On Some Numbers Related to Extremal Combinatorial Sum Problems

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.