A linear algebraic approach to directed designs (Q2712517)

Order lexicographically all \(t\)-tuples and \(k\)-tuples of a totally ordered \(v\)-set, \(V\). A \(t\)-tuple appears in a \(k\)-tuple if its components occur in the same order in the \(k\)-tuple, though they are possibly no longer adjacent. The \(t\)-inclusion matrix \(D^v_{t,k}= [d_{ij}]\) is the \(t!{v\choose t}\times k!{v\choose k}\) matrix with \(d_{ij}= 1\) if the \(i\)th \(t\)-tuple appears in the \(j\)th \(k\)-tuple, and \(0\) otherwise. A column vector \(F\) with \(k!{v\choose k}\) integer entries represents a \(t\)-\((v,k,\lambda)\) signed directed design if \(D^v_{t,k} F=\lambda e_t\), where \(e_t\) is the column vector consisting of \(t!{v\choose t}\) ones. The authors determine the rank of \(D^v_{t,k}\) for \(0\leq t\leq 4\), and give a semi-triangular basis for the null spaces of \(D^v_{t,t+1}\), for \(0\leq t\leq 3\), and for \(D^{t+1}_{t,t+1}\). They then establish that the obvious necesary conditions for the existence of \(t\)-\((v,k,\lambda)\) signed directed designs are also sufficient, provided \(0\leq t\leq 4\).