Abstract
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 $W_{tk}(v)$.
Highlights
Integers t, k, and v with 0 v-set, and letX i t k v − t are considered
In passing we note that the matrix E contains (v − k − t) copies of intersecting submatrices Wt−1,k−1(v − 1)
Around 1980, Graham, Li, and Li [7] presented a right inverse for W with a closed formula
Summary
The inclusion matrix Wtk(v) (known as Wilson matrix) is defined to be a v t by v k (0, 1)-matrix whose rows and columns are indexed by (and referred to) the members of the electronic journal of combinatorics 21(2) (2014), #P2.53. In the set of our notations, for any matrix M , the free Z-module generated by rows and columns of matrix M will be denoted by rowZ(M ) and colZ(M ), respectively, and nullZ(M ) will be the free Z -module orthogonal to rowZ(M ). Hartman [9] stated the following conjecture which is known as the halving conjecture: For 0 i t, there is a (1, −1)-vector in nullZ(W ) if and only if v−i k−i. In [1] the columns of S23(v) have been classified into five classes and by utilizing these classes the correctness of the halving conjecture has been established
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