Directed packings with block size 5 and even ν

Abstract

Let ν andk be positive integers. A transitively orderedk-tuple (a ₁,a ₂,...,a _k) is defined to be the set {(a _i, a_j) ∶ 1≤i<j≤k} consisting ofk(k−1)/2 ordered pairs. A directed packing with parameters ν,k and index λ=1, denoted byDP(k, 1; ν), is a pair (X, A) whereX is a ν-set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ν) is called packing number and denoted byDD(k, 1; ν). It is shown in this paper that$DD (5,1; v) = \left\lfloor {\frac{v}{5}\left\lfloor {\frac{{2(v - 1)}}{4}} \right\rfloor } \right\rfloor$ for all even integers ν, where [x] is the floor ofx.