More on the Wilson $W_{tk}(v)$ Matrices

For integers 0 6 t 6 k 6 v − t, let X be a v-set, and let Wtk(v) be a ( v t ) × ( v k ) inclusion matrix where rows and columns are indexed by t-subsets and k-subsets of X, respectively, and for row T and column K, Wtk(v)(T,K) = 1 if T ⊆ K and zero otherwise. Since Wtk(v) is a full rank matrix, by reordering the columns of Wtk(v) we can write Wtk(v) = (S|N), where N denotes a set of independent columns of Wtk(v). In this paper, first by classifying t-subsets and k-subsets, we present a new decomposition of Wtk(v). Then by employing this decomposition, the Leibniz Triangle, and a known right inverse of Wtk(v), we construct the inverse of N and consequently special basis for the null space (known as the standard basis) of Wtk(v).