Disjoint odd integer subsets having a constant even sum
The authors show that if \(n\) and \(k\) are positive integers and if \(m\) is a positive even integer then the following two statements hold: (i) the odd integer set \(\{1,3,5,\dots, 2n-1\}\) contains \(k\) disjoint subsets having a constant sum \(m\) if and only if \(4k\leq m\leq n^ 2/k\), \(n^ 2- mk\neq 2\) and either \(n\neq 4k\) or \(m\neq 16k-2\); (ii) the odd integer set \(\{1,3,5,\dots,2n- 1\}\) contains \(k+1\) disjoint subsets \(A_ 1, A_ 2,\dots, A_ k\) each with sum \(m\) and \(B\) with sum \(m/2\) if and only if either \(m\) is congruent to \(0\text{ mod }4\) and \(4k+ 8\leq m\leq 2n^ 2/(2k+ 1)\) or \(m\) is congruent to \(2\text{ mod }4\) and \(4k+ 2\leq m\leq 2n^ 2/(2k+ 1)\), \(n^ 2- (2k+ 1) m/2\neq 2\) and \(n\neq 4k+ 3\) or \(m\neq 16k+ 14\). From this result follows a conjecture made by \textit{K. Ando}, \textit{S. Gervacio} and \textit{M. Kano} in [Discrete Math. 82, No. 1, 7-11 (1990; Zbl 0732.05002)] namely that if \(n\) and \(k\) are positive integers and if \(m\) is a positive even integer then the odd integer set \(\{1,3,5,\dots, 2n-1\}\) contains \(k\) disjoint subsets having a constant sum \(m\) if and only if \(4k\leq m\leq n^ 2/k\), \(n^ 2- mk\neq 2\) and either \(n\neq 4k\) or \(m\neq 16k- 2\).
scientific article; zbMATH DE number 719459