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

TL;DR

This paper studies special families of subsets with specific incomparability and size properties, introduces a parameter to analyze their structure, and provides bounds and estimations for related combinatorial numbers.

Contribution

It introduces a new parameter for these subset families, characterizes those with minimal parameter value, and estimates the number of such families and their combinatorial counts.

Findings

Most families have the parameter value ${n-1race k}$

Families with minimal parameter value have a specific structure

Provides an upper bound for the combinatorial numbers of these objects

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 $A'\in F$ that either $A\subset A'$ or $A'\subset A$; 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 ${n-1\choose k}$. 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.