On average and typical values of sums of pairwise distances for subsets of vertices of the $n$-dimensional unit cube

We study the question on average and typical values of sums of pairwise Hamming distances for subsets of vertices of the $n$-dimensional unit cube. We suggest an approach to the problem of evaluation of average and typical values of arbitrary functionals defined on subsets of a finite set as the sum of values assigned to ordered pairs of elements of this set; general formulas for this case are obtained. We find average and typical values of sums of pairwise distances in the case of all subsets of vertices of the $n$-dimensional unit cube and of sums of pairwise distances for subsets of vertices of fixed cardinality.
This research was supported by the Russian Foundation for Basic Research, grant 01–01–00266Б.