Uniform matroid

In mathematics, a uniform matroid is a matroid in which every permutation of the elements is a symmetry.

The uniform matroid ${\displaystyle U{}_{n}^{r}}$ is defined over a set of ${\displaystyle n}$ elements. A subset of the elements is independent if and only if it contains at most ${\displaystyle r}$ elements. A subset is a basis if it has exactly ${\displaystyle r}$ elements, and it is a circuit if it has exactly ${\displaystyle r+1}$ elements. The rank of a subset ${\displaystyle S}$ is ${\displaystyle \min(|S|,r)}$ and the rank of the matroid is ${\displaystyle r}$.

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