#### Title

Bounding the number of bases of a matroid

#### Document Type

Article

#### Publication Date

6-1-1995

#### Abstract

Let b(M) and c(M), respectively, be the number of bases and circuits of a matroid M. For any given minor closed classℳ of matroids, the following two questions, are investigated in this paper. (1) When is there a polynomial function p(x) such that b(M)≤p(c(m)|E(M)|) for every matroid M inℳ? (2) When is there a polynomial function p(x) such that b(M)≤p(|E(M)|) for every matroid M inℳ? Let us denote by MMn the direct sum of n copies of U1,2. We prove that the answer to the first question is affirmative if and only if some MMn is not inℳ. Furthermore, if all the members ofℳ are representable over a fixed finite field, then we prove that the answer to the second question is affirmative if and only if, also, some MMn is not inℳ. © 1995 Akadémiai Kiadó.

#### Publication Source (Journal or Book title)

Combinatorica

#### First Page

159

#### Last Page

165

#### Recommended Citation

Ding, G.
(1995). Bounding the number of bases of a matroid.* Combinatorica**, 15* (2), 159-165.
https://doi.org/10.1007/BF01200752