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Partition matroid
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Partition matroid : ウィキペディア英語版
Partition matroid
In mathematics, a partition matroid or partitional matroid is a matroid formed from a direct sum of uniform matroids.〔.〕
==Definition==
Let B_i be a collection of disjoint sets, and let d_i be integers with 0\le d_i\le |B_i|. Define a set I to be "independent" when, for every index i, |I\cap B_i|\le 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 B_i are called the blocks of the partition matroid. A basis of the matroid is a set whose intersection with every block B_i has size exactly d_i, and a circuit of the matroid is a subset of a single block B_i with size exactly d_i+1. The rank of the matroid is \sum d_i.〔.〕
Every uniform matroid U{}^r_n is a partition matroid, with a single block B_1 of n elements and with d_1=r. Every partition matroid is the direct sum of a collection of uniform matroids, one for each of its blocks.
In some publications, the notion of a partition matroid is defined more restrictively, with every d_i=1. The partitions that obey this more restrictive definition are the transversal matroids of the family of disjoint sets given by their blocks.〔E.g., see . uses the broader definition but notes that the d_i=1 restriction is useful in many applications.〕

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