Partition matroid

In mathematics, a partition matroid or partitional matroid is a matroid formed from a direct sum of uniform matroids.

Let ${\displaystyle B_{i}}$ be a collection of disjoint sets, and let ${\displaystyle d_{i}}$ be integers with ${\displaystyle 0\leq d_{i}\leq |B_{i}|}$. Define a set ${\displaystyle I}$ to be "independent" when, for every index ${\displaystyle i}$, ${\displaystyle |I\cap B_{i}|\leq d_{i}}$. Then the sets that are independent sets in this way form the independent sets of a matroid, called a partition matroid. The sets ${\displaystyle B_{i}}$ are called the blocks of the partition matroid. A basis of the matroid is a set whose intersection with every block ${\displaystyle B_{i}}$ has size exactly ${\displaystyle d_{i}}$, and a circuit of the matroid is a subset of a single block ${\displaystyle B_{i}}$ with size exactly ${\displaystyle d_{i}+1}$. The rank of the matroid is ${\displaystyle \sum d_{i}}$.

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