# zbMATH — the first resource for mathematics

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
Full Text: