Minimum range sequences of all k-subsets of a set

Abstract The purpose of this paper is to find sequential orderings of the class Sk of all k-element subsets of the v element set V = {1, 2, …, υ} with certain minimizing properties. A sequential ordering of Sk is just a numbering of the (vk) k-element subsets of V, where the first one is s1, the second s 2 and so on. Let f(s) denote the position of the k-element subset s in the ordering. For each element e of V the range Rf(e) of e in the sequence f is given by R ⨍ ( e ) = max e ϵ s , s ′ |⨍( s )−⨍( s ′ )z.sfnc; The total range Rf of a sequence f is R ⨍ = ∑ e ϵ v R⨍( e ) and the bottleneck range Bf of a sequence f is R ⨍ = max r ϵ v R⨍( e ) Let F denote the set of all sequences (for fixed values of v and k). In this paper we show there exist sequences f that simultaneously minimize Rf and Bf, give a construction of such sequences for arbitrary v and k, and give exact formulae for minfϵ FRf and minfϵFBf as functions of v and k. These problems arise in experimental designs involving human subjects and in the linking between two stages of switches in multistage switching networks.