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Eine Bemerkung über ein kombinatorisches Problem. (Russian. English summary) Zbl 0134.25004
English summary: Let \(n\), \(k\) be natural numbers such that \(n\equiv k \pmod{2k}\). Denote \(m = 2n + n/k\) and denote
\[ M = \{1, 2, \ldots, m - 1\} - \{2k + 1, 2(2k + 1), \ldots, (n/k - 1)(2k + 1)\}. \]
The set \(A = \{a_1, a_2, \ldots, a_N)\subset M\), \(N\le n\), is said to be of the type \((k)\), if \(\sum_{i=1}^N a_i\equiv 0\pmod m\) holds and if \(a_i + a_j\not\equiv 0\pmod m\) holds for all \(i, j = 1,\ldots,N\). If the set \(A\) with \(n\) elements is of the type \((k)\), then \(q_i^{(A)}\) denotes the number of its subsets of the type \((k)\) with \(i\) elements. The following theorem is valid: The sum \(\sum_{i=1}^N q_i^{(A)}\) does not depend on the choice of the set \(A\), but only on the number of its elements \(n\) and on \(k\) (this sum is denoted by \(W(n, k)\)). Further the formula for determining \(Q(n, k)\), the number of different sets of the type \((k)\) with \(n\) elements, is given.
It can be shown by the method analogous to one the authors used in [Mat.-Fyz. Čas., Slovensk. Akad. Vied 15, 49–59 (1965; Zbl 0128.26801)], that the following relation holds:
\[ Q(n,k) = W(n,k) + 2. \]

MSC:
11A07 Congruences; primitive roots; residue systems
11B75 Other combinatorial number theory
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