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

Keywords: Left inverse, Signed $t$-design, Leibniz Triangle, Standard basis, Right inverse, Root of a block, $\mathcal{R}$-ordering, $B$-changer
For integers $0\leq t\leq k\leq v-t$, let $X$ be a $v$-set, and let $W_{tk}(v)$ be a ${v \choose t}\times{v \choose k}$ inclusion matrix where rows and columns are indexed by $t$-subsets and $k$-subsets of $X$, respectively, and for row $T$ and column $K$, $W_{tk}(v)(T,K)=1$ if $T\subseteq K$ and zero otherwise. Since $W_{tk}(v)$ is a full rank matrix, by reordering the columns of $W_{tk}(v)$ we can write $W_{tk}(v) = (S|N)$, where $N$ denotes a set of independent columns of $W_{tk}(v)$. In this paper, first by classifying $t$-subsets and $k$-subsets, we present a new decomposition of $W_{tk}(v)$. Then by employing this decomposition, the Leibniz Triangle, and a known right inverse of $W_{tk}(v)$, we  construct  the inverse of $N$ and consequently special basis for the null space (known as the standard basis) of $W_{tk}(v)$.