integer parameters n, d, and r. In this paper we continue their approach on this problem and we prove the following results: (i) for all values of n, d, and r such that ; (ii)

1. Introduction

Let , , and be three fixed integers such that ,. We set and The elements of are called n-weight functions and, if , we set . For example, if , and , then . If , we also set (we call -subset of a generic element of ), , and These numbers were introduced in [1] in order to refine the study of a conjecture of Manickam-Miklós-Singhi (for further information on this conjecture and on its links with the numbers see [1–4]). The complete determination of the numbers is a very difficult task and actually they are known only for a relatively small range of the integer parameters ,, and . In [1–3] some of the numbers have been determined, and we report these values:  if ,  if and ,  if and ,  if ,  if and ,  if and .In particular, in [3] the authors prove the last of these results using Hall’s matching theorem.

Also, in [2, 5] the numbers were linked within the context of the combinatorial order theory. More in detail, in [2] the authors introduce two new classes of lattices of signed integer partitions, and , and they show that the numbers can be interpreted as the cardinality of particular types of up-sets in the previous lattices.

On the other hand, the lattices and can also be considered as particular types of discrete dynamical systems. In this context many properties of the numbers can be related to the evolution rules that characterize and as discrete dynamical systems (see [6, 7]). For very recent studies concerning the discrete dynamical systems see [8–12].

In this paper we determine some new identities and new bounds for the numbers . In particular, we show that(i) for all values of , and satisfying (Corollary 5),(ii) in the case and (Proposition 8).

Finally we provide a combinatorial interpretation of the inequality .

The remaining part of this paper is structured as follows. In Section 2 we provide the necessary notations for the sequel. In Section 3 we establish our results and, finally, in Section 4 we briefly describe conclusions and possible future research approaches.

2. Notations

In the sequel, we will assume that a generic weight function , with , has the form with Let us call the indexes the nonnegative elements of and the indexes the negative elements of . The real numbers are said to be the nonnegative values of and the numbers are said to be the negative values of .

If are nonnegative elements of and are negative elements of , with and , a subset of is said to be of type if is made of elements chosen in and elements chosen in .

Let be a finite set of integers. If is an integer less than or equal to , we call -string on a sequence , where are distinct elements of such that . In this paper, each subset of with elements will be identified with the -string of its elements ordered in increasing way. When are nonnegative elements of and are negative elements of , with , the -stringwill be written in the form (thus ).

For example, if and , the 4-string will be written in the form .

Using the string-terminology instead of the set-terminology, in the sequel, we will call a -subset of a -string of .

3. The Results

The range of values of the integer parameters ,, and that we are going to study is the following: . As reported in the first section, we already know that the next result holds for in the case . We state it referring to its proof.

Proposition 1. If and , then .

Proof. See [1].

Therefore we concentrate our attention on the case . In order to examine it, we start by considering the partition of the real interval : The following proposition establishes when an interval determined by contains an integer.

Proposition 2. If and if , there exists a unique integer such that and coincides with . Furthermore if , no integer satisfies (8).

Proof. Let and set . Since the interval has length , there is at most one integer that satisfies (8). Let us now write in the form where , are integers such that . Let us suppose now that ; that is, ; then we have .
Let . We show that satisfies (8).
Firstly, the second inequality is straightforward; secondly, for the first inequality we observe Furthermore Therefore , since .
If , that is, , in (9), we have and (8) becomes Note that (12) has no integer solutions.

Lemma 3. Let be a positive integer such that Then there exists a unique positive integer that satisfies

Proof. By construction of partition , as in (7), there exists a unique such that (14) holds.
We now show that cannot exceed .
Firstly, we suppose that . Then, we write in the form , with integer such that . Since satisfies (14), we have that is, Since and , there is no positive integer that satisfies (16).
Secondly, if , (14) becomes that is, contradicting the hypothesis (13).

Using the previous lemma, we find a useful and appreciable upper bound for . This is one of our important steps to establish, under suitable hypotheses for ,, and , an exact value for .

Proposition 4. Let be a positive integer that satisfies then

Proof. Since , we construct a weight function , with , such that This is sufficient to prove the (20).
Let . Let be a positive real number. In order to simplify the notation, we call the number , in such a way that holds.
At this point we define the function
We now show that, for sufficiently small, that is, is a weight function that satisfies (21).
In fact,
(a)  the denominator of , due to Lemma 3, is a positive number. Furthermore the numerator of is a positive number if and only if . Therefore (24) and the definition of assure that is a weight function.
(b)  Having is equivalent to require
This condition assures that the subsets of the typeare -subsets of .
(c)  Firstly we note that the requirement is equivalent to require Lemma 3 assures the existence of a such . Note that (28) is equivalent to which assures that the -strings of are only of the type (26). Therefore we have constructed a weight function with being nonnegative elements which satisfies (21).

We now concentrate our attention on the subinterval . By Proposition 2 there is a unique value of contained in this subinterval. For this value formula (20) becomes simpler as expressed in the following result.

Corollary 5. Let be a positive integer such that and . Then

Proof. The result follows directly from Proposition 4 since .

The aim of the next proposition is to try to individuate a lower bound for starting from . Recalling that , we have the following result.

Proposition 6. Let r be a positive integer such that . Let , with , as in (3). If then

Proof. We can consider the d-strings of of type where are chosen in .
By virtue of (31), each string of the type (33) is a -string of .
On the other hand, since , each string of type where are chosen in and in , will be a -string of .
The distinct strings of the type (34) are exactly . There are moreover all the -strings of that are the -strings on . This proves the first inequality in (32). Moreover, since , we also have . Therefore Thus the second inequality also holds.

Proposition 6 leads us to state the following conjecture.

Remark 7. We conjecture that where and .
Note that, in order to prove (36), by Corollary 5, it is sufficient to show
In the particular case and the previous conjecture has been proved in [3].

In the next result we show that our conjecture (36) is true when and .

Proposition 8. If and , then

Proof. We observe that if and , then . Hence, by Corollary 5,
Therefore, in order to have the thesis, we must prove that Let us note that as a direct consequence of Corollary 5 and Proposition 6 it follows that if is a positive integer with such that , then
Therefore by (41) inequality (40) is equivalent to the following:
So, to prove inequality (40), it is sufficient to prove that, for each such that and , we have . We setand let be all the -strings in . At this point we consider the following configurations: where . Since is a weight function and for , it follows that all the -strings are nonnegative -tuple of the form associated with . Therefore . This proves (42), and hence also (40) is proved.

In Table 1 we list all the known values of in the range . Let us note that when the integer is fixed and runs in , by Proposition 2, we find at most an integer value such that .