Typical property of one class of combinatory objects and estimation from above corresponding combinatory numbers

Abstract

We investigate properties of families $F$ of subsets of a finite set in a situation where subsets are incomparable by the binary inclusion relation and a) for any $A\notin F$, there is such set $A^{\prime }\in F$ that either $A\subset A^{\prime }$ or $A^{\prime }\subset A$; b) for any $A\in F$, $\mid A\mid \in \{k,k+1\}$. For these families we introduce one parametre and we show that for almost all families $F$ the value of this parametre is $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$. We show that families with the minimum value of the entered parametre have certain structure and we find also number of such families. At last, we find an estimation from above for combinatory numbers of considered combinatory objects.