Bounding the number of bases of a matroid

Abstract

Letb(M) andc(M), respectively, be the number of bases and circuits of a matroidM. For any given minor closed classℳ of matroids, the following two questions, are investigated in this paper. (1) When is there a polynomial functionp(x) such thatb(M)≤p(c(m)|E(M)|) for every matroidM inℳ? (2) When is there a polynomial functionp(x) such thatb(M)≤p(|E(M)|) for every matroidM inℳ? Let us denote byM Mn the direct sum ofn copies ofU 1,2. We prove that the answer to the first question is affirmative if and only if someM Mn 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, someM Mn is not inℳ.

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

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Mathematics Subjects Classification (1991)

  • 05 B 35
  • 05 C 55
  • 05 C 10