A Class of lattices and boolean functions related to a Manickam-Mikl"os-Singhi Conjecture

The aim of this paper is to build a new family of lattices related to some combinatorial extremal sum problems, in particular to a conjecture of Manickam, Mikl"os and Singhi. We study the fundamentals properties of such lattices and of a particular class of boolean functions defined on them.

💡 Research Summary

The paper introduces a novel family of lattices, denoted S(n,r), designed to capture the combinatorial structure underlying a class of extremal sum problems, most prominently the Manickam‑Miklos‑Singhi (MMS) conjecture. Starting from two fixed integers n ≥ 0 and 0 ≤ r ≤ n, the authors consider real weight vectors a₁ ≥ … ≥ a_r ≥ 0 > a_{r+1} ≥ … ≥ a_n and define a partial order ⊑ on the power set of {1,…,n} by the rule: for subsets X and Y, X ⊑ Y iff the sum of the a_i over X does not exceed the sum over Y. This order makes the whole Boolean lattice distributive; restricting to a carefully chosen sub‑poset yields S(n,r), whose elements are strings of the form i₁…i_r | j₁…j_{n−r} with symbols drawn from a totally ordered alphabet that separates “positive” and “negative” indices. The paper proves that (S(n,r),⊑) is a distributive, graded lattice, closed under meet and join, and possesses an involution given by set complement (X⊑Y ⇒ Yᶜ⊑Xᶜ).

The authors develop several structural results: a recursive formula for the number of elements at each rank, an algorithm to generate the Hasse diagram uniquely, and a description of covering relations (the “|” symbol plays a central role). They also show that the lattice is self‑dual and that its rank function coincides with the number of non‑zero symbols in the string representation.

The second part of the paper connects this lattice to extremal sum problems. For a weight function f ∈ W_n(ℝ) (i.e., Σ_{i=1}^n f(i) ≥ 0) they define f⁺ = |{i : f(i) ≥ 0}| and consider two quantities:
 γ(n,r) = min_{f∈W_n, f⁺=r} α(f), where α(f) counts all subsets Y⊆