Inclusion Matrices for Rainbow Subsets

Abstract

Let $\text {S}$ be a finite set, each element of which receives a color. A rainbow t-set of $\text {S}$ is a t-subset of $\text {S}$ in which different elements receive different colors. Let $\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right)$ denote the set of all rainbow t-sets of $\text {S}$, let $\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right)$ represent the union of $\left( {\begin{array}{c}\text {S}\\ i\end{array}}\right)$ for $i=0,\ldots , t$, and let $2^\text {S}$ stand for the set of all rainbow subsets of $\text {S}$. The rainbow inclusion matrix $\mathcal {W}^{\text {S}}$ is the $2^\text {S}\times 2^{\text {S}}$ (0, 1) matrix whose (T, K)-entry is one if and only if $T\subseteq K$. We write $\mathcal {W}_{t,k}^{\text {S}}$ and $\mathcal {W}_{\le t,k}^{\text {S}}$ for the $\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right)$ submatrix and the $\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right)$ submatrix of $\mathcal {W}^{\text {S}}$, respectively, and so on. We determine the diagonal forms and the ranks of $\mathcal {W}_{t,k}^{\text {S}}$ and $\mathcal {W}_{\le t,k}^{\text {S}}$. We further calculate the singular values of $\mathcal {W}_{t,k}^{\text {S}}$ and construct accordingly a complete system of $(0,\pm 1)$ eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let $\mathcal {D}^{\text {S}}_{t,k}$ denote the integral lattice orthogonal to the rows of $\mathcal {W}_{\le t,k}^{\text {S}}$ and let $\overline{\mathcal {D}}^{\text {S}}_{t,k}$ denote the orthogonal lattice of $\mathcal {D}^{\text {S}}_{t,k}$. We make use of Frankl rank to present a $(0,\pm 1)$ basis of $\mathcal {D}^{\text {S}}_{t,k}$ and a (0, 1) basis of $\overline{\mathcal {D}}^{\text {S}}_{t,k}$. For any commutative ring R, those nonzero functions $f\in R^{2^{\text {S}}}$ satisfying $\mathcal {W}_{t,\ge 0}^{\text {S}}f=0$ are called null t-designs over R, while those satisfying $\mathcal {W}_{\le t,\ge 0}^{\text {S}}f=0$ are called null $(\le t)$-designs over R. We report some observations on the distributions of the support sizes of null designs as well as the structure of null designs with extremal support sizes.